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IN '4; L J'. «‘J: :J-‘a'JJJ' ‘4 «“Ji' {3’4 3J4 7H7: Date 0-7639 LIBRAI" ‘1’ Michigan ‘5 rate University This is to certify that the thesis entitled SIMULATION AND OPTIMAL CONTROL OF ECONOMIC GROWTH SYSTEM: APPLICATION OF SYSTEM THEORY TO THE DESIGN OF ECONOMIC POLICIES presented by Sung Joo Park has been accepted towards fulfillment of the requirements for Ph Dr degree inWence 7 . 6"" '"’ {314 W 1 GM aux/lexi- Major p ofessor \\\ OLA/xxx ( Cl (C! 7 8y ‘ llllllllllllllllllllllllllllll 3 1293 01085 5678 SIMULATION AND OPTIMAL CONTROL OF ECONOMIC GROWTH SYSTEM: APPLICATION OF SYSTEM THEORY TO THE DESIGN OF ECONOMIC POLICIES By Sung Joo Park A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Electrical Engineering and Systems Science T978 C/N aha“ ABSTRACT SIMULATION AND OPTIMAL CONTROL OF ECONOMIC GROWTH SYSTEM: APPLICATION OF SYSTEM THEORY TO THE DESIGN OF ECONOMIC POLICIES By Sung Joo Park Application of system theory to the problems of socio-economic systems can be interpreted as an effort to find the analogies between the two different systems--engineering systems and social or economic systems. The thesis is an attempt to apply the system theoretic approaches to an economic system, an economic growth system in particular, reflect- ing the increasing concerns of economic growth among LDC's under highly interdependent recent world economic situations. The economic growth system as explained in the modern theory of economic growth is a closed system with capital being the system state and with the two basic economic processes of capital accumulation and production. Closed two sector (dual) economic growth system has been modeled based on the theory of economic growth and disaggregated into agriculture and nonagriculture which is different from the conventional way of disaggregation for the economic growth model--consumption- and capital-goods sectors. The dual economic growth system with thiscfichotomyconsists of two system state equations and a measurement equation, and comprises non- linear dynamic system which can be solved numerically. Sung Joo Park Optimal control of the dual economic growth system with saving rate being control variable was performed based on Pontryagin's maximum principle. Objective function--social welfare--was defined in terms of the combina- tions of consumption and capital to conduct the optimal control where constant exponents (weights) on the consumption and capital were used to avoid the possible existence of singularities. The sufficient conditions for the existence of the optimal control also were derived for the case of bounded control and no terminal constraint. Numerical solution of the optimal trajectories could be obtained efficiently by variation of extremals with a simple adjustment scheme--with the constant costate influence function matrix--for the case of the Korean economy. In the open economic growth system, two components--trade and balance-of—payments--were added to the closed system with further modifi- cations. Income distribution and foreign indebtedness were also con- sidered in the open system along with economic growth. Optimal policies of saving rate and import were formulated and derived by general optimization method with a piecewise quadratic objective function and orthogonal function (Legendre) approximation of the policy paths during a time horizon. The open model was further simulated for the case of alternative objective functions and alternative policies on investment and grain prices. Considerations on the external food shock has been added to the open economic growth system to investigate the effects of the shock to the internal economic variables and to design feasible policies to mitigate or eliminate the negative effects of the external shock. Sung Joo Park Further modifications on the aggregate production function, saving behaviors and investment, price mechanisms, labor (migration), tax policy, trade, foreign capital movements, and inclusion of uncertainties will make the model close to the real situation. The results of the study (with applications to the case of the Korean economy) demonstrated the practical usefulness of the theory of economic growth for the design of economic policies with respect to the transitory behaviors of an economy during a certain (finite) planning period. In sum, the study showed the possible applications of system theory to an economic system to bridge the gap between the two disciplines, and thus to provide more insights in understanding the dynamics of economic systems. To my parents and grandma, for their understanding, encouragement, and love. 1'1 ACKNOWLEDGEMENTS I wish to express my sincere thanks to my thesis advisor and the Chairman of the Guidance Committee, Professor Thomas J. Manetsch, for his patient and inSpiring guidance in this endeavor and throughout my doctoral program. His unfailing faith in me and encouragement have been the light to the deemed unending periods of frustration. I also wish to thank Professors Gerald L. Park, Robert O. Barr, and Anthony Y. C. Koo, for serving on my guidance committee, and for their concerns on the work. Special thanks are due to Professors George E. Rossmiller and Michael E. Abkin for their interests and counsels. I also want to express special appreciation to Mrs. Judy Duncan for her kindness and help in typing the drafts of the thesis. A sincere thanks goes to Drs. Nam Kee Lee and Dong Hi Kim for initially providing the opportunity to pursue the advanced study in the United States. My gratitude for the spiritual support of my family, especially my parents, defy description. They have been the unceasing source of encouragement through all the work. TABLE OF CONTENTS Page DEDICATION ii ACKNOWLEDGEMENTS iii LIST OF TABLES vi LIST OF FIGURES vii Chapter I. INTRODUCTION l 1.l Background and Needs of the Study 3 1.2 Scope and Objectives of the Study 7 11. GENERAL CONCEPTS: BACKGROUND INFORMATION l0 II.l System Theory l0 11.2 Simulation l2 11.3 Optimal Control T4 111. ECONOMIC GROWTH MODEL l8 111.1 One Sector Growth Model l8 111.2 Two Sector Growth Model 22 111.3 Modified Dual Economic Growth Model 23 111.4 Analysis of Economic Growth of Korea 29 IV. OPTIMAL ECONOMIC GROWTH 35 1V.l Economic Objectives of a Society 35 1V.2 One Sector Optimal Growth Model 38 1V.3 Two Sector Optimal Growth Model 4l 1V.4 Numerical Procedures of Optimal Control 46 1V.5 Optimal Growth of Korean Economy 49 V. GROWTH MODEL OF THE OPEN ECONOMY V.l Need for the Open Economic Model 58 v.2 Description of the Components 6l iv Chapter VI. VII. VIII. SIMULATION OF THE OPEN MODEL (:QITT- Design of Economic Policies . Empirical Application to the Korean Economy FURTHER ANALYSIS OF THE OPEN ECONOMY UNDER THE EXISTENCE OF EXTERNAL FOOD SHOCK V11.l Introduction VII.2 Analysis of Shocks VII.3 Storage Rules VII.4 Application to the Open Model of Korea SUMMARY AND CONCLUSIONS V111.l Summary of the Results VIII.2 Further Research and Recommendations V111.3 Conclusions APPENDIX A. BASIC DATA APPENDIX B. ESTIMATION OF CAPITAL STOCK APPENDIX C. COMPUTER PROGRAM BIBLIOGRAPHY Page 89 100 130 133 136 138 154 160 165 168 171 175 182 Table (DNNOWU'I CDJ>J>J> DON LIST OF TABLES Results of the Optimal Control Aggregate Production Functions Results of the Optimization Dependency on the Grain Import Parameters for the Grain Storage Rules Comparisons of Performance Indices: Utility Payoff Matrix Gross National Production Population Capital Formation Capital Stock vi Page El 62 102 l32 l39 166 168 169 170 174 Figure 43.000410) #00“) hhh-fi-b 0501wa -—‘ LIST OF FIGURES Block Diagram of One Sector Economic Growth System Block Diagram of Dual Economic Growth System Input Responses of Dual Economic Growth System State Trajectories of Dual Economic Growth System Numerical Procedure for the Optimal Trajectory of the Economic Growth System Optimal Trajectory, k(t) Costate Trajectory, 5(t) Optimal Control, §(t) Trajectory of Consumption per Labor, c(t) Trajectory of Output per Labor, y(t) Major Components of the Open Economic Growth System Simple Feedback Loop for the Changing Proportion of Income Transfer Monotonic Decreasing SCURVE for B = 0.2, C = 0.5, and ID = 2 The k-th order Distributed Delay with Gain Balance-of—Payments Component Price-Quantity Determination with Perfectly Inelastic Supply Possible Paths of Constraints for the Case of Heaviside Unit Step Penalty Function Optimal TOSR (total desired saving rate) vii Page 20 25 32 34 48 53 54 55 56 57 6O 66 78 80 82 84 94 104 Figure uooowosmb \l 0‘ 0'1 05 01 0‘0 O3 03 ON 01 0’3 03 OS 01 OS 03 Ch 0‘ O O O O O O O 0501-wa .11 .12 .13 .14 .15 .16 .17 .18 .19 .20 viii Optimal TMPI (the marginal propensity to import) Gross National Product Real GNP Growth Rate Debt Service Ratio Foreign Borrowing Total Debt Aggregate Consumption per Labor Rate of Change of Consumption per Labor Income Distribution Ratio Domestic Saving Rate Foreign Saving Rate Real GNP Growth Rate Debt Service Ratio Foreign Borrowing Total Debt Rate of Change of Consumption per Labor Income Distribution Ratio Participation Ratio World Price of Wheat Petroleum Price: Saudi Arabia (Ras Tanura) Different Patterns of Shock Real GNP Real GNP Growth Rate Debt Service Ratio Foreign Borrowing Page l05 llO lll ll2 ll3 ll4 ll5 ll6 ll7 ll8 ll9 123 l24 125 l26 127 128 129 l31 l3l 134 142 143 144 145 Figure 7.8 7.9 7.10 7.11 7.12 7.13 7.14 7.15 8.1 ix Domestic Rice Price Domestic Barley Price Domestic Wheat Price Reserve Stock Level of Wheat Amount of Wheat Import Dependency on the Grain Import Availability of Wheat in World Market World Price of Wheat Fixed Capital Consumtion Page 146 147 148 149 150 151 152 153 173 CHAPTER 1 INTRODUCTION Recent trends in applying system theory to the problems of socio- economic systems have been rooted to the question: "If system theory can improve the guidance of airplanes and spacecraft, can it also be helpful_in devising the policies for solving the problems of an econ- omyflg:,§_§9§i§§¥2" Obviously, a society or an economy is not a con- crete moving object which can be measured precisely, like an airplane or a spacecraft, and thus numerous assumptions and simplifications are needed to find analogies between the two. Economjgfisystems, as well as other social systems, can be charac- terized by the three basic properties; uncertainty, dynamics, and the existence of feedbacks. In fact, the classical economic theories have failed to take into account of the three properties: they assume perfect information or full knowledge of the parameters of the economic system, static or comparative static analysis has been dominantly used with the "ceteris paribus" conditions, and the nature of the adaptive economic process, where decision makers increase their knowledge by the cumulative experience of "doing while learning", has not been fully explored in general. With the lack of ability of the traditional methodologies in economics to handle these problems, new methodologies from other disci- plines have been sought and applied gradually as a part of diffusion processes among the different disciplines. Although there were earlier attempts [P4], [$8], [$9]. [$15], the pioneering work of applying system theory to economics can be attributed to Arnold Tustin [T16], who viewed economic system as interdependent dynamic systems and analyzed their time responses to exogenous shocks using the transfer function method which was fOUnd to be very successful in designing servomechanisms. A. W. Phillips, best known for the Phillips' curve, also applied about the same time the theory of servomechanisms to business cycles and stabilization policies of macroeconomic models [P5]. These earlier works, however, were not successful because of two reasons; the technical reason of computational restrictions, and the circumstan- tial reason of the inclination of the interests towards the general com- petitive equilibrium models those days [01]. With the advent of computers, the first obstacle was removed, and more complex models have been emerged [H6], [M2], [M7]. There are two ways of handling complex and/or large scale systems in general. The first is analytical by aggregation [M1] or decomposi- tion [H5], and the second is by simulation. Each of the two methods has its own merits: by aggregation and decomposition, the model can be sim- plified enough to be handled by analytical tools and may achieve analyti- cal preciseness which most theoreticians believe to be of foremost impor- tance, while the simulation approach allows one to investigate the inter- actions and linkages of elements of the system at the detailed level of the real situation with less analytical preciseness. The actual use of any one of the methodologies in modeling depends on the nature of the problem, the use of the model, the availability of information, and the desired level of accuracy. National economic planning is not only a economic but also a polit- ical process: it includes the definition and choice of goals, identifying the economic variables and interrelationships, designing policies, chart- ing the possible paths of economy with respect to the feasible policies, and selecting the best policy to meet the goals. The problem of design- ing policies for economic growth under different circumstances has been /* the central core of the thesis; however, the motivation was from the // recognition that actual society is too complex to be solved by any one of the methodologies, and there may be certain gains from looking at the problem from different sides using different methodologies, and thus to advance system theory in the analysis of dynamic economic models. 1.1 Background and Needs of the Study In recent years, the world has experienced increasing interdepen- dency of international economy and a series of shocks--most notably oil crisis and food shortages--which affected virtually every country in the world. Even though the initial motivation of the interdependency originated from the mutual benefits of trade, it created high levels of insecurity as some countries became heavily dependent on others for certain products which are essential for their continuing growth or existence. The immediate concerns for the highly dependent countries, lflOSt of the developing countries, are the deepening of foreign indebted- ness as a result of the high prices of the essential products in the inorld market, and their effects to economic growth which may relieve them from the excessive indebtedness. Economic growthqha§_been one of the principal objectives of the economic policies both in advanced and in less developed countries, ..-v .u— .g.....-- believing that growth can be a solution to a variety of other economic problems such as indebtedness of a country, inequitable distribution of income, and so forth. The theory of economic growth,1 which tries to explain the primary causes of production and their interrelationships over time periods, has been developed for decades. However, the "state-of-the-art" of the theory is not satisfactory: too many controversies exist over the basic assumptions, heavy reliance of the growth models on the balanced growth, golden-rule paths, or steady-state growth prevents empirical application "2 aloof to a specific economy, thus creating a model of "mythical states from the reality. If a theory has some value and thus deserves to be called a "theory", it should be able to answer questions raised from the real situation at a certain "admittable level" of preciseness. The existing models of economic growth have failed in this sense by staying a safe distance from reality. The thesis is an attempt to modify the existing models of economic growth with the following questions in mind: (1) what are the appropriate descriptions of an economy where the tran- sitions and/or interactions between the primitive and advanced sectors are more significant than those between the consumption- and capital-goods sectors? 1The modern theory of economic growth is meant here instead of other types of growth theory such as the grand theory of economic growth or the theory of economic development [J1], pp. 4-6. 