'6 km .- .o;&;i . a ." "'- v' F 1-15 thesis ex; 2211:; for internal :23: control thea Tiree cases ar- 12 egg-on enema :79;H“ ! ‘ mun Balance ' :a: the 0:311:31 3‘ :1:;uely deter: ‘L ‘ ht: Hie QUECET ABSTRACT AN APPLICATION OF OPTIMAL CONTROL THEORY TO POLICIES FOR INTERNAL AND EXTERNAL BALANCE BY Nguyen Thi Bich Thuy This thesis examines the dynamic problem of quantitative economic policy for internal and external balance within the framework of optimal control theory. Three cases are considered: the one-country, the two—country with common external balance, and the two-country with linear dependent external balance. When applying Chow's control approach, it is found that the optimal solution for achieving the joint balance can always be uniquely determined whenever Tinbergen's principle on the equality between the number of independent targets is met; however, there is no guarantee that the solution is feasible for a given economy. In fact, when the optimal fiscal and monetary policies for achieving internal and external balance of the U.S. and Canada are amalyzed for the period of 1961-1970, it is found that they are incon- sistent and a trade-off between the attainment of joint balance and the feasibility or consistency of policy-instruments must occur when limits are imposed upon the instrument-magnitudes. .-V¢:!: . h I It is also four. szzire responses fr: :azezts ard growth 3:: lager appropr mice. 1: encourage fu 33th linear qu; lzisfcnd that the szategies are alh'ai ‘Glo 4e: I: the search reexamine stra: Nguyen Thi Bich Thuy It is also found that if more complicated assumptions, such as active responses from the second country, or conflicting balance of payments and growth targets are added, the optimal control framework isrualonger appropriate for the analysis of internal and external balance. To encourage future research, a two-player multistage non zero sum game with linear quadratic system.and perfect information is presented. It is found that the non-cooperative (or so-called Nash) equilibrium strategies are always inferior in the case of two controllers which leads to the search for a non-inferior solution, in particular the Pareto-Optimum strategy. gear-=- AS APPLICA'. in] AN APPLICATION OF OPTIMAL CONTROL THEORY TO POLICIES FOR INTERNAL AND EXTERNAL BALANCE BY Nguyen Thi Bich Thuy A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Economics 1974 I "Can c5 chi in can hqc tép, Mtg, Ira (Lang gang mic £0 106m, Cha con dd méz:z& fidu, Mqi dddng c5 Mg»chu 15am cho con." (T. T. Ba, 1966) Con th L51 Me Day Va XLn Sufit you; Ghi An. N. T. B. T. 11 PM? ‘ . l ' ' :...::=3r mtnony ‘r . F vi; p '1 4 _ lash to exprzh Ziegzun Lee. I w. szggesziag the dire. 5;»;2rt. Professor preparation of this :2 greatly arm :15 assistance of P that of the Inte .tapgropriate e 1‘ :. . anesse des Bernard Ja: H . -. c aide inesti :5? v- .‘ , 4 tEtHErcne. I .‘3. c“. a. ‘ to’e‘JOurs prc "“J ‘4 “b. Nu tIEIl-d me ..._u iSt FE ACKNOWLEDGMENTS I wish to express my deep gratitude and sincere appreciation to Professor Anthony Y. C. Koo, Professor Robert H. Rasche, and Professor Kwang Yun Lee. I would like to especially thank Professor Koo for suggesting the direction of the thesis and for his encouragement and support. Professor Rasche was of invaluable assistance throughout the preparation of this study. The time and attention he devoted to the work is greatly appreciated. In addition, I would like to acknowledge the assistance of Professor Lee. Special thanks are due to Dr. Thai Van Can of the International Monetary Fund for his suggestions in regard to an appropriate econometric model of this study. J'adresse des remerciements sinceres au professeur Alain Haurie et a M. Bernard Jacquet de l'Ecole des Hautes Etudes Commerciales pour l'aide inestimable qu'ils m'ont apportée en des étapes critiques de ma recherche. Je m'en voudrais de ne pas mentionner 1e soutien que m'ont toujours procurée Lise et Yves Lacroix. Je leur dis sincérement merci. SchlieBlich empfinde ich in nicht geringem MaBe Dankbarkeit und groBe Wertschatzung meinem Lehrer gegenflber, der ebenfalls mein bester Freund wahrend meiner positivsten sowie auch negativsten Erfahrungen geworden ist. Herr Doktor V. Leroy Name ist es, dem ich ftir seinen uberaus hilfreichen Einsatz fur mich, seine Unterstutzung und Emutigung danke, die mich dazu anspornten den wesetnlichen Wert und die Schbenheit iii ' "fig-I gal-4""N L '.