2A balanced growth, or golden-rule path, "thus indicating that it represents a mythical state of affairs not likely to obtain in any actual ecbnomy,"'[R5], p;99. (2) what is needed in the model to take into account the interactions of an economy with the world in the light of trade and capital flows? (3) what are the primary linkages between foreign borrowing or foreign capital flows and economic growth? (4) what are the effects of external shocks to economic growth and iother domestic economic variables? éEEj) what are desirable and feasible growth policies, and how can the ' best policy be chosen? Also, the study was motivated, in part, by a desire to test empirically the basic formulation of the theory of economic growth, its applicability to the LDC's (less developed countries), and its ability to cope with problems relating to external effects. The empirical study has been done on the economy of Korea for illus- rative purposes. Korea, one of the fast growing developing economies among LDC's, has experienced the "take-off" and the “acceleration" stages of development during the past decades. The economic growth has been phenomenal; the average real growth rate during 1965 to 1975 was 10.2 percent, the export has grown about 29 times with an average annual growth rate of 22.6 percent during the same period. High level of average education and technology transfer from the advanced countries enhanced the growth, however, the basic forces for growth came from the increase in investment and the migration of labor from traditionally latent agricultural sector to the industrial sector. Investment, which has risen from 0.151 (gross investment ratio) in 1965 to 0.314 in 1974, has been financed substantially from foreign sources-~foreign loans, direct foreign investment, use of international reserves, and net foreign transfers. As a consequence, the burden of foreign indebtedness, which has accumulated to an estimated total debt standing of six billion dollars at the end of 1975 (from the negligible debt in 1965) with 700 million dollars of debt service payment during 1975,1 draws heavy attention along with the costs that the Korean economy has to pay as a result of urban migration. Clearly, these costs may play a hindering role in the pursuit of continuing growth. This burden has been acutely, though not devastatingly, experienced in Korea when the oil crisis and food shock struck the whole world. Coupled with the fact that Korea is entirely dependent on overseas for the oil supply, and is chronologically depen- dent on imports for about a quarter of its total grain requirements. The main “questions, then, become more clear. First, what are the possible ways of investing to further economic_growth and to relieve the burden of excessive indebtedness? In other words, what are wise foreign borrowing schedules if the desired investments exceed the domestic savings? Secondly, what are the consequences of the possible paths of growth for other aspects of the economy-~foreign trade, level of consump- tion, distribution of income, etc.? Thirdly, what will be the effects of the probable future shocks to the economy and what policies can dampen or eliminate the negative effects of the shocks? Fourth, is there any policy_which is superior to the other available policies and will lead tot thedhigher economjc_gr_owth, sound indebtedness, more equitable dist- ribution of income, and a higher level of consumption? Even though the questions are from a specific case, these are not the questions of only Korea but also typical to most of the LDC's. 1.2 Scope and Objectives of the Study The remaining chapters of the thesis can be divided into two equal parts; the analysis and control of the closed economy, and the analysis and control of the open economy. The high level of aggregation of the closed economy enables one to use analytical tools--state-state repre- sentation of the economic system and optimal control of the system model using Pontryagin's maximum principle-~while the complexities of the closed model do not allow the application of analytical tools used in the open model. Chapter II will describe the basic analytical tools. Chapter III starts from the neoclassical model of economic growth which explains the basic causal loops for production in a one sector aggregate economy where the saving (investment) and production mechanisms govern the whole econ- omy. A Meadean type two sector model, which disaggregates economy into consumption- and capital-goods sectors, is also examined and is modified into a two sector model of primitive and advanced sectors, more specifi- cally, agricultureand nonagriculture. Chapter IV concentrates on finding the optimal policy for a closed 7~—_.__._._~—._‘ hfi economy in terms of the path of saving rate over the planning horizon which maximizes the "alternative" desired social welfare functions. Direct application of the maximum principle has been made, the necessary and the sufficient conditions for the existence of the optimal control have been derived, and a numerical algorithm has been developed for the specific case of an objective functional without terminal conditions. The closed model has been further modified into the open model in Chapter V. This model includes trade and balance-of—payments components in addtion to the other components of the dual economic growth model. Designing the economic policies of the open model can be quite different from that of the closed model; it deals not only with the questions of the saving rates and capital formation but also with those of the foreign trade, foreign capital flow and indebtedness, foreign currency reserve, etc. These added features make the model not readily applicable for the analytical tools such as the optimal control theory as in the closed model. "Creative" trial-and-error or rule-of—thumb methods may provide certain insights for the design of better policies whereas an attempt to obtain theglbestf policy applying optimization techniques with a given objective function yields a set of optimal solutions of complex system which may be infeasible in strict mathematical sense but may have practi- cal usefulness if the definition of the constraints can be relaxed. In this case, a set of orthogonal functions such as Fourier series or Legendre polynomials can be used to approximate the dynamic paths of the policy variables. Determining objective function is not an easy task and may constitute conceptual difficulties. Piecewise quardratic objec- tive function can be used more generally than quadratic objective func- tion in order to remove some of the difficulties. One of the prominent advantages of the open model over the closed model is its ability to respond to external shocks and to investigate their effects on internal economic variables. High grain prices in the world market have been used for an external food shock to the open model. Implications for the management of the domestic grain stock to dampen the shock to the internal mechanism have been analyzed in Chapter VII. The final chapter concludes the thesis with a summary, further research areas, and possible gains from applying different methods to the problem of economic growth. 9 fi__k‘._._—_—_..__ . or a model which can be used for the investigation of economic growth, designof policies for the desired growth or sustained growth, both with or without external effects. More specifically, the objectives are to design an analytical framework, to be illustrated by the Korean economy case, which permits one to investigate: (1) growth and consumption paths, and saving rate of a closed economy where the interactions between agriculture and nonagriculture are significant (2) the best policy for saving rates which will maximize the desired social welfare functions (3) the effects of alternative growth strategies on foreign indebted— ness, distribution of income, consumption and economic growth in the open model which includes foreign trade and capital movements (4) the effects of external shocks to internal economic variables such as the growth rate, consumption, foreign debt, food prices and distribution of income ( the design of policies for the desired or sustained economic growth with or without external shocks, while satisfying constraints on 1 the control and/or the state variables of the economy. CHAPTER 11 GENERAL CONCEPTS: BACKGROUND INFORMATION Any problem solving (in the physical world) necessarily deals with models. By definition, diflflflél'ls an abstract representation of the actual system. It can be physical, conceptual, verbal, graphical, or mathematical with the purpose to help one to understand the system oper- ation, and to predict its behavior under certain conditions. The model used in the study is based on the theory of economic growth. In this chapter, the basic tools used to deal with the model--system theory, simulation, optimal control--will be discussed in brief. 11.1 System Theoryw A system is a set of interconnected entities, conceptual or physi- cal, organized toward a goal or set of goals. This concept is quite general and broad in nature, and thus the word "system" has been used to mean many different things for many different purposes. System the- ory, which deals with "system", also has been defined in diverse ways.1 Among the many possible difinitions, system theory here will be confined to a tool (in engineering sense) to design the "best" system to satisfy a certain purpose. Often, it uses mathematical models in the 1Ludwig von Bertalanffy even goes further by stating that General System Theory (GTS) is the only possible way for the unification of sci- ences, which encompasses large branches of science such as "classical" system theory, computerization and simulation, compartment theory, set theory, graph theory, net theory, cybernetics, information theory, theory of automata, game theory, decision theory, queueing theory, etc. [B4]. 10 11 form of algebraic equations, difference equations, ordinary or partial differential equations, and functional equations. System theory starts from the system equations such as (for the case of ordinary differential equation model),1 Him. no.1?) (2.1) G(7(t), 17(1), '17) (2.2) Silt) 70:) where 2'15 the system state vector, U'is the input vector, y'is the out- put vector (or observations), 1 and V represent possible vectors of sys- tem disturbance and measurement error, and F and G identify the state and measurement equations (or operators). Four basic problems arise in the system theory: modeling, analysis, estimation, and control [55]. Modeling, developing a model which adequately represents the physi- cal situation, is undoubtedly the most critical step, since if the model is not adequate, the subsequent mathematical or computer manipulations are meaningless. To help to preserve the adequacy of a model, several steps of modeling procedure have been developed which will be discussed in the next section relating to simulation. Analysis determines the system output y given system input U and system structure F, G, and x} Quantitative analysis determines the pre- cise behavior of the output, such as trajectories of the output, and qualitative analysis determines general properties of the behavior of the output such as stability. This is only possible under the assumption that the system structure as given is perfectly known. 1Notation: the upper bar and the lower bar will indicate vector and matrix, respectively, throughout the theses unless otherwise specified. 12 Estimation raises question about the system structure. It uses the observations of y and U'to estimate the properties of the actual system. Three types of estimation problems can be defined; state estimation, identification, adaptive estimation. State estimation is to estimate 1 using observation 1 with given system structure. Identification refers to the estimation of parameters (or new states in a sense) using the observations, (original) system states, and system structure, and thus completes the estimation of the model. Adaptive estimation is state estimation combined with identification, i.e., the state estimation prob- 1em with the combined state of original state and new state (or parameter). In the control problem one determines the input U given the desired output y and the system structure. There are three types of control; open-loop, closed-loop, and adaptive. Open-loop control means a control expressed in terms of the initial state of system and independent vari- able(s) such as time. Closed-loop control is expressed as an explicit function of the observed variables, and thus feeds-back the output to the input. Adaptive control is a more complicated closed-loop control where adaptive estimation is carried out simultaneously. All the basic problems related to modeling are closely interrelated: any one of input, output or system structure can be determined only by assuming the others are perfectly known. Iterative procedure of analysis, estimation, and control--which will hopefully converge-~may be used to complete the system modeling. 11.2 Simulation Although the concept of simulation can be traced back to long ago, the modern version of simulation, defined as "a numerical technique for 13 conducting experiments on a digital computer which involves certain types of mathematical and logical models,"1 has its origin in the work of von Neumann and Ulam in the late 1940's when they applied a method based on the random numbers (which later was named as "Monte Carlo Method") to study neutron diffusion and to determine the thickness of reactor 2 The initial motivation of the von Neumann and Ulam's work was shields. to incorporate stochastic uncertainties in a deterministic model, but they also recognized that the method is capable of dealing with problems of a more complicated nature which can't be solved by conventional ana- lytical methods. Uncertainty (randomness) and complexity form a basic motivation for simulation, thus the simulation model can be far more complex than the models that system theory (as defined in the preceding section) usually deals with. The basic problems of simulation are the same as those of system theory-~modeling, analysis, estimation, control. Modeling is the core of the simulation because of its critical importance to the rest of the procedure. To minimize the risk of model being inadequate for the real situation, a conceptual procedure can be followed as a helpful guide to a general problem solving. The general procedure (or system approach) is an iterative learn- ing process with set of steps (or phases) as following [M3]: (1) feasibility evaluation (2) abstract modeling 1[N1], p.3. 2Part of the original thoughts of von Neumann suggested by Ulam can be fbund in a von Neumann's letter to R. D. Richtmyer. R. D. Richtmyer and J. von Neumann, "Statistical Methods in Neutron Diffusion," [Tl], PP. 751-764. 14 (3) implementation design (4) implementation (5) system operation The major phases of systems approach may comprise sub-phases. Fea- sibility evaluation consists of six sub-phases; needs analysis, system identificaiton, problem formuation, generation of system alternatives, determination of physical, social, and political realizability, and determination of economic and financial feasibility. Abstract modeling may be followed through sub-phases such as; selection of feasible system alternatives emerged from feasibility evaluation, choice of specific type of abstract representation-~static, dynamic, micro, macro--, com- puter implementation, validation, sensitivity analysis, stability analy- sis, and model application. The above procedure as suggested in [M3] is, by no means, complete. and the validity of the procedure can only be judged on purely pragmatic grounds. In essence, problem solving (using simulation) is a continuous process of modification and adjustment. 11.3 Optimal Control Theory Optimal control theory, which is a dynamic extension of static optimization, comprises two major branches; dynamic programming [B2] and Pontryagin's maximum principle [P8]. Dynamic programming is a computational technique which extends the decision making concept to sequences of decisions which together will define an optimal policy and trajectory based on the principle of optimality:1 1[32], p. 83. 15 An optimal policy has the property that whatever the initial state and initial decision are, the remaining must constitute an optimal policy with regard to the state resulting from the first decision. While dynamic programming finds many applications in discrete systems, there is one serious drawback; the "curse of dimensionality" which indicates the exponentially-increasing size of computer memory require- ments as the number of decision variables increases. On the other hand, Pontryagin's maximum principle was originally developed for continuous systems and will be used here in the closed economic growth system. Pontryagin's maximum principle, in essence, consists of a set of necessary conditions that must be satisfied by optimal solutions. All of these necessary conditions originate in classical calculus of varia- tions, but are formed in a more or less systematic way by use of a l Hamiltonian function. The problem is to find an admissible control 6* which satisfies the system in) = 3136(1). at). t) (2.3) and to minimize an objective functional (performance index), J(U) = h(X(t ). t ) + I 9(X(t), U(t), t) dt (2.4) f f to with or without additional constraints such as 32*(to) (2.5) Wtf) N X 0 where 2'15 the system state, u'is the control input, t0 and tf are the 1Asterisk will be used to denote the optimal values. 16 initial and the final times, respectively. An augmented scalar function, Hamiltonian, will then be defined as mm), Um. p(t). t) 2 gm). at). t) +flnnann.mu.n1 at) where E'is a costate vector or a vector of Lagrange multipliers which corresponds shadow prices for the case of cost minimization. Pontryagin's maximum principle, then, states that an optimal control must minimize the Hamiltonian, i.e., a necessary condition for 6* to minimize the func- tional J is H(7*(t). Wt). Wt). t) ; H(3<'*(t). mt). 3*(t). t) (2.7) for all t e [to, tf] and for all admissible controls. Thus, the necessary conditions for 6* to be an optimal control can be derived as _a_H,- — X*(t) - 3_ WM. u*(t). 3*(t). t) (2.8) P pn)=-¥xeuxuuo.purt) as) X H(?*(t). Wt). 'p'*(t). t) _<__ H(3<'*(t). 'u‘(t). 3*(t). t) for an_ admissible u(t) (2.10) for t 8 [1:0, th’ and boundary conditions [39;(§*(tf). tf) - amp)T a}, + tH<2*(tf). Ur 0, then the growth of the capital stock can be specified by the differential equation km = s(t)A(t)F[K(t), L(t)] - uK(t) (3.3) Assume that N(t) is the current size of the population and that population growth is independent of the economic variable and growing at a constant rate N(t) = nN(t) (3.4) where n is the rate of the population growth. Assume further that the number of workers (labor) is a constant fraction 0 < w < l of the total population L(t) = wN(t) L(t) = nL(t) (3.5) If the technological change can be assumed as an autonomous growth at a fixed rate, a, then A(t) = aA(t) (3.5) where a is the rate of the technological growth. The preceding equations from (3.1)to (3.6) form the basis of the aggregate economic growth system (Figure 3.1). If the neoclassical assumptions on the production function--constant returns to scale in 20 s(t) , .. +flz\ K(t) I K(t) ’\\_// r. _. _. _____________ l I PRODUCTION l Y(t) l I : . F[K(t).L(t)l l | A(t) L(t) I I f I l l l 0 o I a—al 11 A t L(t) 11 é—n L ._ _ _ _ _ _____ _ _____ .1 Figure 3.1 Block Diagram of One Sector Economic Growth System 21 capital and labor, i.e., homogenous of degree one in capital and labor --can be made, the model (neoclassical growth model) can be converted into the equation in the per labor quantities.1 [311] Let the vari- ables be defined as: output per worker : y(t) = Y(t)/L(t) capital per worker : k(t) = K(t)/L(t) consumption per worker : c(t) = C(t)/L(t) investment per worker : Z(t) = Z(t)/L(t) Then the system equation and the production function can be reduced to k(t) = s(t)y1 11 PRODUCTION I Figure F11.) PRODUCTION II 2 K111) 1111:) s11t) ( S(t) Z - 52(1) 12(1) 1 K20) Fem 3.2 Block Diagram of Dual Economic Growth System 26 121(1) 11(t) - u1K](t) (3.19) K2(t) = 12(1) - u2K2(t) (3.20) where ui is the depreciation rate of capital for each sector. These equations can be rewritten using the equations (3.15)-(3.18), K1(t)= i(t)sl(t)Y](t) + i(t)sz(t)Y2(t) - u1K1(t) (3.21) K2(t) = [l-i(t)]51(t)Y1(t) + [l-i(t)]52(t)Y2(t) - u2K2(t) (3.22) The block diagram for the above equations, (3.21) and (3.22), is shown in Figure 3.2. Retaining the assumptions of the Meade and Uzawa's model--neoc1a- ssical assumptions--, the system state equations can also be converted into per labor variables. Thus k1“) = -tu,+g,(t)11<,(t) + 1(t)s,(t)y, (4.6) bu)=-%¢-mauuwum*t+mohuwwoa1 (4n 1 The notations will be used as: k(t) 2-%% , y'(.) 2-31 39 o=§§=mu)-€“uu1 (4m then, p(t) = e'rt. (4.9) Since the Hamiltonian (4.5) is a linear function of input, s(t), this problem constitutes a "bang-bang" control problem. The optimal control for 0:; s(t) ;=l can be obtained as, r 1 if p(t) > e"‘t s(t) = ( indeterminate if p(t) = e-rt (4.l0) L o if p(t) < e"”t Because of the indeterminacy of the control, the optimal control doesn't exist for a time interval [ta, tb]1 such that p(t) = e-rt 9 t S [ta: th- To avoid the possible existence of singularity, an alternative obj- ective utility function can be suggested as U[c(t)] = c(t)a (4.11) where a is a constant such that 0 < a < 1. It can be shown that all the desirable conditions of utility function are satisfied by the above utility function. Using the utility function (4.ll), the Hamiltonian can be MJ=ELMUFHUa€m+thMUflU-9MU] mam 1This interval is called singular interval, and the indeterminacy condition is called singular condition. 11" 3M “WI ‘- ! Ir;- «1,1 40 and the necessary conditions will become uo=suwu)-nu) MAN MH=ELMUPaMUW4€m+pHHd0Mh)-fl MAM o=-druor*wo%*t+mowo MAM rt/aJl/(a-l) thus, s(t) = 1 - 91m [p(t)e (4.15) The sufficient condition for the maximum can also be obtained from 32H a-2 a -rt ——2—= a(a-l)[l-s(t)] y(t) e (4.l7) as which is negative for 0 < a < l, and s(t) < 1. Since the control has to be constrained--corresponds to the conceptual or practical constraints --to assure the existence of the optimum, the optimal control can be obtained as f SM if y](—fl-(:p(t)ert/a]1“a.1 )él-SM S(t) = < 1 - fipunrt/ajma'” if l-sM< " <1-sL cSL if " ;l-SL (4.l8) where 5L and SM indicate the lower and the upper boundaries of saving rate such that SL’ SM 2: (O, l). The sufficient conditions assure the strict concavity of Hamiltonian with respect to s(t), and thus the above optimal control is possible. In essence, the alternative utility function (4.ll) removed the possibility of singularity. 4T 1V.3 Two Sector Optimal Growth Model The optimal economic growth of one sector model will be extended into two sector model in this section. To simplify the total welfare with intertemporal discounts, a modified Hamiltonian will be defined as1 um +BT Jim + as) U(.) + [9(51: + g?) (4.19) H(.) 5 H(.) e"t 1Nhere the system state equation is from the equation (3.25), and p rep- resents a modified costate variable. Then, the necessary conditions using the modified Hamiltonian can be derived as l 12‘: 3’7 (4.20) 3P ~ _ 3H + ~ p _ - .3? rp (4.21) —_ 3H 0 - 5? (4.22) The above equations enable one to use Hamiltonian without discount rate, and vvill simplify the expression and computation. l\ natural extension of one sector optimal control problem with consunuation-oriented utility function into two sector problem leads to the ‘formulation of utility function a a U(- ) = c1 1c2 2 (4.23) where c1 is the consumption per labor of agricultural product c2 is the consumption per labor of nonagricultural product a1 and a2 are the relative (preference) weights between the consumption of two aggrigate commodities. l . . . . . v . For notat1onal conven1ence, t1me t w1ll not be expressed 1n the a"lables in this section. 42 It can be shown that the utility function (4.23) satisfies the Conditions for the "well-behaved" utility function. Using the system state equation for the two sector model, (3.25), and the utility function (4.23), the Hamiltonian can be given by1 H(.) = c1 c2 + p a a _ _ 1 _ 2 ‘ [(1 S1)y1] [(1 $2)y2] + p1(V11k1 + v12y151 + V12y252) I p2(V21k2 + v22y151 I szyzsz) (4°24) v12 = 1/11 v21 = ’(“2 I 92) v22 = (l - i)/l2 *witfi the definitions as in the equation (3.25). The necessary conditions are also given by r. k ;_ v + v y s + v y s k := ll 1 l2 1 l 12 2 2 (4.25) (_21k2 + V 22y151 + v225’252 31"] a A 1a)[(]-Sl)y1] (1'51)y‘ll[(1‘52)y2] 2 + p](v]]+v]2yi5]fl + p2V22yi51 -1 + ”9 a2 a2[(l- s )y1]a[(l-sz)y2] (l- -s 2)y2 + 91v12y25 2 T ”2(V21IV22y252) _JL (4.26) The modified Hamiltonian and the modified costate will be used wlthout tilde for the rest of this chapter unless otherwise specifies. 43 a -l a 1 2 3H ~a]y][(l-S])y]] [(l-sz)y2] + P1V12Y1 + p2V22y1 0=3§= a a -l l 2 -a2y2[(l-s])y1] [(1-52)y2] + p1V12y2 + pzvzzyz (4.27) The sufficient condition for the optimality also can be obtained from the Hessian matrix 32H 5‘1“1 a2‘]. g§§”' a132y1y2[(1‘51)y]] [(1‘52)y2] a -l y (l-s )y I 1 ._1 2 2 1 a2 2 (1'515Y1 32'] yz (1'51)y] -a, yg'11-521y2 L(4.28) Since 0 < a], a2 < l, and 0 < s], 52 < l, the scalar term of (4.28) is positive. Furthermore, 31'1 Y1 (1'52)y2 . . . . .5E_'§EDTT:§TTYT < 0, and the determ1nant of the matr1x 1n (4.28) 15 l - (a + a l 2) a132 Therefore, the Hessian matrix (4.28) is negative definite if and only if a1 > 0, a2 > 0, and a1 + a2 < l. (4.29) The sufficient condition (4.29)--similar to (4.17) for the one sector case--assures the existence of global optimum. The optimal control, s(t), can be obtained by solving the simultaneous equation (4.27) such as 44 1 1 1 1"“1‘32 5‘ = 1 - 33- 1 1'32 1 + a2 [E;§;(p1v12y1+92V22y1)] [5;y;(p1v12y2 "2V22y2)] (4.30) 1 1'31""2 5:]-L 2 y2 1 + a1 1 + 1"3'1 [a1y,(p1v12y1 92V22y1)] [a2y2(plvl2y2 pz"224"2)J (4.3l) Hence, the optimal saving rates will be given by (4.30) and (4.3l) if these values are within the constraints between 0 and 1. Upper limit or lower limit of the constraints will be given whenever the unconstrained optimal saving violates the constraints. This is possible because of the strict concavity of the Hamiltonian with respect to s when the sufficient conditions are satisfied. Finally, the case of combined consumption-capital utility function will be considered. The utility function can be given as (4.32) Then, the Hamiltonian becomes a a b b v k +v s +v s H(.) = [(1-51) y11 l[(1_52)y2] 2k] 1k2 2 + p1 11 1 12y1 1 12y2 2 v21k2+V22y1S1+V22y252 (4.33) and the necessary and sufficient conditions also can be obtained as k = fl_k'+ §_§ , which is the same with the equation (4.25) 45 _ a1 32 b1 b2 1 . — [(l-s])y]] [(l-sz)y2] k] k2 (ay1/y]+b1/k])+p](v]]+v]2y1s]) ;. + p2V22y1'51 p = - a1 a2 b1 b2 . [(1-51)y1] [(l-sz)y2] k1 k2 (a2y2/y2+b2/k2) + p1V12’252 _ + p2(V21+V225’252)._ + rfi' (4.34) a1-l a2 b1 b ._ 8H -a]y][(l-s1)y1] [(l-sz)y2] k1 k 0:“: ._ 2 + p1V12y1 + p2V22y1 as = a1 a2-1 51 b -a2y2[(l-s1)y]] [(l-sz)y2] k1 k2 + p1v12y2 + pzvzzyg‘ (4.35) Solving the above simulataneous equation to get the optimal saving rates, _ 1 l-a -a 1 2 l 2 y1 m ._a II —J I l—a W _4 x N l-a 1 _EE§§;(p1V12Y1+p2V22y1)] 2 1 a2 [5E§E(p1v12y2+p2v22y2)] _ (4.36) 1 1- b b -—————— S = 1 _ 1__ 1 2 2 y 2 1 a1 1 1'a1 fairy](F’1"123’1”':’2"225’1)J [35§;(p1v12y2+92V22Y2)] (4.37) Then the sufficient conditions will also become 46 32H 31'] 32'] b‘ b ‘23—:““1"“23’13’2m'51)3’1:l [(1'52)y2] k1 k2 2 (31'1)Y1(1'52)YZ 32 yZ(]'S])y] (az‘])y2(1'51)y] 3] y1(1-52)y2 thus, the Hessian matrix [aZH/agz] is negative definite if and only if a.I + a2 < l and 0 < a], O < a2 (and s1, 52 < l) The above condition guarantees the global maximum at any instant of time, i.e., for each fixed sets of state and costate variables. When a1 > 1 and a1 + a2 > l (and a2 > O), the Hessian matrix is positive definite; for all the other cases, it becomes indefinite. 1V.4 Numerical Procedures of the Optimal Control In order to determine the optimal controls, (4.36) and (4.37), explicitly, state and costate equations should be solved which yields a nonlinear two-point boundary problem. Analytical solution of this kind is impossible in general, and thus necessarily rely on numerical procedures. The basic task of the numerical procedure is to find the control so as to satisfy all the necessary conditions of optimal control during the time period. Since the final time is fixed by the given planning horizon, the necessary conditions can be summarized as following for the case of free final states: i(t) = 311/215 (4.38) i(t) = -aH/aE+ r'p‘ (4.39) 47 0 = aH/a? (4.40) p(tf) = ah/ak ltf = 0 Generally, the computational procedure can be given as following: 1) use initial guess on any of the variables to obtain the solu- tion to a problem in which one or more of the necessary condi- tions is violated 2) adjust the initial guess in an attempt to make the next solution come closer to satisfying all of the necessary conditions 3) repeat the preceding steps until the iterative procedure conver- ges, and thus all the necessary conditions will be satisfied. Various methods have been used for numerical solutions in practice such as Ritz's method, dynamic programming, the gradient method, the method of steepest descent, variation of extremals, the method of quasi- linearization, and the method of invariant imbedding [Sl], [K4]. The method chosen in the study is a variation of extremals which starts from the initial guess of the costate variables and solve the system equa- tions iteratively1 by changing the initial costate variables without storing the whole paths of the variables until the final costates come closer to zero, since there is no terms of final states appear in the objective functional. The choice of the method can be justified by the linearity in the 1Simultaneous linear differential equation solver by the method of Bulirsch-Stoer, DREBS, in IMSL (International Mathematical & Statistical Libraries) has been used. 48 C > Initial Guess 3(0) A! Initialization Parameters Time Advance Time in No Compute current 3” ”12’ V22 Optimal Control S Solve State & Costate Equations (DREBS) Final Time? Converge? c 3 Change P(0) by Adjustment Schemes No Figgure 4.l Numerical Procedure for the Optimal Trajectory of the Economic Growth System 49 initial and the final costates, i.e., the changes in the initial costates correspond linearly to the changes in the final costates, which comes from the physical meaning of the costate in the case of economic growth system. The costate implies the social demand price of a unit of invest- ment in terms of a currently foregone unit of consumption, thus the costate will be positive for all normal economy, and will be decreasing monotonically towards the final time as the opportunities for the invest- ment are decreasing as time passes. This concept is just an extention of the Lagrange multipliers--shadow prices in the case of static optimi- zation--along the time scale. With this specific property of the costate variable, a numerical algorithm given in Figure 4.1 can be used efficien- tly to solve the problem. The adjustment scheme is the key in this method which can simply be given by (as constant costate influence function matrix), 3(0) = Blmom - ALPH-S(tf) (4.41) where ALPH is a constant adjustment coefficient--diagonal element of costate influence function matrix--such that O < ALPH < l. For nega- tive final costate, the new initial costate should be increased from the old initial value, and the new initial costate should be decreased for the positive final costate until the norms (or square of Euclidean norms) of the final costate variables are within the given error limit. IV. 5 Optimal Growth of Korean Economy In applying the preceding discussions to Korean economy to obtain the optimal growth paths, one has to determine the relative weights in the objective function. This is not an easy task even for the alternative 50 objective functions, because there are virtually innumerable combinations of the ralative weights of each consumptions and capitals--a1, a b 2’ l’ b2--which satisfy the sufficient conditions of optimality. It also can be inferred from the discussions in chapter 1V.l that the higher the sum of the relative weights of consumptions (or closer to one), the longer the periods of maximum savings waiting for the final consumption spree. Thus, the probability of getting saturated optimal control--bounded optimal saving rates--will be increased for the higher values of the sum. Although the relative weights between the goods for consumption or capital may change the optimal paths of the state and costate variables not less significantly, the main concern in the theory of optimal growth lies on the consumption-investment decision, i.e., determining the rel- ative weights between the a's and b's. Thus, the relative weights for the alternative objective functions are given in such a way to reflect the alternative decisions between consumption and capital investment by varying the relative weights of these rather than varying the rel- ative weights between the goods of each sector. Table 4.l summarizes the values of the relative weights used for the alternative objective functions, initial conditions, parameters, values of the final states, initial values of the costates (the results of the numerical procedure), final costates and its norm, the number of iterations, consumption and output at the final time, and the values of the overall performance index (social welfare). Because of the definitions of the alternative objective functions, the direct comparisons of the performance indices are meaningless; however, the comparisons of the performance indices for the alternative policies 51 TABLE 4.1 RESULTS OF THE OPTIMAL CONTROLT 1 2 3.‘ 4' 5 5' U(.)T 5.800 5.423 4.435 3.575 3.092 2.530 y(lO) 0.5158 0.5874 0.5043 0.5115 0.5155 0.5182 c(lO) 0.5513 0.7457 0.7585 0.7783 0.7838 0.7873 51,52 0.2 0.2 0.2 0.2 0.2 0.2 51,52 0.0 0.2 0.4 0.5 0.8 1.0 k1(l0) 0.3204 0.4014 0.4212 0.4297 0.4344 0.4374 k2(lO) 1.2040 1.5498 1.5395 1.5787 1.7008 1.7150 p1(0) 1.1573 2.9047 3.7444 4.0559 4.0980 3.9785 p2(O) 0.4127 0.8552 1.1185 1.2375 1.2782 1.2735 p1(lO) -0.0047 0.0159 0.0145 0.0189 0.0184 0.0155 p2(lO) 0.0100 -0.0103 -0.0078 -0 0095 -0.0094 -0.0082 ||p(1o)|l:Z 0.0001 0.0004 0.0003 0.0005 0.0004 0.0003 ITER§ 7 5 8 8 8 8 +Initial Conditions and Parameters: p2(O) = 0.8, ALPH = 0.5, ERR = 0.00l P1(0) = 2.59 TThe value of U(.) is the value of social welfare function using the modified Hamiltonian, i.e., the value of non-discounted welfare. §The number of iterations for the square Euclidean norm to be within the error limit (ERR). 52 with the identical objective function may have some value as an indicator for the effectiveness of the policies. As has been discussed, the numerical algorithm in Figure 4.l shows fast convergency of the final costates to the origin with less than eight iterations from arbitrary initial guesses for the convergency error limit of 0.00l (for square Euclidean norm). This also can be observed in Figure 4.3 which shows the costate trajectories of each goods for alternative objective functions to converge to the zero from the positive initial values. Figure 4.2 shows the optimal trajectories, k, of each sector with respect to the different objective functions given in Table 4.l. The corresponding costates and control histories are shown in Figures 4.3 and 4.4. It can be noticed that the change of saving rate from the upper limit to the lower limit (switching for the case of "bang-bang" control) occurs later in the period as the relative weights on capitals are in- creased since accumulating capital is more important, and the maximum saving will persist till the last minute of impulse consumption for the extreme case of the infinite relative weights on capital. The inflections in the optimal trajectories occur when the society decides to save less and consume more, which also can be shown by sudden rises of consumption trajectories in Figure 4.5. Figure 4.6 shows the trajectories of the output per labor for corresponding objective func-' tions. 53 you ‘0“ U1 A n won per labor Ch hfi" m *7 kzm m—uw—ull' I.” —l 1 7 I , (l970) constant millio has k hfl‘ L Figure 4.2 Optimal Trajectory, k(t) Note: The numbers 1 to 6 correspond to the cases in Table 4.1. 54 4;” 4,40 ‘b’m 1 .oo 2.00 aim 4'- 00 5100 5100 TIME IN YERRS Figure 4.3 Costate Trajectory, at) lwa. g‘l 0,50 55 52(t) A r l 4 Bl 3. 9) 1700 2100 3200 4:00 5200 3:00 1100 0200 5100 TIME IN YERRS Figure 4.4 Optimal Control, s-(t) 111.00 56 1 .Igfll, «93 _zmu «96 see Loam, cog co: copppws ucmumcou .u 4100 5100 0100 7100 0100 0100 Th .00 TIME IN YERRS F 3. r 2.00 r I. Trajectory of Consumption per Labor, c(t) Figure 4.5 57 new «one 8.4. 3.... 5mm gone, Log co: seep—we accumcou .x 7.00 1500 Y 9. T I 8.00 i: TIHE IN YEARS, ?00 32m {mo E00 30.55 Trajectory of Output per Labor, y(t) Figure 4.6 CHAPTER V GROWTH MODEL OF THE OPEN ECONOMY V.l. Need for the Open Economic Model One of the major weaknesses of the model which has been developed in the preceding chpater -is its closedness, which neglects foreign trade and its effects on the other economic variables. Since World War II, the whole world has experienced fast growing world trade and increasing dependency of total production on foreign trade. With the growing importance of foreign trade, it is essential to open the- economy to include foreign transactions--not only products but also services, foreign transfer, and capital flows--to look into the whole picture of the behavior and interactions of the economic variables. Nevertheless, not much has been known of the forces and interactions within the variables for the foreign trades and in relation to the domestic economic variables. One obvious reason for this is the complexities of the international economic situation--many of the economic problems among nations can be solved via political means rather than economic policies-- and the decreased emphasis on laissez faire in international markets adds to the above complexity. Consequently, the approach which has been taken in this chapter is more or less normative, i.e., stress is more on the behavioral linkages than the estimation of functional forms in strict sense. Basically, two components were added to the closed economic model: these are foreign trade and balance-of-payments. While the foreign trade component is for transactions of products and services, the balance-of- 58 59 payments component is for the movement of foreign capital and foreign currency. The production component is the same except for a few modifications. It has been disaggregated into non-agricultural, non-grain agricultural, and grain production--anticipating the further analysis in a later chapter to investigate the effects of the world food situation. Also, saving and consumption components were extended to include the different saving ratios for the wage share and profit share of the non-agricultural production, and different savings from the non-grain agricultural production and grain production. Income distribution was considered in terms of the ratio of the per capita farm income to the per capita non-farm (urban) income. This is another important indicator of the total welfare of the economy and should be included in investigating overall economic policies. Grain market mechanisms--price mechanism, grain stock policy, grain import policy-- were also included for the case of world food problems (food shocks). Figure 5.l is an overall block diagram for the open model. 