‘:::scha:'tswis se I ”- air-fies udten zu e on site: geteinsam ve- st: eineo bleiben; der Wirtschaftswissenschaften und des Lebens wahrend meiner Auslandsstudien zu entdecken. Erinnerung an und Verpflichtung fur die schbnen gemeinsam verbrachten Tage, wo immer ich auch sein werde, haben einen bleibenden EinfluB auf mein ganzes Leben. iv 1.1mm " . on... . 2.3L : V. ‘i Filitiillh . . . . F SSKEY 0F LIT L1. 0n the I Intern. 12. On the to Mari ore-comer r 2.1. reSEnI 9‘ J '9 Optima. on Ins ’9 o W . U.S. 0: Balanc Optima REsults , COUCIus'lOn ' o ‘ ‘ on p0 and E of ti TABLE OF CONTENTS Chapter Page INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . l I SURVEY OF LITERATURE. . . . . . . . . . . . . . . . . . . . 3 1.1. On the Theory of Economic Policy for Internal-External Balance . . . . . . . . . . . . . . 3 1.2. On the Application of Optimal Control Theory to Macroeconomic Stabilization Policy . . . . . . . . 9 II ONE-COUNTRY MODEL 0 o o o o o o o o o o o o o o o o o o o o 15 2.1. Presentation of Votey's Model . . . . . . . . . . . . 15 2.2. Optimal Control Problem Without Constraints on Instrument—Variable Magnitudes . . . . . . . . . . 19 2.3. U.S. Optimal Policies for Internal and External Balance: Appraisal and Amendment of the Optimal Solution. . . . . . . . . . . . . . . . . . . 32 Results . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . 47 I I I Two-COWTRY MODEL 0 o o o o o o o o o o o o o o o o o o o o 5 0 CASE A: PASSIVE RESPONSES AND COMMON EXTERNAL BALANCE 3.1. Presentation of Votey's Model . . . . . . . . . . . . 50 3.2. Optimal Control Problem without Constraints on Policy Variable Magnitudes . . . . . . . . . . . . 54 3.3. U.S. and Canada Optimal Policies for Internal and External Balance: Appraisal and Amendment Of the Optimal SOlut ion 0 O I O O O O O O O O I O O C 65 in! ‘9' . w . . - zmr.' gf- . T THO-COWIRY 3' GSEB: PAS: ECEGAL EAL. Ll. Present cation: 2 5 ' .. ‘05. 09:12:53 A on Pol $.l U.S. a' and Ex of the HEPCOCXTRY I tn (3 in: CC}; 12. Deter: 3'3' 0n the TVs Pl L6. Deter: 6'1' SUDna: 6.2, Flirt}: Chapter Page IV Two- C OUNTRY MOD EL 0 O O O I O O O O O O O O O O O O O O O O 8 8 CASE B: PASSIVE RESPONSES AND LINEAR DEPENDENT EXTERNAL BALANCE 4.1. Presentation of Votey's Model with Modifi- cations on the Foreign Sector . . . . . . . . . . . . 89 4.2. Optimal Control Problem Without Constraints on Policy-Variable Magnitudes . . . . . . . . . . . . 93 4.3. U.S. and Canada Optimal Policies for Internal and External Balance: Appraisal and Amendment of the Optimal Solution . . . . . . . . . . . . . . . 111 V Two-COUNTRY MODEL 0 o o o o o o o o o o o o o o o o o o o o 149 CASE C: CONFLICT OF INTERESTS AND GAME THEORETICAL APPROACH 5.1. Formulation of the Problem. . . . . . . . . . . . . . 150 5.2. Determination of a Nash Equilibrium Set (61, fiz). . . 152 5.3. On the Inferiority of Nash Equilibrium for a Two Player Multistage Game. . . . . . . . . . . . . . 160 5.4. Determination of the Pareto-Optimal Set . . . . . . . 163 VI CONCLUSIONS AND RESEARCH RECOMMENDATIONS. . . . . . . . . . 170 6.1. Summary of Conclusions. . . . . . . . . . . . . . . . 170 6.2. Further Research Recommendations. . . . . . . . . . . 