6O POPULATION Labor V Capital PRODUCTION Domestic Saving INVESTMENT l (T ' 1 L _______ _I I I w l __ __ .. _J CONSUMPTION .. ._ —— _ -) ECONOMIC Foreign POLICIES __ __ __ Saving ‘1 1 I ’F l l | I 1 I 1 I 1, . 1 1 ‘"‘““‘"‘J L'“—‘"‘ BALANCE TRADE Payment 0F Receipt , PAYMENT Export Import Debt Borrowing Service WORLD FINANCIAL MARKET MARKET Figure 5.1 Major Components of the Open Economic Growth System 6l v.2. Description of the Components Production The production component computes the aggregate outputs of agriculture and non-agriculture, total GNP, and its growth rates. The use of the aggregate production function with capital and labor as the factors of production has been used widely for empirical studies regardless of its conceptual difficulties. The basic difficulty--measuring the heterogeneous outputs and inputs in one unit or the validity of the assumed homogeneity-- has been the subject of unending disputes.1 However, it is clear that the quantity of output produced (regardless of whether it's homogeneous or heterogeneous) by any economy is constrained by the available supplies of capital and labor. Thus, an aggregate production function may be expressed by: Y = F[K, L] (5.l) where Y is the aggregate output, K and L are the amounts of capital and labor, respectively. Specific forms and characteristics of the production function commonly used are summarized in Table 5.1. Although the Constant Elasticity of Substitution (CES) production function [A5] and the Transcendental Logarithmic (TRANSLOG) production function [C4] are more general, the Cogg— Douglas production function has been used most frequently because of its 1J. Robinson pointed out the difficulties inherent in measuring the quantity of capital by a single number (or index), and suggested to measure capital in labor units [R4]. Against Robinson's view, D. G. Champernowne showed that a chain index of the capital stock can be formed when output per labor decreases as the rate of interest increases [Cl], and R. Solow discussed further necessary and sufficient conditions for the aggregation. [SlO]"Of course, it's not true that only one kind of capital good exists, but then there's also more than one kind of labor," Ibid., p. 101. Fugcpmmm+emmg+ x: me+._: Pva+Nmu> xcpmmm+mamg+mm+Pm 4:_mm+¥:Pmmm+_mn= e 1 m 1 m 1m e m m a N: a N> a >: GHN+H>+=H>2 a on m+ m+ m mgcpumxcpu o+Nm4:FH m+ cw? .ucmumcou .H m H>+mq>s .s:_ ::5 .g:: > > a > fix:.um:+s:FN:+x:_P:+o:u>:_ .mucmwumwwwmoo any mo mm=_m> mnp :o mucmgmu gm wobmz 5:855:55 up": 3.3.435;M x\:+_>x: E: mm: .4 _ a:_m:5:a:w " A s\> : ¥\> : :4 a: < u > mcwmmocomc ” v MM acawmcou n—ua+m ma mum3021nnou .m 8 ucmamcoo a m on + gm 0 > an :mm:_4 .N o pampmcou a go o a go o Hon .me cw: u > Acmwacomo go u:nu:o-u=quv ma cowpgoaoga umx_m ._ cowuzuPumnzm V mpmom on: gm: cowpoczm co xuwowummpu op mcczamm moppmwcmuumcmsu mzoahozau zomhuaoomm mh0 (5.22) 72 where s : the export share of the focus country 0, Q : total exports of the focus country and the world, respectively c, C : "competitiveness" of the focus country and the world, respectively. The equation (5.22) can be rewritten as <1=SQ then, 4=sé+oé=s0+or() new where q : total export growth of the focus country s O : world export growth effect Q 5 : competitiveness effect The appropriate measures of relative competitiveness may be relative prices, quantity improvements, improvements in servicing, changes in trade policy, etc. Export shares have to be expressed in terms of quantity in order to satisfy the condition that export shares vary directly with relative com- petitiveness, since an increase in relative competitiveness could lead to a decrease in export shares if export value shares are used when an elasticity of substitution is less than one in absolute value.1 In practice, however, export value shares have been used largely because of the absence of reliable quantity data and measure. The basic assumption of the CMS analysis--a country's export share in world markets 1Elasticity of substitution is defined as the percentage change in relative quantities demanded divided by the percentage change in relative prices. 73 should remain unchanged over time--has been set for retrospective purposes; to explain the causes of the export changes in the past. It will be modified for the projective purpose that the export (value) share is assumed to follow the logistics curve with the belief that there may exist upper limit for the growth of export share (or competitiveness) of a country.1 The logistics curve, which has been used for the population growth, can be given by v = 17155x + g] (5.24) where a, b, g are constants and |b| < l. As the independent variable x goes to infinity, Y approaches to l/g. Thus, l/g is the upper limit of Y, YM. If g or YM is given, other constant parameters can be Obtained by using OLSE (ordinary least square estimation) such as %-- g = abx (5.25) ln[l/Y - g] = ln a + x ln b O?‘ Y* = 5* + b*x. 2 Often g or YM is not given, thus, it can be varied to obtain one which 2 yields the best fit--not only in terms of R but also the significance of the t-test--since Y can be viewed as a function of YM. 1When all the countries reach to their (asymptotic) limits, or reach to the steady state in world trade, the assumption which the CMS analysis is based on holds true. 2The gompertz curve can be used for the same purpose: x y=gab . 1b1<1 x bX or ln y = ln 9 + b ln a = g* + a* . 74 The logistics curve can be generalized further as 3 xn x x2 x Y = l/[aoa1 a2 a3 ...an + g], Ianl < l (5.26) as x + w , Y + l/g. The equation also can be rewritten ln[l/Y - g] = lna0 + xlna1 + lena2 + ... + xnlnan or 2 n * + * + * + ... + * , a0 a1x a2x anx Y* = If the total world export grows exponentially, and the export share of the focus country grows along the (generalized) logistics curve, the export growth of the country can be given by q = s Q = [boeblt]/[a0a]ta2t2...antn + g] Retaining to the second order of the generalized logistics curve, EXPO(t) = [BO*EXP(B]*T)]/(AO*A]T*A2T2 + 5] (5.27) where EXPO : export of the focus country B 0, B1, A0, A], A2, G : constant parameters to be determined T : time in year Import will be divided into two parts, import content of export and the others. Import content of export represents the direct and indirect import needs of foreign natural resources, raw materials, or machineries to produce the goods which are to be exported. The other imports will generally compete with the domestic goods, and will be labeled as compressible import since these are subject to the government import control or policy. Thus, the total import can be TIMPO(t) = EMCONT(t) + CMPO(t) (5.28) 75 where TIMPO : total import of goods EMCONT : import content of export CMPO : compressible import. The import content of export and the compressible import can be determined by EMCONT(t) = CONMX(t)*EXPO(t) (5.29) CMPO(t) = TMPI(t)*GNP(t) + C (5.30) where CONMX : import content of export ratio TMPI : the marginal propensity to import (for compressible import) C : a constant There is invisible trade, other than the visible trades of goods, which includes foreign travel, transportation, insurance, investment income, government transactions, donations, and miscellaneous services. Considering the relative insignificance of the net gap, it will be assumed that the net gap is bounded by the lower and upper boundaries such as NIVT(t) = Max [NIT(t), LINVT], if NIT(t) < LINVT (5.3T) Min [NIT(t), UINVT], if NIT(t) > LINVT NIT(t) = RECIT(t) - PAY(t) where NIVT : net invisible trade NIT : expected invisible trade gap RECIT : receipts of invisible trade PAY : payments of invisible trade LINVT, UINVT : lower and upper limits of invisible trade gap. 76 The receipts and the payments of the invisible trade will be given using the trend lines. Balance-of-Payments The balance-of—payments component determines the foreign currency reserve, desired and actual foreign borrowings, total foreign debt, and debt service ratio. From the basic foreign currency equation, BOP(t) = (g [NTRD(t) + NCI(t)] dt (5.32) or, BOP(t+DT) = BOP(t) + DT*[NTRD(t) + NCI(t)] where BOP : foreign currency (or its equivalences such as gold, SDR's, reserve position in the IMF, etc.) reserve NCI : net capital inflow and NTRD(t) = NVT(t) + NIVT(t) (5.33) NCI(t) = FBR(t) - DBS(t) (5.34) where FBR : foreign borrowing at time t 085 : debt service at time t The foreign borrowing is determined by the amount of the desired foreign borrowing and the world financial market situation. FBR(t) = MFLA(t)*DBR(t) (5.34) where DBR : desired foreign borrowing MFLA : multiplier for the availability of foreign loans. 77 The multiplier for the availability of foreign loan, which reflects the world financial loaning situation, will be derived from the monotonic decreasing or downward leping S-shaped curve believing that countries facing critical foreign indebtedness may not be able to obtain the sources of foreign loan as much as they want, i.e., they have to cut down the consumption level and economic growth to overcome the vicious circle of deepening indebtedness. Frequently, S-shaped curves are given by the TABLE (table look-up) functions [F2], [M3]. Table function, which assigns the values to the points of interest by interpolating or extrapolating the observed data points, is not restricted to specific shape of the function. For the conceptual or hypothetical functions without observed data, the flexibility to test the sensitivity is essential. The use of table function is cumber- some, if not inflexible, in this respect. To remove this inconvenience, the combined negative exponential function will be used for the S-shaped curves in general. For ID = l, SCURVE(A,B,C,T,ID) = l - B*EXP[A*(T-C)], if T < C (5.36) ‘1 - 8*[2 - EXP[-A*(t-C)]], if 1 > c where A,B,C : parameters for S curve T : independent variable 10 : index indicating ID = l : monotonic increasing 2 : monotonic decreasing 0,00 0,00 0,40 0,00 ":50 78 iv _a _a \IUTUJ-‘lb AM” 0110 0100 0130 0140 1;.00 0100 0170 0100 0100 1100 Figure 5.3 Monotonic Decreasing SCURVE for B=O.2, C=O.5, and ID=2 79 For ID = 2, SCURVE(A,B,C,T,Z) = 2*[l - B] - SCURVE(A,B,C,T,l) The SCURVE for B = 0.2, C = 0.5, ID = l, and for different values of A are shown in Figure 5.3. Thus, MFLA will be calculated by MFLA(t) = SCURVE(AMF,BMF,CMF,DSR,2) (5.37) where AMF,BMF,CMF are the parameters reflecting the world financial situation. The desired foreign borrowing is DBR(t) Max [0, SLK(t)] (5.38) DBOP(t) - BOP(t) SLK(t) where DBOP is the desired foreign currency reserve. The desired foreign currency reserve increases as the national economy or the total amount of trade grows DBOP(t) = K]*[EXPO(t) + TIMPO(t)] (5.39) where K : constant parameter for the proportion of desired foreign currency reserve to the total trade The process of the accumulation of debt is a delay process with borrowing as input, debt service as output, and the loaning period as average delay, which may be approximately described by the distributed delay [F2], [M3]. One of the basic properties of the distributed delay process is the conservation of flows--every single entity which enters 80 into the delay process should come out eventually without loss or gain. In practical problems like the maturation or population growth models, the distributed delay process with loss (or attrition) has been used. [V ] [A ] However, the basic feature of the debt accumulation process is the gain in the delay, the accrual from the interest on capital, which can be formulated by using accrual rate opposite of the attrition rate in sign in the model of the distributed delay with attrition as shown in Figure 5.4. Thus, it can be formulated as 501(1) dt = r'i..1(t) ‘ r‘1-(11) + Gi(t)’ and 011:) = 1,5, 0,11). x111=r0(t1 ~ u(t) 0,11) 0211) r2(t) rk_1(t‘ (5.40) (5.4l) 6km - rk(t)=y(t) 0,111 A, Figure 5.4 The k-th order Distributed Delay with Gain 81 where : the i-th storage in the cascaded stages r. : the flow rate out of the i-th storage G. : gain at the i-th stage 0 : storage in the k-th order delay process. The basic assumptions relating the storage, flow rate, and gain of the delay process are r1.(t) £01m, i = l,2,...,k (5.42) CD A fl V II > n A fl v O A c+ V II 3; O O A fl V (5.43) DEL : average delay AC : accrual rate. The equations can be rewritten in terms of the flow rates which is more convenient in simulation, 3,, = fi'g—L 0,4111 - [1 - Acm D—E—L-T r (t)], i = 1, ..., k (5.44) i The number of stages, k, and the average delay can be determined by the grace period and maturity of a loan. The accrual rate equals the interest rate, the output of the delay process is the debt service, and the sum of the storages is the total debt. TDBT(t) = 21:, 01(t) (5.45) DBS(t) = Rk(t) = Y(t). (5.46) The overall block diagram of the balance-of—payments component is shown in Figure 5.5. 82 EXPO(t) 11 6—k1 TIMPO(t) + BOP t w (1- X? SLK(t) I DBR(t) ? O if SLK(t) ;,0 DBR(t)=SLK(t) f if SLK(t) > 0 1+k3 -——) Tr DSR(t) DBR(t) " MFLA(t) [::: Figure 5.5 Balance-of—Payments Component 83 Grain Component Grain component has been added to examine the external and internal effects of the grain policies to the total economy. The agricultural production has been disaggregated into non-grain and grain agricultural production, grain consumption demand and price mechanisms, and grain storage subcomponents are added. The grain production in physical quantity can be given by QGi(t) = AREAi(t)*YLDi(t) (5.47) where 0G1 : grain production in MT AREA, : area planted for the i-th grain YLDi : yield of the i-th grain i=1 : rice 2 : barley 3 : wheat. Generally, the planted area is a function of the expected price (and capital investment for the case of the land reclamation, land development), and the yield is a function of capital investment, labor, and technological progress such as new variety, fertilizers, herbicides, etc. Retaining the Cobb-Douglas form as in the equation (5.2), 051(t) = Ai*Ki(t)ai*Li(t)bi*TTi(t-l)ci, i = 1,2,3 (5.48) where TT, : one year lagged terms of trade which is the one year lagged i-th grain price divided by the overall price index. 84 In the grain component, it is assumed that the government has complete control over the supply or the prices of grains. It should be noticed that the demand of grains cannot be directly controlled, i.e., the govern- ment cannot set the supply and prices of the grains at the same time, and thus the individual preferences not directly controlled by government policy. This is the case with the perfectly inelastic supply which is shown in Figure 5.6. Figure 5.6 Price-Quantity Determination with Perfectly Inelastic Supply 85 The demand functions for grain can be given in different forms; the linear-expenditure system, the system of double logarithmic functions, the Rotterdam differential model, and the indirect addilog system. [Yl], 2 [Pl]]. Though there are several shortcomings, the system of double logarithmic function will be used because of the relevance in the case of market-clearing situation. ln[D(t)/POPL(t)] = A + [_3_ 1n P(t) + 6 1n I(t) (5.49) where 5': demand per capita "0| : price of grains I : income or GNP per capita : constant vector IJ> §_: matrix for the price elasticity of demand [C : vector for the income elasticity of demand. Equation (5.49) can be used in two ways for the market-clearing situation: first to determine the amount of demand given prices, secondly to determine the (market clearing) prices of grains if the supply fails to meet the demand. For the case of determining the prices of grains given supply, SUP (which is not equal to the demand), equations can be inverted such that 1The linear-expenditure system (or Stone-Geary expenditure system) has been pointed out to be superior to others in terms of the desired properties of the demand system such as income constraint, invariance with respect to increasing transformation, Slutsky condition, and Ville's condition. [Y1], However, this system has a serious weakness in the estima- tion procedure since the additive disturbance terms are implicitly assumed to be mutually dependent. [Pl]. 2Problems relating to the use of this form has been discussed in pp. 83-84 of [A2]. 86 ln P(t) 8" [1n SUP(t)/POPL(t) - A - E15 1(5)] (5.50) given the elasticity matrix B_is non-singular. Grain storage in terms of annual carry-over stock is determined by the grain production, demand, storage capacity, and desired stock levels. The desired grain stock will be given by the proportion of total demand DGSTKi(t) = CSTKi(t)*DMGRN1(t) (5.5l) where DGSTK : desired grain stock level CSTK : desired proportion of grain stock to the total demand DMGRN : demand of grains The change in grain stock and the demand—supply gap can be DSTKi(t) = DGSTKi(t) - GSTKi(t) (5.52) DSGAPi(t) = DMGRNi(t) - QGi(t) where DSTK : changes in the grain stock GSTK : existing grain stock DSGAP : demand-supply gap of grains. Grain import requirements (or increases in the stock level) are determined by the net shortages (or surpluses) GQIMPTi(t) = DSGAPi(t) + DSTKi(t) (5.53) GSTKi(t) = GSTK1(t) + DT*DSTKi(t) where GQIMPT : grain import requirement to meet the demand (consumption demand plus storage demand) If net surplus occurs, i.e., DSGAP plus DSTK is negative, the stock will be piled up. 87 GSTKi(t) = GSTKi(t) - DT*DSGAPi(t) (5.54) GQIMPTi(t) = O. The actual import is determined by the import requirements and grain import policy parameter. GQIMPi(t) = CGMi(t)*GQIMPTi(t) (5.55) where GQIMP : actual grain import CGM : grain import policy parameter which may be a function of domestic and/or world grain prices. The total supply of grain is given by the production, import (or export), and changes in stock. SUPGRNi(t) = QGi(t) + GQIMPi(t) - DSTKi(t) (5.56) where SUPGRN is the total annual supply of grain in year t. The dependency on the grain import can be calculated by 3 ._ GQIMP.(t) DPGIMP(t) = 2"] ' (5.57) 3 21:]SUPGRNi(t) where DPGIMP is the dependency on the grain import. Other Components Population and labor, price indices, and the world market conditions are given exogenously by fitted trend line in the model. Total labor force is given by the trend with the constant growth rate, and the pro- portions of the agricultural and non-agricultural labor are given by a linear function of time. Urban and rural populations are calculated in the similar way except the proportions are given by the monotonic increasing negative exponential function. 88 Price indices are given by the trend projections except the grain prices for the case of supply shortage or surplus. The overall price indices are where y 3 P5.( “*05 (t) PIG(t) = 3 2i=1 QGi( ) (5.58) PIA(t) = wa1*PIG(t) + w *PING(t) a2 PI(t) = w *PIA(t) + w *PINA(t) 2 l PIG : price index of grains PIA : price index of agricultural product PING : price index of non-grain agricultural product PINA : price index of non-agricultural product PI : overall price index 1 wi : the weight of price index for each product. Two kinds of world market conditions are considered. The first is the financial situations which are taken into account in the multiplier for the availability of foreign loans, parameters in the debt accumulation process--grace period, maturity,and interest rate. The second is the world grain &.Set prices, which are not known generally, but will be assumed to follow of scenario. CHAPTER VI SIMULATION OF THE OPEN MODEL VI.l. Design of Economic Policies The need for macroeconomic policies arises because of the inadequate self-adjustments of the economic system to the shocks it is constantly subjected. Thus, economic policy consists of the deliberate manipulation of a number Of means 51151551 to it in order to attain certain aims. (iggég; may be lowered to stimulate employment; social security may be introduced to further an equitable distribution of the national product, and so forth. In a strict sense, the policies can be discriminated into three different kinds according to the nature of the means_used;(reforms, qualitative .. *1...- policy,fiand quantitative_policy. RefOFmS} the most far-reaching types of policy, refers to changes in foundation or introduction of new systems of policies, qualitative policy means changes in structure such as a change in_the number of taxes,and quantitative policy is the changes that can be brought about in the values of the instruments of economic policy.1 Most of the policies experimented with in the models are of the quantitative type because they are particularly used to quickly adapt the position of the economy to variations in the frequently changing environment. In general, designing policies can be hindered by the complexities and the high degree of uncertainties in societal systems not only with regard to the structures but also to the aims or objectives of the society. 1[T3], p. 7. 89 90 Because of these obstacles and the interdependence between most economic phenomena, the policy and the desired states of the economy at a certain time can't be solely determined; it is necessary to consider them as a (coherent) whole. Examples of the goals of an economy might\be: (l) high, \ stable, and growing level of real income, (2) stable prices or low inflation \ rate, (3) a balance~of~payment surplus, (5) equitable or balanced distri- bution of income over various groups of the population--social, industrial, regional--etc. The instruments or instrument variables may be tax rates, public expenditures, the rate of discount, reserve ratios, foreign exchange rate, wage rates, etc. In fact, all these goals and instruments are inter- related in such a way to produce both "goods" and "bads." Among other possible instruments, the desired total saving rate and the import of compressible goods, i.e., controllable import, are chosen to investigate the behavior of the model in a search for optimal policies. Higher total saving rates will increase the total production and the growth rate, while it may raise the foreign indebtedness if the country fails to provide with domestic savings, or, at the same time, higher domestic saving can translate into lower consumption levels. Import of goods has a similar effect on the economy; higher imports mean higher consumption but an unsound balance-of—payments and probably increase in the foreign debt. What a country wants is conflicting; high production and economic growth, low foreign indebtedness, and high consumption level. This is only possible for a country with abundant sources of production. The matter of deciding the social goals is not simple; it involves both material and spiritual 1~ell-being and is complicated by political factors. 91 However, interactions between decision makers and models may help a great deal in determining goals and once the (conflicting) goals are given, a feasible policy can be defined as one which can trace a path along the time scale that satisfies the "desirable" or reasonable boundaries (con- straints) of the state variables and achieves the goals. Normally, this has been done by the analysis of a system which starts from a set of policies and examine their consequences--paths of variables and certain performance indices which the policy makers may be interested in. One inherent weakness of this approach is that it may be too costly to find a satisfactory dynamic path of policy which satisfies all the desirable requirements, since one doesn't know in advance how the changes in input will be translated into the changes in the paths of other economic variables. Conceptually, this problem can be handled inversely, i.e., by the control of a system which starts from the desired performance indices and constraints to determine the inputs. However, technical difficulties in solving the optimization for control in a complex system, and the con- ceptual difficulties in constructing the objective function, which usually is not known initially, have been the barrier for the control concept to be applied in complex systems. Since the conceptual difficulties can by no means be completely removed so as those in constructing the policies for the analysis, the concept of "alternative objective functions" can be used to lessen the difficulty where the policies can be determined using a set of possible objectives. 92 The remaining portion of this section will be devoted to the discussion of technical difficulties in practice, and the ways of removing these difficulties which can lead to the design of dynamic policies in an open economy. Optimization of a Complex System Finding the best economic policy of a complex system, i.e., determining the values of instrument variables that render a certain welfare function a maximum, is quite complicated in mathematical nature and impossible in general if not only the instrument (input) variables but also the state variables are subject to certain boundary conditions. The usual constrained optimization problem can be stated as follows: Optimize f(7) (6.l) subject to l. §_gi(¥) < u. 1 _ 1, 1:], one,” where f(7) represents a certain objective (or welfare) function, 91(Y)'s are the functions of control (instrument) and state variables that have to be constrained, and Y'is a vector of instruments such as saving rates, tax rates, public expenditures, etc.,for the case of an economic system. For special cases of f(7) and gi(7), linear and nonlinear programming techniques can be used to solve the problem. If the constraints are on the independent variables which can be transformed in such a way not to alter the objective function, the unconstrained optimization routines such as search techniques and gradient methods can be applied depending on the nature of the model. When constraints exist on the state variables which cannot be controlled directly but derived by the control variables, solving the optimization is impossible in a mathematical sense where only feasible solutions are qualified to optimize the Objective function. But in practice, the constraints may not be strict as in a mathematical sense, i.e., some of the constraints may be relaxed to get the optimum solutions. Certain 93 feasible boundaries for the income distribution ratio, the debt service ratio, and the level of consumption may be desired to ensure the total economy working without drastic chaos, but one may not be sure of the exact boundaries. Due to these complex implicit constraints, search or gradient methods cannot be directly used to solve the problem; however, one possible way may be the method of peanlty function which includes the constraints as a penalty function in the objective criterion. The objective function can be: Optimize Z = f(7) + P[g(Y)] (6.2) where f(7) is the constrained objective function, P[g(7)] is the penalty function which is the function of the constraints, and Z is the objective function of unconstrained optimization, so that the general unconstrained optimization routines can be applied. The penalty is given in such a way as to make the value of Z higher (or lower) for violating the constraints for the case of minimization (or maximization). One of the commonly used penalty functions is in the form of the Heaviside unit step function which gives penalty or no-penalty according to whether the constraints are violated or satisfied. One weak- ness of using the Heaviside unit step penalty function is its rigidness, i.e., the values of the objective function may not be sensitive to the changes in the constrained state variables which are improved from the previous states, especially when the boundaries are not strict in a mathematical sense but rather can be considered as "desired boundaries" which can be violated by a certain amount. This is shown in Figure 6.1 for illustration, with the case of the error-minimization and one-sided constraints given as 94 X Figure 6.1 Possible Paths of Constraints for the Case of Heaviside Unit Step Penalty Function 91(7) §=ui’ i = l,..., n. (6.3) The cases of A and B yield the same value of the penalty function regardless of the closeness of B to the boundary. Moreover, the state C may be the best among the three while it yields the highest value of the objective function if the Heaviside unit step penalty function is used. The preceding illustration leads to the formulation of penalty functions in such a way as to reflect the sensitivity of closeness to the desired boundaries. This can be achieved by using a penalty function as follows instead of the Heaviside unit step function. 95 Thus, the penalty function can be r mi-G[gi(7) - 1i]’ if 1i 3_gi(7) (6.4) P[9(X)] =« o . 151,: 9,170 _<_ u, where G(.) is a function of the deviations from the desired boundaries, mi and “1 are the penalty weights for the violation of the lower and upper boundaries, 1. 1 and ui, respectively. The distinction from the former, in essence, lies in the choice of the functional form of G; linear, quadratic, etc. Another distinction that can be made between the normal optimization problem and that of a complex system is the formulation of the objective function. The most commonly used form is the quadratic objective function with the error terms to be minimized which implies underachieving a desired level or path, and overachieving are penalized equally. This has to be modified in many real economic problems, since overachieving and under- achieving may have different values. For example, the potential undesirable implications of any given surplus of balance of payments may differ sub- stantially from those of a deficit of equal magnitude. The undesirability of underachieving and overachieving the target growth rates will be significantly different. It is in this sense that the piecewise quadratic function offers a more general framework for policy optimization.1 f(7) + P[917)] (6 5) a'Y + 1119(7)] + l/ZIY'AY + 9171' 55(7)] Optimize Z dT+v2Tg' 1[F ], pp. 183-195. 96 where 741‘, 3001', c=[a. 51. “E931 and the elements of the matrix C, Cii’ can be given by f C11 If yi 5 ”(Y1) = {yilyi > Y1} ch- =1 0 if y.- e N(yi) = {yily} 3y.- :yli‘} \c}, if y, e L(yi) =1y,1y, < y11 where U(yi) : the upper extreme set for yi L(yi) : the lower extreme set for yi M(yi) : the set of y. which satisfies the lower and the upper boundaties cgi, y: : the upper boundary value and the point 1 1 Cii’ yi : the lower boundary value and the point The preceeding considerations on the optimization of complex systems—- redefining the strictness of the constraints and the use of a piece wise objective function--allows one to apply control concepts to the complex system, and general Optimization routines such as search techniques can be used for this purpose. Orthogonal Representation of Policy Variables Policy making is dynamic in nature, i.e., one has to decide policies at every specific point in time to produce the overall desired goals during a certain time period. In the closed model, a policy could be expressed as a function of time which was obtained by solving the optimal control problem. With the 97 complexities in the open model, optimal control cannot be directly applied.1 Even in the complicated model, however, there are two ways to fulfill this purpose: first is the determination of the discretized points of the paths of policy variables as a whole, and the second is to find or fit a specific function which will trace the path satisfactorily with the unknown coef- ficients to be determined. Within reasonable numbers of the discretized points to be determined, the general optimization routines can be used for the first,however,as the number of variables (points) increases, it is impossible to get solutions in general at reasonable cost. The second way can be applied more generally in this sense. It can be observed that the basic idea of this is the same with the function representation of signals in engineering. The analogy is not only that the signal forces the system to take the action necessary for accomplishing the desired objective but also with the generally very complicated realized shapes. Because of complexity which can be caused by enormous numbers of sources, the signal is represented as a function of time, and thus any quantitative representation of a real signal is necessarily an approximation. The usual way of representing an arbitrary time function is to use a linear combination of a set of elementary time functions which form a basis in the functional space of a certain dimension. Among other pro- perties, one important desirable property of the basis functions is 1If the system state equations can be maintained as in the closed model, the approximation of function can also be used for optimal control where the problem is to find the polynomial of degree n (or others) which has the least mean square deviation from the given function on a certain interval, i.e., the classical problem of finding the Fourier coefficients in the expansion of the function in Legendre polynomials. [P8], pp. 197-213. 98 orthogonality which will allow one to determine any given coefficient without the need for knowing any other coefficient--the finality of coefficients--which can be illustrated as follows: Let f(t) be an arbitrary function Of time to be represented by the linear combination of orthogonal (basis) functions f(t) = 2,20 a, x,(t) (5.5) where xi's are the orthogonal functions, i.e., t [t2 Xi(t) X-(t) dt 0 , for l f 3 (6.7) 1 J = 1., for i = ' J J Multiply both sides of (6.6) by xj(t), xj(t) f(t) = xj(t) [1.20 a1. x1.(t) dt (5.8 then, t2 f _ n t2 It] xj(t) (t) dt - {i=0 ai It] xj(t) xi(t) dt (6.9) = aj NJ. Thus, t2 aj =1/1jft] xj(t) f(t) dt (6.10) This shows the finality Of coefficients as described before. The possible forms of arbitrary functions can vary. The Fourier series, which uses sinusoidal elementary functions as the basis, and the polynomial function with the Legendre polynomials as the basis are commonly used. Any one of the two functions can be more efficient in a specific case in the sense of better approximation to the real values and better convergence in carrying out optimization. Legendre polynomials which are a particular solution of the Legendre equation, will be used here. If an arbitrary function is f(X), then f(X) = [1:0 a, P,(X) (5.11) 99 where P.'s are the Legendre polynomials such as 1 P0(X) = 1 (5.12) P1(X) = X P2(X) = 1/2 [3x2 - 1] P3(X) = 1/2 [5x3 - 3X] Pk+](t) = l/(k+l) [(2k+l) X Pk(X) - k Pk_1(X)], k=2, 3, 4, ... It can also be shown that the Legendre polynomials satisfy the orthogonality relation within X c [-l, 1] such that f1} Pm(X) Pn(X) dX = O , for m 7 n (6.13) = 2/(2n+1), for m = n In the actual application of designing policies, the domain is the planning period of time, t1 to t2. Thus, by changing the variable as x = [2(t - t])]/(t2 - t1) - 1 (6 14) or t = [(t2 - t])/2](X + 1) + t] then, 1 [_1 Pm(X) Pn(X) dX (5.15) t = 2/(t2-t1) It? Pm(t) Pn(t) dt = 0 , for m f n = 2/(2n+l), for m = n hence, t 2 _ It] Pm(t) Pn(t) dt - O , for m # n (6.16) (tZ-t])/(2n+1), for m = n. The equation (6.16) ensures the orthogonality of the basis (Legendre polynomials) for t c[t1,t2], and thus the arbitrary function (6.11) can be used to approximate the policy paths during the planning period from t1 to t2. 