172 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 APPENDICES o o o o o o o o o o o o o o o o o o o o o o o o o o o o 186 vi grunt: “ o .1 Cases of int Soninal value 51' :crical V3 1961-1970 . . Jae-country 1 deviations o: :crinal Valm Stzizal v5 1., Mia: 1951 Table 1.1 2.1 2.2 LIST OF TABLES Cases of internal and external disequilibrium . Nominal values for state variables - U.S.: 1961-1970. Historical values for exogenous variables - U.S.: 1961-1970 0 o o o o o o o o o o o o o o o o o o o o One-country model: penalty weights attached to the deviations of state and control variables from their nominal values. . . . . . . . . . . . . . . . . . . Nominal values for the state variables — U.S. and canada: 1961-1970 0 o o o o o o o o o o o o o o o 0 Historical values for the exogenous variables - U.S. and canada: 1961‘1970 o o o o o o o o o o o o o o o o Two-country model (Case A): penalty weights attached to the deviations of state and control variables from their nominal values. . . . . . . . . . . . . . . . . Nominal values for the state variables - United States and canada: 1961-1970 0 o o o o o o o o o o o o o 0 Historical values for the exogenous variables — U.S., Canada and EEC: 1961-1970 . . . . . . . . . . . . . Two-country model (Case B): penalty weights attached to the deviations of U.S. and Canadian balance of payments from.the equilibrium . . . . . . . . . . . Two-country model (Case B): penalty weights attached to the deviations of control variables from their nominal values. . . . . . . . . . . . . . . . . . . vii Page 35 36 45 70 71 81 115 116 117 125 F 1.... . 1 "fl '“'4 '31 d ‘6’ V '- s. LS. Governm with actual . LS. interes and nominal r r '~- . optical actual and pa 1 F-S. balance the actual p. 3.3. interes 3 .1331 rate the Optimal :“O‘Coun t r). tures: the Paths . . cptb‘al Pat ‘VO‘COUUtn, capital St: I; | 9 F, A‘ 9CDuI-‘tr ‘3 u “‘1 Pa IkO‘CO untr ”final Pa 120%“ tu Es- 1’,‘ 1"0 l’ 01 Figure 2.1 2.2 LIST OF FIGURES U.S. Government expenditures: optimal paths compared with actual and nominal paths (one-country model) . . . . U.S. interest rate: optimal path compared with actual and nominal paths (one-country model) . . . . . . . . . . U.S. optimal GNP net of capital stock compared with the actual and potential GNP (one-country model). . . . . . . U.S. balance of payments: optimal paths compared with the actual path (one-country model) . . . . . . . . . . . U.S. interest rate: optimal path compared with the nominal rate (one—country model). . . . . . . . . . . . . Two—country model (Case A) - U.S. Government expenditures: the optimal path compared with actual and nominal paths . Two-country model (Case A) - Canadian government expendi- tures: the optimal path compared with actual and nominal paths 0 O O O O O O O O O O O O O O O O O O O O O O O O O Two-country model (Case A) - U.S. interest rate: the optimal path compared with actual and nominal paths . . . Two-country model (Case A) - United States: optimal GNP net of capital stock compared with the actual and poten- tial GNP. O O O O O O O O O O O O I O O O C O O O O O O O Two—country model (Case A) - Canada: optimal GNP net of capital stock compared with the actual and potential GNP. Two-country model (Case A) - U.S. balance of payments: optimal paths compared with the actual and nominal paths. Two-country model (Case A) - U.S. Government expenditures: optimal paths compared with nominal path. . . . . . . . . Two-country model (Case A) - Canadian government expendi- tures: optimal paths compared with nominal path. . . . . viii Page 38 39 40 41 42 73 73 74 75 76 77 83 83 3‘; Ive-country optimal path Ive-country between the cezt expendi and nominal Ten-country betseen the rent expendi and noninal Ive-country the two exte ntiml path l‘u'o-conntry between the C pajraents: no:inal path TVO‘COLmtrv Jalance of p- aC:Llal and n [a \ F :VO'COuntn. F\ O “'“F tra:‘ectc :11 '"‘ (trade-c :1 - x .— ,3 n O n 9 5‘ n H J \ If. -‘I’ ) Figure 3.9 4.1 4.2 4.3 4.:a 4.6 (A) 4.7(A) 4.8(A) 4-9(A) 4.10 (A) 4- 11 (A) Two-country model (Case A) - U.S. interest rate: optimal paths compared with nominal path. . . . . . . . . . Two-country model (Case B) - Effects of trade-off between the two external targets on the U.S. govern- ment expenditures: optimal paths compared with actual and nominal paths . . . . . . . . . . . . . . . . . . . . . Two-country model (Case B) — Effects of trade-off between the two external targets on the Canadian govern— ment expenditures: optimal paths compared with actual and nominal paths . . . . . . . . . . . . . . . . . . . . Two-country