100 V1.2 Empirical Application to the Korean Economy The considerations of the preceding section, design of dynamic pol- icy using optimization of a complex system, will be applied to the case of Korea with the open model. More specifically, the problem can be stated as following: find the coefficients of the arbitrary time func- tions of Legendre polynomials for the total desired saving rate (TOSR) and the marginal propensity to import of compressible goods (TMPI) in such a way to maximize the economic growth subject to the desired boun- daries for foreign indebtedness and consumption level. As already been observed, the use of the same functional form for the objective and the penalties for the constraints allow one to handle them in a unified fashion. Thus, using the piecewise quadratic form, the overall objective function (performance criterion) is given by t2 2 Minimize u = ft {[CW](RGNY(t)-DRGNY(t))] + 1 [CW2(DSR(t)-DDSR(t))]2 + [CW3(CONR(t)-DCONR(t))]2} (5.17) where RGNY : real GNP growth rate DRGNY : desired real GNP growth rate DSR : debt service ratio,1 i.e., debt payment divided by export DDSR : desired debt service ratio CONR : rate of change of aggregate consumption DCONR : desired rate of change of aggregate consumption 1Although the debt service ratio has been commonly used to represent the foreign indebtedness as an indicator of debt servicing capacity of a country, other indicators such as the ratio of non-compressible import to total imports, export fluctuation index, growth rate of export, etc. can also be used. [F3] 101 CW1's are the relative weights of the objective function and the pen- alty functions such that cw1 = N1 u[DRGNY(t) - RGNY(t)] (6.18) cw2 = ”2 p[DSR(t) - DDSR(t)] (5.19) cw3 = W3 u[DCONR(t) - CONR(t)] (5 20) where u(.) is the Heaviside unit step function and Wi's are constants. The policy variables, TDSR and TMPI, are expressed in the linear combination of the Legendre polynomials up to the second order, thus TDSR(t) = SRA + SRB X(t) + [SRC/2][3X(t)2 - 1] (5.21) TMPI(t) = TMPA + TMPB X(t) + [TMPC/2][3X(t)2 - 1] (5.22) where X(t) = [2(t - t])]/(t2 - t1) - 1 (5.24) and SRA, SRB, SRC, TMPA, TMPB, TMPC are the coefficients of the poly- nomial to be determined. Optimization routine COMPLEX,1 which searches the minimum (or max- imum) of the multivariable, nonlinear objective function sequentially with nonlinear inequality constraints, has been used to find the opt- imum values of the coefficients, i.e., the values of the coefficients which minimizes the objective function (6.17) satisfactorily with a cer- tain convergence criteria. Different optimization results have been obtained using the alt- ernative objective functions with different sets of relative weights 1[K9], pp. 358-385. 102 TABLE 6.1 RESULTS OF THE OPTIMIZATIONS I II III IV V VI Parametersi CW2 l 10 10 l l 1 DRGNY 0.1 0.1 0.05 0.1 0.1 0.1 DDSR 0.2 0.1 0.1 0.2 0.1 0.1 . Condns1 TMPA 0.15 0.15 0.15 0.15 0.15 0.15 0.25 0.25 0.25 0.25 0.20 0.19 TMPB -0.05 -0.05 -0.05 -0.05 -0.015 -0.015 0.05 0.05 0.05 0.01 -0.003 -0.003 TMPC -0.05 -0.05 -0.05 -0.05 0.001 0.001 0.05 0.05 0.05 0.05 0.005 0.005 SRA 0.2 0.2 0.2 0.2 0.1 0.1 0.4 0.4 0.4 0.4 0.35 0.3 SR8 -O.l -0.1 -0.1 -0.1 -0.125 -0.125 0.1 0.1 0.1 0.01 -0.015 -0.03 SRC -0.1 -0.1 -0.1 -0.1 0.005 0.01 0.1 0.1 0.1 0.1 0.025 0.025 TMPA+ 0.15 0.l5 0.15 0.17 TMPC 0.25 0.25 0.25 0.23 SRA+ 0.2 0.2 0.2 0.245 SRB 0.4 0.4 0.4 0.352 TMPB/ -6.0 -6.0 TMPC -3.0 —3.0 SRB/ ~6.0 -6.0 SRC -3.0 -3.0 . Condns TMPA 0.2 0.2 0.2 0.2 0.187 0.187 TMPB 0.0 0.0 0.0 0.0 -0.01 -0.01 TMPC 0.0 0.0 0.0 0.0 0.0033 0.0033 SRA 0.3 0.3 0.3 0.3 0.284 0.284 SRB 0.0 0.0 0.0 0.0 -0.05 -0.05 SRC 0.0 0.0 0.0 0.0 0.0l6 0.016 Results TMPA 0.24965 0.22308 0.2180l 0.21718 0.19991 0.18995 TMPB 0.01335 0.00877 0.00021 0.00977 -0.01490 -0.01068 TMPC 0.00033 0.02509 0.03180 0.01266 0.00490 0.00306 SRA 0.39917 0.39990 0.39871 0.39988 0.34953 0.29990 SRB 0.06617 0.07330 0.09941 -0.00250 -0.01575 -0.03028 SRC -0.00l79 -0.00578 -0.04457 -0.09983 0.00509 0.01006 U 369.759 404.819 356.271 435.618 670.794 919.208 Iterations 143 118 89 73 96 68 1.Other parameters: CW1 = 1, CW3 = 1, DCONR = 0.1, DT = 1.0 ALPHA = 1.3, BETA = 1.0, GAMMA = 5, DELTA = 0.001 fBoundary conditions. The first number is the lower boundary and the second number is the upper boundary. 103 and desired values, and using the alternative implicit and explicit boundaries which will determine the maximum and minimum of TDSR and TMPI. Table 6.1 summarizes the different parameters used for the alter- native objective functions, the different boundary conditions and ini- tial conditions, the results, and the number of iterations to meet the convergence criteria of the COMPLEX algorithm specified by the parame- ters, ALPHA, BETA, GAMMA, and DELTA. The constraints on the coefficients of the polynomials (which are the variables in the COMPLEX algorithm to be searched) are given to limit the values of TDSR and TMPI within a reasonable boundaries. These have been constrained by the intervals [0.25, 0.48] and [0.18, 0.26] in general for TOSR and TMPI, respectively. For the last two cases, V and VI, the shape of the functions also has been restricted to reflect a certain pattern of path (or policy) which a country may prefer; such as a policy to decrease its dependency on the foreign ca- pital and the import of compressible goods. The monotonic decreasing values of TDSR and TMPI can be attained by imposing additional const- raints on the coefficients as follows: f(X) a0 + a1X + (52/2)[3x2 - 1] [3a2/2][X + 51/352]2 + a0 - 32/2 - 55/552 (5.25) then, the conditions will be 82 > 0, and X = -a]/3a2 ;:1, or equivalently, a1/a2 ;:-3. (6.26) The optimal paths of TDSR and TMPI with respect to alternative obj- ective functions and constraints are shown in Figures 6.2 and 6.3, res- pectively. Policy path I shows increases in both total saving and imp- 0,10 0,00 0 .04 104 VI 41.00 r 1.00 2100 Figure 6.2 3100 5100 0100 0100 T I ME I N YEHRS Optimal TOSR (total desired saving rate) 0,22 0.00 0:01 .20 4A 0017 105 0 0 1) o 41.00 Ti 1.00 Figure 6.3 €00 r f T T 0100 4.00 15.00( 0.00 1100 0.00 0100 Optimal TMPI (the marginal propensity to import) 10.00 106 ort thus implying the high economic growth and high foreign indebted- ness since it is unrealistic for a country (like Korea) to achieve the saving rate of 0.46 from only domestic sources. Policy II shows an increasing path similar to I for TDSR but a significantly different path for TMPI. This comes from the higher wei- ghts on the foreign indebtedness relative to the others for policy II, i.e., heavier penalty on foreign indebtedness by changing the weight of CW2 from 1 to 10 and lowering the desired debt service ratio to 0.1 from 0.2. Thus, policy II exhibits less import of compressible goods while maintaining about the same level of foreign capital borrowing. This is conceptually correct, in other words, if a country wants to lower the foreign indebtedness, it has to restrict the import of comp- ressible goods first of all rather than to restrict the production cap- acity by lowering the foreign capital borrowings. Policy paths, V and VI, reflect the policy towards self-sufficiency in capital formulation and reduction in the import of compressible goods, which, in turn, will lower the consumption level. Other paths, III and IV, show policies in between the above two groups. The preceding considerations of the shape of the policy paths, TOSR and TMPI, can be more clearly shown by the paths of economic variables of interest--GNP, growth rate, debt service ratio, annual foreign bor- rowings, total debt standing, aggregate consumption per labor, income distribution ratio, domestic and foreign saving ratio. These are plot- ted in Figures 6.4 to 6.13. Generally, more than twice the real GNP of 1975 (4200 billion 1970 won) can be achieved in 1084 by any of the six policies. A high level of GNP during the planning period (1975 to 1984), could be attained by 107 policy IV which borrows more than the other policies for the first half of the period. But its growth rate drops significantly below 6 percent during the last period because of the early fast growth and the de- creased foreign borrowings in the latter period. While policies I, II, and III show the similar high level of GNP and GNP growth rates (about 10 percent throughout), policies V and VI show the lower growth rates due to the lower levels of total desired saving rates. However, these curves still exhibit 8 to 9 percent average growth rates and about 7 percent in 1984. Debt service ratio, current foreign borrowings, and total debt combined can show a clear picture of foreign indebtedness. With the high borrowings and imports, policy I yields the highest and fast in- creasing foreign indebtedness. Policies II, III, and IV show similar paths of debt service ratio and total debt while the sharp decrease in the growth rate of the foreign borrowings can be attributed to the lower total desired savings for the second half of the planning period. Pol- icy V maintains a current debt service ratio of 0.11 to 0.12 throughout, and increase the foreign borrowings and total debt by relatively moder- ate rates. Policy VI decrease the debt service ratio substantially to about 0.07, maintains almost current level of foreign borrowings, and shows slowly increasing total debt in current terms, thus providing a sound basis of economy from a possible vicious circle of indebtedness. Annual foreign borrowings in 1984 can be varied from the current level of 3500 million dollars in 1975 to more than ten times, and the total debt may increase two to nine times the 1975 debt standing of about six billion dollars. Aggregate consumption per labor which is one of the most important 108 Measure for the social welfare is increasing about one and half to two times the 1975 level in 1984 with different increasing rates. For high- er production and import, like policies I and IV, the consumption level during the period is high. For the policies with lower foreign indebt- edness, like V and VI, the consumption level is relatively low. Paths of the consumption growth rates show more clearly the characteristics of the consumption behavior during the planning period with respect to the different policies. The real consumption growth rate will be about 6 to 7 percent annually with policy I, and about 4 to 3 percent with policies V and VI, respectively. Other policies show significant chan- ges in consumption behavior: policy IV reaches about 8 percent then drops to 4 percent, policies II and III Show rapid increase from 2 and 4 percent to about 9 percent annual consumption growth. Income distribution is another important consideration. It is mea- sured by the income distribution ratio, defined as the per capita farm real income divided by the per capita urban real income. The improved income distribution ratio (Fig. 6.11) is mainly because of the internal feedback loop through migration and transfer of income from urban to rural via the nonagricultural sources. The distribution ratio ranges from 0.9 to 0.95 by 1984, and the best policy for equitable distribu- tion is VI which shows higher ratios most of the time. Policy V shows slightly less values of distribution ratio than VI, and policy IV shows relatively low income distribution ratio throughout the planning period. As has been mentioned, there are two sources of savings-~domestic and foreign savings. If domestic saving is given, then the foreign sav- ing can be determined by the difference from the total savings which can be given by a specific policy. Domestic saving rate has been given at 109 about 24 percent during the planning perod. The least equitable income distribution policy IV can achive the highest domestic savings, and the most equitable distribution policy VI yields the lowest domestic savings. Foreign saving, implicitly assumed to be under the government reg- ulation or policies which can control the flow of the foreign savings, can be determined as the net of the domestic savings from the total savings needed. The overall paths of the foreign savings will therefore become similar to those of the total savings given the steady domestic savings. Each of the policies represents the optimal policies which opti- mize the "alternative" objective functions and constraints. Nevertheless, each policy may have substantially different implications. Policy I shows high and increasing debt service ratio, high foreign borrowing and total debt, lower income distribution ratio, and high economic growth and consumption. Policy II represents higher growth, foreign indebtedness, and consumption but a lower income distribution ratio. Policy III exh- ibits higher growth and foreign borrowing, medium debt service ratio and consumption, and lower income distribution ratio. Policy IV displays high economic growth and consumption, higher foreign indebtedness, and low income distribution ratio. Policy V presents lower growth, foreign indebtedness, consumption, and the higher income distribution ratio. Policy VI results in low economic growth, foreign indebtedness, and high income distribution ratio. In sum, policy I represents a program which focuses on enjoying affluence with foreign borrowings, but results in the least equitable distribution of income. Policy VI represents an austerity program on consumption towards self-sufficiency in capital formation, and thus provides a sound basis of foreign indebtedness and 110 8.31. 35$ 3.1... 8..“- 8&0 co: cowpp_n pcmgmcoo Acumpv swap 8. .... » 09mm 104» €50 TIME IN YEARS 0km r 0.00 Gross National Product Figure 6.4 111 ‘94» Figure 6.5 T 04» £00 EARS 0100 0100 TIME IN Y Real GNP Growth Rate I'— I 0.00 1h¢n A] 900 0.01 I 001‘ 4 0:00 112 I 1100 2100 3100 4.00 .00 0100 1.00 0100 0.00 10.00 Figure 6.6 Debt Service Ratio 113 III II IV I v 10.00 7100 0100 0100 0100 IN YEHRS r 0.00 TIN! 0100 r 3. r 2. 1100 8.3“ 8.0% 8.8.. 86* mew—pot cowppws 86m“ 3.340 8.2a .o.. d >86? 8.8. 8.2: 8.?“ 8. ”0.00 Foreign Borrowing Figure 6.7 114 VI 0100 0100 1100 0100 0100 10.00 TIME IN YEARS 0100 1100 865.1 86$. 86mm 86mm 86m- mgwppou cowppwe 86m. Shawl ”36% 86mm 86w. 86% 55.55 Total Debt Figure 6.8 olfl abor 2:5 per 1 E constant million won 0.00 5:01 0‘.“ .1 0n; 115 ‘p.26 j f 2.00 8.00 Figure 6.9 4100 0100 0100 TIME IN YEHRS Aggregate Consumption per Labor 0.00 Titan 0.1: 116 Iv _ _51 \ \ 1. 0 61 8. 51.00 1100 2100 0100 4100 0100 0100 7100 0100 To) TIME IN YEARS Figure 6.10 Rate of Change of Consumption per Labor 10.00 117 «'1 0,01 0,00 0,02 8 ’ 2100 0100 50° '1'” 511.00 mo ‘10:: 111 1511110 Figure 6.11 Income Distribution Ratio 0100 1.00 0.00 0100 1100 118 0 ~27 043 A4 930 1001! 1 > 0—0 H 0 088 9 .24 J D 35.55 1:55 2100 3:55 0100 0100 0100 11170 0100 0100 15.55 TIME IN YERRS Figure 6.12 Domestic Saving Rate 0,10 119 III (D (b 4 51.00 Figure 6. IV V - VI Rho Thu: 01 TIME IN YEARO 13 Foreign Saving 0.00 0.00 111.00 120 equitable distribution of income. Other policies exhibit results be— tween the affluence and austerity. Although it has not been included in the objective function (6.17) explicitly, the distribution of income is an importan objective of a society, and thus will be investigated under the different policies which will gear to the equitable distribution. There are two channels of income transfer from urban to rural in general: first, increase in quantity and/or value of agricultural pro- duction, secondly, farm income from nonagricultural sources which are the earnings of farm labor participating in the production of nonagri- culture. More investment to agriculture for land developement, irriga- tion, researches for new varieties, etc., will increase the physical production of agriculture, while price increase (or control) of agri- cultural product will increase the values of the agricultural produc- tion relative to nonagricultural production. The second channel of income transfer, which involves the complex interactions of socio-econ- omic policies, is possible as a result of the rural industrialization. Assuming the policy VI in the preceding discussion for the total desired saving and the marginal propensity to import reflecting the foreign indebtedness, seven policies will be designed in addition to look into the effects of income transfer. Policy l : normal trend of grain prices and other variables Policy 2 : linearly increasing agricultural investment from 0.1 in 1975 to 0.15 in 1984 Policy 3 : lower grain prices, i.e., 10 percent lower than the normal increasing rate of grain prices Policy 4 : higher grain prices, i.e., 10 percent higher than the normal increasing rate of grain prices 121 Policy 5 : encouragement of rural industrialization by making the feedback gain for farm labor participation into nonagri- cultural production higher, thus making it respond more sensitively to inequitable income distribution Policy 6 : combined policy of 3 and 5 Policy 7 : combined policy of 2, 3, and 5. The last two policies are added to investigate the possibility of main- taining the income distribution ratio by agricultural investment and rural industrialization. The paths of the key economic variables are plotted in Figures from 6.14 to 6.16. Effects of the policies on GNP are essentially the same except for possible inflationary and deflation- ary effects. Grain price control is the most significant factor for the transfer of incomes: however, in the long run, other policies--agricultural in- vestment and rural industrialization--may be more effective, i.e., may provide the sound basis for the continuing equitable distribution of income. High grain price policy, policy 4, leads to the transfer of in- come from urban to farm, but it also leads to low domestic savings since more income has to be spent on food for the urban people which will yield higher foreign borrowings to meet the total desired savings. Consumption paths, like those of GNP, are the same except for cer- tain discrepancies due to inflationary or deflationary effects. For the policy of low grain prices (which is likely to be pursued in Korea), the greatest concern is how to provide farmers with certain channels which will force the income transfer to keep a level of normal income distri- bution ratio. Policy 6, which specifies low grain prices and rural ind- ustrialization, more than restores the level of income distribution to the normal trend by 1984. Policy 7, which includes more agricultural investment than policy 6, recovers faster than policy 6, and it reaches 122 the balanced state of income distribution. Without any additional ac- tions for the case of low grain price policy (policy 3), the distribu- tion of income is far less than the normal path. No one policy dominates the others at every point in time, i.e., none can be ranked in a strict preference over others for all the ele- ments of the performance criteria at every point in time. Some policy may be strictly superior to the others for a certain objective but may be inferior for the other objectives or a certain policy may dominate the others during some period of time but not during the rest of the planning period. There is no gain without cost. One has to decide what to sacrifice in order to obtain the others, in other words, one has to decide a pref- erence set (or weights of the objectives) from the possible sets of preferences of time and objectives. Trade-offs are essential in this sense to determine a policy which will actually be selected and imple- mented: the decision making processes will enter the scene for this purpose. 123 0012 J 8 91.00 1100 2100 3100 0100 10: 0100 11700 0100 0100 10 .00 Figure 6.14 Real GNP Growth Rate gas 0.18 .4 l 0.10 0.03 0.00 0,00 001 124 ‘94» fl, 1000 £00 31-” 4'0 Figure 6.15 00 8-00 0000 TIME IN YERRS I 7 r Debt Service Ratio 7 00m 04» 10.00 125 55.50 r 0.00 sin TIME IN YEARS ‘7 3km 11 86:1 8.2.. 86m. m::__ee :e___:: 4mm... >88 Foreign Borrowing Figure 6.16 126 1,2,5 3,6,7 / T 0.00 IN YERRS 01710 TIME £00 £30 8.0mm 86mm 8610.» 8.2.1» 8.3.u 86mu| 86m. .o.. 86m. 86m. > 8&- 84m 50.00 Total Debt Figure 6.17 127 0,00 0,10 0,10 0,00 Y T T .00 1100 2.00 3.00 4.00 £00 41.00 1.00 0100 0.00 10 .00 Figure 6.18 Rate of Change of Consumption per Labor 128 92‘ 0,01 0 .7. on." 0‘... 0‘.“ 0 .18 T .00 1100 5100 330° " J .10 m rm 8100 TIME IN TERMS Figure 6.19 Income Distribution Ratio 0%» 1b 00° 129 5! .1 31 3.. 8 ..1 3., / 2 L 2 2:12. / I" // 4 > 1!. // 3. ‘9‘: cl 3. a] 3;. O 3. $.00 :15 2.00 3100 .00 0100 0100 1100 0100 0100 111.00 TIME IN YEARS Figure 6.20 Participation Ratio CHAPTER VII FURTHER ANALYSIS OF THE OPEN ECONOMY UNDER THE EXISTENCE 0F EXTERNAL FOOD SHOCK VII.l. Introduction For most of the decades before 1972, world grain markets were soft with real prices slowly declining, (Figure 7.1) and the grain stocks in a certain part of the world were accumulated to be burdensome. The trend was reversed in 1972 when severe weather struck some parts of the world causing bad harvest and massive grain purchases by the Soviet Union (which had been a major exporter before) creating highly unstable and in- secure world grain situations. The buffer of the remaining North American reserve was inadequate to absorb all the pressures, and the world grain reserve stocks declined far below the minimum contingency levels, only 31 days amount remaining by 1975-1976. Thus in fact the entire world was "living hand to mouth." [88] Food aid was reduced and some exporting countries placed restrictions on commercial grain sales abroad. The waves of the shock were immense--wide spread malnutrition, famine, and starvation-—and consequently, it awoke people from the myth that "the world is capable of feeding itself forever." For the countries who depend on the outside for much of their food needs, the effects of the world food shortage can be very costly; it directly affects foreign exchange and affects the development programs by forcing reductions in the volume of nonfood imports. Economic stability is affected by way of heightened inflationary pressures and the levels of food consumption through cutbacks in cereal imports, which in turn may 130 131 "'3 Canada (Fort William) 2.1 m 3 fl ”9%. Australia (Sydney) U.S. Gulf Ports ,/ . 8 I .1 3- \ 8. 01 11 3. $100 2100 0100 0100 0100 111.00 11.00 11.00 17.00 15.00 20.00 TIME IN YERRS Figure 7.1 World Price of Wheat E 8 t ”1 ..Q \ “9!! 31 3. p >- 3. .1 s. 001 4 + A A A 8 $.00 1100 0100 0100 {00 15.00 11.00 11.00 10.00 15.00 20.00 TIME IN YEARS Figure 7.2 Petroleum Price: Saudi Arabia (Ras Tanura) . _ 132 TABLE 7.1 DEPENDENCY ON THE GRAIN IMPORT+ Year Rice Barley Wheat Total Grain1 1965 0.0 0.041 0.622 0.088 1966 0.009 0.0 0.596 0.079 1967 0.029 0.0 0.819 0.143 1968 0.057 0.05 0.695 0.182 1969 0.191 0.031 0.845 0.284 1970 0.123 0.0 0.804 0.229 1971 0.19 0.0 0.784 0.28 1972 0.134 0.122 0.877 0.317 1973 0.102 0.169 0.854 0.308 1974 0.044 0.143 1.024 0.253 1975 0.089 0.157 0.931 0.266 +Source: MAF, Food Bureau Grain Balance Sheets. The dependency is in terms of the ratio of import to the total utilization which includes seed, feed, industrial consumption, loss, and human consumption. Dependency = l — self sufficiency. 1Total grain is the sum of rice, barley, and wheat in metric ton. 133 result in a spreading and worsening of malnutrition, famine, and starva- tion. For some of the LDC's who can afford the foreign currency needed for the import of foods and have been pursuing sustained economic growth, like Korea, the main concern will be the effects of the external obstacle to the continued growth and effects to other internal economic variables. During the course of economic development in Korea, the structure of the economy has changed drastically from a traditional agrarian economy to a semi-industrialized economy, and thus the nation became highly dependent on the imports for its grain needs. Table 7.1 shows the trends of import dependency of grains during the past decade. While the imports of rice and barley fluctuated but stabilized at low levels, the import of wheat increased almost to the total consumption level and thus contributed to make the total dependency from about 9 percent in 1965 to 27 percent in 1975. The main purpose of this chapter is to analyze the effects of the uncertain world grain market and corresponding grain prices on economic growth and various internal economic variables using the open growth model, and to design possible grain reserve rules to lessen the effects of the shock on domestic prices and the total economy. VII.2 Analysis of Shocks Although there are qualitative aspects such as psychological effects, the shock (of food or oil) can immediately be observed in the form of sharp price increases as long as the markets are still functioning. (Figures 7.1 - 7.2) Generally, there are three patterns Of price increases, namely, stable, unstable, and asymptotic stable shocks. "Stable Shock" implies 134 a price increase which will eventually settle down to a constant price apart from the original price, "unstable" shock means the price increase without bound, and "asymptotically stable” shock indicates a price increase which will return to the initial price eventually. (Figure 7.3) PPICE Unstable 5155—125 Asymptotically Stable Time Figure 7.3 Different Patterns of Shock 135 As was seen in Figures 7.1 and 7.2, the oil and food price increases were stable (almost unit step) and asymptotically stable shocks, respectively. There may be various ways of expressing the shocks; using differential equations as in the dynamic system theory, TABLE look up function, or a certain form of algebraic function which can approximate the shape of a shock. One possible use of an algebraic function for an asymptotic stable shock of grain price is squared sinc function. The sinc function, performs ideal low-pass filtering when it enters into convolution, i.e., removes all components above cutoff frequency and leaves all below unaltered (since the Fourier transform of the function can be expressed in the form of box-car, II(x)). It can be given as follows:1 Sin nx nx sinc x = with the properties that l sinc 0 sinc n 0, for all nonzero integer n fix sinc x dx 1. The square of sinc function is . 2 _ sin nx 2 51nc x — (-—;;f—) (7.2) which represents the power radiation pattern of a uniformly excited antenna, or the intensity of light in the Fraunhofer diffraction pattern of a slit. Also the properties are sincZO = 0 '[85], pp. 52-57. 136 sinczn = 0 for all nonzero integer n f: sinczx dx = 1. For the world grain price shock, the equation (7.2) has been modified as ' 2/T n -1 2 A[51"ng n(x£1) )1] (7'3) where A represents the magnitude of shock, and T is the period of the shock. It has been shown in Figure 7.15, for the world wheat price shock, with A and T equal 2 and 4 years, respectively. VII.3 Storage Rules The idea of a food reserve for the buffer between uncertain production and demand has been thousands of years old. Reserve stocks--working stocks and contingency reserve--may serve the purposes to: (1) reduce the danger of food shortages, (2) reduce price variations and protect producers and consumers from unstable markets,1 (3) stabilize farmers' income and the general economy, and (4) assist economic growth. A storage rule, which defines how reserve stocks will be achieved in order to achieve a specified objective, should be precise to Show how much will be added to or taken from reserve stocks in a given period. The more common storage rules used are to make the level of reserve stocks equal to or a function of: (1 a constant target quantity (2 production level 4 price with upper and lower price bounds ( (5 ) ) (3) price, loan rate, target stocks ) ) supply, i.e., beginning stocks plus production 1The benefits of price stability to consumers and/or producers have been a moot subject [W4], [S3]. - 137 All the above rules focus on the stabilization of domestic market with no attention to the foreign market. This is because protecting farmers from low prices and reducing government farm subsidy burden have been the main concern in a country like the United States. However, for a food deficit country like Korea, one of the main concerns is to protect the domestic market and economic growth from the fluctuations of the foreign market. Thus the storage rules should reflect the effects of foreign markets. This leads to the consideration of world price level as a major factor in determining the reserve stock level. Two possible rules were tried in the study: first is a storage rule which accumulates and releases the stock proportional to the amount of price increase above a given normal level, and the second is a storage rule which uses a constant level of price increase above the normal level as a signal to release the below the constant level as a signal to accumulate to specified lower and upper limits. To be more specific, the first storage rule can be obtained by changing the desired grain stock level according to the world grain price such as CSTKi(t) = WSTK*[1. - (WPRi(t) - 1.)/2.] + WSTKL (7.4) DGSTKi(t) = CSTKi(t)*DMGRNi(t) (7.5) WPRi(t) = WPGi(t)/WPGRi(t) (7.6) where CSTK : desired proportion of grain stock to the total demand WSTK : maximum proportion of grain stock to the total demand WPR : ratio of world price to (average) normal price WSTKL : minimum proportion of grain stock (contingency level) to the total demand DGSTK : desired grain stock level 138 DMGRN : demand of grains WPG : world grain prices WPGR : normal level of world grain prices The second storage rule is giVen by WSTK , for WPR.(t) < CWPR (7.7) CSTKi(t) ={ ' WSTKL , for WPRi(t).: CWPR where CWPR is a policy decision parameter to switch the desired stock level to the maximum or minimum limit, and the equations (7.5) and (7.6) are the same as for the first rule. The actual values used for the application to Korean economy will be given in Table 7.2, and the implica- tions of the storage rules will be discussed in the next section. VII.4 Application to the Open Model of Korea A scenario of food shock has been used to show the effects of the shock, which is almost identical to the one experienced during 1972 to 1976, with the peak around 1980. This is a hypothetical setting just to illustrate the mechanisms, however, some studies show that this can be realized regarding the global demand and supply trends. [1 ] Six cases of major economic variables have been generated and plotted for the normal case; no storage rule with food shock and different storage rules with the food shock. Table 7.2 summarizes the cases with different parameters. In carrying out the experiment, further assumptions are needed: first, there exist some limitations on the availability of grains in the world market (at whatever price!) which reflect the difficulties to meet the 139 TABLE 7.2 PARAMETERS FOR THE GRAIN STORAGE RULES l 2 3 4 5 6 Food Shock No Yes Yes Yes Yes Yes WSTK 0.0 0.0 1.0 2.0 1.0 2.0 WSTKL 0.15 0.15 0.05 0.05 0.05 0.05 CWPR - - - - 1.1 1.1 WPGR - 120 120 120 120 120 import requirement as long as the world grain price remains high, e.g., government restrictions of grain exporting countries on the commercial grain sales abroad. Secondly, since the importance of wheat imports relative to the imports of rice and barley is dominantly increasing in Korea (Table 7.1), it will be assumed that the only restriction on grain import is on the import of wheat. In other words, only wheat is imported and only its supply is directly influenced by the fluctuations of the world grain market. Using the previous parameters and assumptions, the paths of GNP, GNP growth rate, debt service ratio, foreign borrowings, domestic grain prices, stock levels of wheat, wheat import, dependency on the grain import, and world price of wheat are obtained as shown in Figures 7.4-7.15. The real GNP shows no substantial change except slight zigzags in the growth rate with respect to the world price shock. This is partly because of the faster growth the total economy relative to the portion of grain production to the total economy, and partly because of the assumed 140 relatively weak effects (linkages) between the saving rate and income transfer which create insignificant changes in the accumulation of capital. Froeign indebtedness, expressed in terms of debt service ratio and foreign borrowings, doesn't show significant change between the cases with no additional reserve and with one or two years' reserve. The results therefore imply that no substantial increase in the burden of indebtedness will occur as a result of increasing the reserve level of grain stock. More clear effects of the shock appear on the domestic grain prices. Due to the nonavailability of wheat in the world market, up to 10 percent shortages Of the total wheat needs, the domestic rice, barley, and wheat prices rose substantially to about 1.6 to 1.8 times of the trend levels which are quite high but may be realistic considering the tripling world price shock. All of the grain reserve rules show damping effects of the price increases, however, the higher prices of the rule 3 and 4 in the after-the-peak period shows the inappropriateness of shortage accumulation when the price is still high in the world market even though it is decreasing. Although rule 6 (two years' wheat consumption reserve with a constant policy decision parameter which will signal the release and accumulation of the stocks) appears to be superior to the others, the lowest price level of storage rule 4 at the peak shock period implies that further damping could be possible by the combination of the two, i.e., release the stocks according to the rule 4 when world price increases but don't start to accumulate the stock until the world price is lower than a certain level. Figure 7.11 shows the actual storage levels of wheat which follow the specified storage rules. According to the different desired level of wheat reserve for each storage rule, the import of wheat will change widely from zero to more than twice of the current import level, thus causing the grain import dependency from near zero to 50 percent of total grain needs l4] (utilization) since the most of the grain imports are wheat. In sum, the model does not show significant effects of the food shock to the economic growth of Korea even without any reserve policies, however, its main effects can be realized as higher domestic grain prices which will affect and spread throughout the total economy with changes in income distribution and higher inflation rates. Considering the relatively small increase in burden of foreign indebt- edness needed to operate a grain reserve and the substantial damping of domestic price increase possible, increasing grain reserve stocks when the world price is low is essential to the stabilization of the total economy. The ideal maximum desired level of grain (wheat) reserve seems to be higher than the two years' total consumption, however, the actual level should be determined by considering the total costs of operating the stocks, cost of building storage, losses, etc., in addition to the benefits. Obviously, the costs of removing the risks for each country will not be necessary if there exist world grain reserve stocks which are enough to dampen any sudden changes in the world supply of grains, thus stabilizing the world grain prices. 142 l,2,3,4,5,6 fimo r 9. imp imp imp TIHE IN YERRS r 4.00 inn V 2.00 :2 em. 8.8...) 8.8m 86% 8.1. :03 comppmn ucmymcou .8;Lr 9965p 21 1 .o u 219* > 86m. 86$ 86m- 8.ew.o 22....‘0 Real GNP Figure 7.4 yen 13:0 \ \\ \\ \\ 143 3 \ - \ T“ \ 6 ‘Vv‘\_\ \ 4' \ r 1.00 T 2.00 flan flan T! Figure 7.5 din 02» NE IN YERRS Real GNP Growth Rate l44 R a£1.00 {.00 2100 3.00 4'. 0 .00 s{.00 7200 3.00 0‘50 1)) .00 Figure 7.6 Debt Service Ratio 339.7 V -.0.00 90.00 490.00 90.00 50.00 00 050.00 790.00 73.00 145 00 7200 0100 0200 70.00 Figure 7.7 Foreign Borrowing 146 $0“ Ith r 9.” 0200 4200 0200 0200 TIN: IN YEARS iwo I 27m, : 8.... Saw 2...... Sum woven ommpv xmucw mowed Domestic Rice Price Figure 7.8 price index 7.00 J 7,00 4,00 Jo” 1200 Y W‘— 2.00 3.00 Figure 7.9 147 00 'flmo 7200 0200 02 TINE IN YEafls Domestic Barley Price T 0.00 1 0.00 717.00 J 1.80 price index 7g» you L qmo 148 ‘;4n flan Figure 7.10 T {um fin din 74m TINE IN YEHRS Domestic Wheat Price T 0.00 I ‘0“ ihmn 149 8.34 8.31. 86% Ram 8.0m» 8.8m mcou qume ccmm302u 8.3% .6—1 212ml > 86.... 8.37— 7.00 0.00 0200 Tom T 7200 TIHE IN YERRS 2200 0200 4200 72 find: Reserve Stock Level of Wheat Figure 7.11 150 117.00 3200 0200 0200 0200 “HE IN YEHRS V 2.00 {.00 ”...—.5”.v U _meE _ucmmaocu. Amount of Wheat Import Figure 7.12 0,20 151 cp.00 fim 2&0 im 32 7h0 fin 00 £50 £00 IIHE IN YEARS Figure 7.13 Dependency on the Grain Import Y 9.00 ’Tan A 0.0‘ _A 0 152 2,3,4,5,6 J06“ V Y .00 2.00 Figure 7.14 :2 00 0200 .0: 0200 72 00 0200 Availability of Wheat in World Market f, 8.00 717.00 153 2,3,4,5,6 ae.emm aa.emw ...amv no.0mn aa.am« aa.emw no.9mm no.9mu ou.amm ao.e . E; » 02m. 43 T [BE IN YERRS W— 6.00 T 3.00 T 22 2330 ‘txn World Price of Wheat Figure 7.15 CHAPTER VIII SUMMARY AND CONCLUSIONS V111.l Summary of the Results The overall process of economic growth as explained in the modern theory of economic growth can be viewed as a closed system of a single system state equation or a set of system state equations with capital per labor being the system state. Following Meade and Uzawa, the traditional way of treating the two sector growth model is to disaggregate an economy into consumption- and capital-goods sectors. From the technical point of view, it can be seen as a scheme to keep the model simple as possible, i.e., the whole model has only one system state equation with the single capital accumulation process of the capital-goods sector. Hence the basic framework of the one sector model can directly be used with only minor changes. The dual economic growth model dichotomized into agricultural and nonagricultural sectors comprises simultaneous differential equations with different saving rates, different depreciation rates of capital, and different production processes. The resulting system state equations are nonlinear depending on the form of production function used, and thus the analyti- cal solutions are impossible in general. To find the optimal capital accumulation programs or paths, social welfare function should be explicitly defined. While there is no way to remove the conceptual difficulties regarding social welfare completely, 154 155 three types of welfare functions (which satisfy desirable properties of a "well-behaved" utility function)--consumption-oriented, capital-oriented, composite--have been tried. The Hamiltonian obtained using a simple linear consumption,being a utility function as in the Meade and Uzawa's model, is linear with respect to the control variables (saving rates) and constitutes a "bang-bang" control problem, and it may contain a singular interval during which the optimal control is indeterminate. To avoid this problem, a utility function with constant exponent (weight) can be used, where the sufficient condition is that (the sum of) constant weight($) should be less than one with each weight being nonnegative. For a finite planning horizon as in practical policy making, capital should also be included in the welfare function. The sufficient condi- tions for the existence of optimal control (saving rate) for the dual economic growth with a composite welfare function are: the constant weights of consumption are nonnegative and the sum of the weights of consumption is less than one regardless of the forms of production func- tion used. (There is no condition on the weights of capital for the existence of optimal control except the conceptual constraints of nonnegativity). The optimal control problem for the optimal capital accumulation yields a nonlinear two-point boundary problem for which it is generally difficult to obtain even numerical solutions. A special property of the costate variables in the case of the economic growth model, namely, the linearity in the changes between the initial values and the terminal values of the costate variables, enables one to use an efficient itera- tive method based on the variation of extremals with a simple adjustment 156 scheme. The property comes from the physical meaning of the costate variable in this case, that is, the costate means the social demand price of a unit of investment in terms of a currently foregone unit of consump- tion (or opportunity cost over the remaining time horizon). Results of the application of the closed model to the Korean econ- omy for both analysis and control can be summarized as follows: First, the basic structure of economic growth model as given can be used to keep track the transitory paths of economic variables with further modi- fications for proper (projected) values of saving rates. Secondly, the special form of the dual economic growth model allows one to investigate the stability properties of the nonlinear system. In other words, the impulse response shows the stability with respect to the impulse input, and thus the overall stability of the system depends mainly on the form of production function. In this regard, the dimini- shing marginal productivity of the production function is the key to ensure the existence of the stability (or steady state) of the econ- omic growth model. Thirdly, the changes (switchings) of saving rate from the upper boundary to the lower boundary occur in most cases of the optimal accumulation of capital. The switchings imply the changes in the social decision on the relative importance of the consumption and investment, i.e., whether to consume more and save less or to save more and consume less at a specific time. Fourth, the heavier the weight of capital relative to consumption, the later the switching occurs. This reflects that the society will build up the productive capacity (capital) as long as possible until everyone agrees to consume more in return for their earlier labours. 157 Fifth, the relative weights of capital and consumption in the social welfare function for the case of optimal capital accumulation affect the control and state variables substantially, and thus the main decision for the policy maker(s) lies in the determining the relative weights of the social welfare function, i.e., the priorities of the social choices. The closed model can be converted into the open model by introducing two additional sectors, namely, trade and balance-of—payments. Export is given by exogenous projections using a generalized logistics curve, and import is divided into two parts--import proportional to export and compressible import determined by the marginal propensity to import which can be controlled by import policy. The balance-of-payments com- ponent includes total foreign debt and foreign borrowings along with the level of foreign currency reserve. A distributed delay process with accrual (or gain) has been used to model the debt accumulation process. These added complexities do not allow one to use the analytical tools as in the closed model, and thus one has to rely on more general methodologies to handle complex models such as general optimization and simulation. Obtaining optimal policies in a complex model is often impossible by the normal optimization techniques. A penalty function method which relaxes the constraints as "desired“ constraints can be used for this purpose. To lessen the conceptual difficulties in the objective func- tion, alternative objective functions in piecewise quadratic form can also be used which offer a more general framework for policy optimiza- tion. Orthogonal representation of policy variables using Fourier series or Legendre polynomials will further simplify the efforts to obtain the dynamic paths of policy. 158 The results of the application of the open model to the case of Korean economy can be summarized as follows: First, the paths of optimal controls can be generated using optimization routines with respect to alternative functions and constraints which represent the alter- native economic policies. Secondly, different paths of the two instrumental variables, saving rate and import (total desired saving rate and the marginal propensity of import of compressible goods), can result in substantially different paths of economic variables which can be grouped into three categories; abstinence policies, prodigality policies, and policies in between. Thirdly, examining the foreign indebtedness paths--paths of debt service ratio, foreign borrowing, total debt-~suggests that the ind- ebtedness can be handled and reduced to more a sound level while retaining higher economic growth at the same time by the abstinence pol- icy which decreases both the foreign saving rate and import of the comp- ressible goods. Fourth, it can be observed that grain price control is the most significant factor for the transfer of income from urban to farm in the model with the grain sector, however, more agricultural investment and rural industrialization may provide the sound basis for the equitable distribution of income without substantial changes in the growth rates and foreign indebtedness. Fifth, with the policy of low grain prices, rural industrialization expressed as a participation ratio should be increased in substantial level to maintain a good income distribution ratio. Increase in agri- Eg. ' 159 cultural investment will aid in maintaining the level of income dist- ribution without substantial increase in the foreign indebtedness. To show external effects on the open model, a scenario of food shock--sudden increase in the world grain prices--, similar to the one during 1972 to l976--has been given to the open model. The food shock, given as an asymptotically stable shock, showed negligible effect to the real GNP paths because of the decreasing imp- ortance of the grain production relative to the total GNP, however, it created high increases in the domestic grain prices and may have a critical effect on the inflation of the whole economy. As can be expected, a grain reserve policy linked to the world grain prices contributes to dampening the effect of the food shock on the dom- estic price increases without further increases in foreign indebtedness. The actual level of grain reserve should be determined by considering other factors also such as storage building costs, stock operating costs, losses of storage, etc. All the above results are based on the assumptions and formulations of the model used. It obviously needs further verification and model- ing efforts for application in practical policy making. 160 VIII.2 Further Research and Recommendations Being specific instead of broad and general always increases the risk of making untrue statements. So does showing the specific paths of economic variables. (And even the optimal paths of an economy!) 0n the other hand, being broad and general loses practicality which is important in the design of economic policies. Further efforts on the modifications, extensions, and refinements of the model should be interpreted as an effort to bring the model away from the two extremes to preserve both rationality and practicality. The Production Function Aggregate production functions are not conceptually justifiable 1 Recent research and but have been used for practical convenience. disputes among economists on the concepts of capital, labor, and tech- nical progress reflect, in essence, the needs for a more refined and justifiable production function. The production potential can be determined by (l) shifts along a given production surface (or an isoquant for the case of constant returns to scale) and (2) shifts of the production surface. The former implies input substitutability and the latter implies technical progress. The original Solow's model assumed that input proportions can vary "2 at any time. It was later termed a "putty-putty model. If equipment 1R. Solow even said that, "I have never thought of the macroecon- omic production function as a rigorously justifiable concept. In my mind it is either an illuminating parable, or else a mere device for handling data, to be used as long as it gives good empirical results, and to be abandoned as soon as it doesn't, or as soon as something better comes along," [512]. 2"putty“ stands for capital in a malleable state which can be made into equipment requiring variable capital labor ratios; "clay" stands for capital in a "hardened" state with a constant capital labor ratio. 161 can have various input proportions (ex ante) and become immutable once the equipment has been set up (ex post), the production process is said to be putty-clay. Another possible case of input substitutability is clay-clay where one technique will always be chosen irrespective of fac- tor prices. Technical progress shifts isoquants inwardly, i.e., the same amount of output can be produced with less inputs. It can be exogenous or end- ogenous according to the way the technical progress enters into the pro- duction function, and can be embodied or disembodied according to whether the technical progress is due to any changes in the factor inputs or not. Neutrality and nonneutrality of technical progress was defined to deter- mine the contribution of specific factor of production to total tech- nical progress, i.e., to determine the direction of the shifts of iso- quants. Hicks neutrality assumes equal contributions of capital and labor to the total progress--output-augmenting. Harrod neutrality means tech- nical progress with only contribution from labor (labor-augmenting), and Solow neutrality is for the shift of isoquants along labor axis (capital- augmenting). These considerations on the refinements of aggregate production function either by redefinition of capital and/or labor or by intro- ducing "pure" technical progress together with further disaggregation of an economy (multi-sector model) will undoubtedly give more insights on the overall economic growth. Saving and Investment Another area of further research relates to the investment decisions. Saving behavior reflecting the distribution of income as given in the 162 open model were largely determined by preassigned values of expected sa- ving rates and parameters for the effects of income distribution. Saving rates out of wages and profits of nonagricultural saving were also given as predetermined. Efforts to refine the basic framework and to gather more information about these are needed for better representation of real behavior. Investment to agriculture and nonagriculture has been determined by an investment-allocation parameter. As mentioned earlier, the cla- ssical economic optimization deals mainly with determining the invest- ment-allocation parameter, i.e., determining "how to allocate the lim- ited resources to different activities in order to achieve a certain objective." In the growth model, it usually is given as a constant to determine the optimum saving rates. Including the allocation para- meter as another control variable (along with the saving rates) in the closed model will complicate the system state equations of (3.25) as follows: 163 where the last term includes multiplicative terms of control (input) variables, which prevents obtaining the optimal solution in general/ Further modifications of the model or theoretical developments may lead to obtaining the optimum saving rates and the optimum investment- allocation simultaneously. Modifications of Open Model 5 Most of the modifications which aided to convert the closed model . — into the open model can be the subjects of further refinements and veri- fications. Trade and foreign capital movements are highly uncertain and may be modified to include internal effects of other economic variables id once more convincing relationships are identified. Labor migration and the closed loop mechanism of income distribution are too simplified to describe complexities in the actual processes. Equitable distribution of income does not necessarily mean equitable distribution of utility; people move for many reasons other than money-~educational opportunities, pollution and ecological problems, conveniences in transportation and other facilities, tradition and cultural influences, etc.--which make a model far more complicated, and will need behavioral assumptions for further refinements. Tax policy have not been explored in detail and will deserve further work to explain the consequences of alternative tax policies and to design the feasible tax policies. Exogenously given price“mechanisms by time trend may also be refined to reflect inflation effects. Uncertainty Both the closed and the open models as given are deterministic; there are no uncertainties and disturbances (noise) introduced in the 164 models. Considering the highly uncertain economic activities and their interrelationships--such as savings behavior, foreign capital flows, labor migration, world financial market conditions, world grain prices, income distribution and its effect to other economic variables, etc.--, it will be a logical step for further study to include uncertainties in the form of specific probability distribution functions. Stochastic models, which introduce disturbances, can not be solved analytically in general unless drastic simplifications are made. Stocha- stic control may be applied to a highly simplified model for the solution of optimal growth. Generally, however, Monte Carlo techniques will pro- vide insights into the behavior of models with statistical disturbances. External Shocks The food shock given in the model is a hypothetical scenario. Addi- tional modificaitons will certainly be needed to investigate more rigor- ously the effects of food shocks on the domestic economy. The areas which need more work are: domestic grain demand and production func- tions, world grain production, consumption, and market conditions, foreign aid trends and prospects of a world grain reserve, and effects of grain prices on the inflation of the total economy. Investigation of the effects of other shocks, such as an energy crisis, may also be possible by adding relevant linkages using a framework sim- ilar to the one used in the case of food shock. 165 VIII.3 Conclusions The research attempted to describe the transitory (dynamic) behav- ior of economic growth and to design economic policies to fulfill the economic objectives during a certain time period using system theory. It has demonstrated the practical usefulness of the modern theory of economic growth by showing the trajectories of economic state and con- trol variables, and optimal trajectories for the case of the Korean economy. No theory or model is perfect. This is especially true for the social systems where no precise information is available unlike phy- sical or engineering systems where relatively precise information can be gathered by experimentation. Due to this imperfection, a model or models can only provide decision makers with a limited set of informa- tion that they must weigh against many other (quantifiable or unquanti- fiable) factors. The point will be illustrated further by the payoff matrix1 of Korean economy with respect to alternative objective func- tions (performance indices) given in Table 8.l. The payoffs indicate the values of the social welfare expressed as discounted sum of weighted capital and consumption over the planning horizon with alternative ob- jective functions and alternative policies (and models). Selecting the largest value (the highest social welfare) is meaningless, since the value may be realized if both the (uncertain) objective and the model are true. In other words, one still has to weigh the uncertain- ties and thus needs a certain bases-~decision criteron such as Laplace 1The basic idea of the objective payoff matrix is in [C3]. 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