model (Case B) - Effects of trade-off between the two external targets on the U.S. interest rate: optimal paths compared with actual and nominal paths. . . Two-country model (Case B) - Effects of trade-off between the two external targets on the U.S. balance of payments: optimal paths compared with actual and nominal paths . . . . . . . . . . . . . . . . . . . . . Two-country model (Case B) - Effects of trade-off between the two external targets on the Canadian balance of payments: optimal paths compared with actual and nominal paths. . . . . . . . . . . . . . . . . . Two-country model (Case B) - United States: optimal GNP trajectories compared with the desired and actual GNP (trade-off experiment A where q33 = 10-4, q44 = 105). Two-country model (Case B) - Canada: optimal GNP tra- jectories compared with the desired and actual GNP (trade-off experiment A where q33 = 10-4, q44 = 105). . . . Two-country model (Case B) - U.S. balance of payments: optimal paths compared with the equilibrium (trade-off experiment A where q33 = 10‘4, q44 = 105) . . . . . . . . . Two-country model (Case B) - Canadian balance of payments: optimal paths compared with the equilibrium (trade-off experiment A.where q33 = 10‘4, q44 = 105) . . . . . . . . . Two-country model (Case B) - U.S. government expendi- tures: optimal paths compared with nominal path (trade- off experiment A where q33 = 10-4, q44 - 105) . . . . . . . Two-country model (Case B) - Canadian government expendi- tures: optimal paths compared with nominal path (trade off experiment A where q33 = 10‘4, q44 = 105) . . . . . . . ix Page 84 119 119 120 121 121 126 127 128 128 130 130 . . a a.” in” 1 -|‘ d $5.? ins-country . t J \_ ,r ‘1' D \ ,. C I \__. i:- 'V CAI .g5) :11 lanere q33 F ' ", Two-country ' / .EC‘CO ‘mt Ive-country i paths onpar trajectories ererinent C jectories coi experiment C Two-country :gtinal pat'n experizent C‘ Two-country 1 :ents: 09:1 31.: experimea 'Ij fro-country tures: ogti 9'; v 3“ eneflze ‘ k '. :‘D-C0‘mt ry tures: 0».- PM oi exPerize C “hare q. 33 “Mount“, trajeCtorie rs experiIreq 36c: ‘. li0\ kO‘ ,. D‘s. Jutr ""u 5,- Figure 4.12(A) 4.6(A) 4.7(C) 4.8(C) 4.9(C) 4.10(C) 4.11(C) 4.12(C) 4.6(B) 4-7(B) 4.8m) 4.9(B) 4-10 (B) Two-country model (Case B) - U.S. interest rate: optimal paths compared with nominal path (trade-off experiment A where q33 3 10‘4, q44 = 10-) Two-country model (Case B) - United States optimal GNP trajectories compared with the desired GNP (trade-off experiment C where q33 = 105, q44 = 10‘4) . . . . . . Two-country model (Case B) — Canada: optimal GNP tra- jectories compared with the desired GNP (trade—off experiment C where q33 = 105, q44 = 10‘4) . . . . Two-country model (Case B) - U.S. balance of payments: optimal paths compared with the equilibrium (trade-off experiment C where q33 = 105, q44 = 10-4) . . . . . . . Two-country model (Case B) - Canadian balance of pay- ments: optimal path compared with the equilibrium (trade- =10-) o o o o o o 0 off experiment C where q33 = 105, q44 Two-country model (Case B) - U.S. government expendi- tures: optimal paths compared with nominal path (trade- off experiment C where q33 = 105, q44 = 10‘4) . . . . . . . Two-country model (Case B) — Canadian government expendi— tures: optimal paths compared with nominal path (trade- off experiment C where q33 = 105, q44 = 10‘4) . . . . . . Two-country model (Case B) - U.S. interest rate: optimal paths compared with nominal path (trade-off experiment C where q33 = 10 , q44 = 10 ). . . . . . . . . . . . . . Two-country model (Case B) - United States: optimal GNP trajectories compared with the desired GNP (trade-off experiment B where q33 = 105, q44 = 105). . . . . . . Two-country model (Case B) - Canada: optimal GNP tra— jectories compared with the desired GNP (trade-off experiment B where q33 = 105, q44 = 105). . . . . . Two-country model (Case B) - U.S. balance of payments: optimal paths compared with the equilibrium (trade-off experiment B where q33 = 105, q44 = 105). . . . . . . . . Two-country model (Case B) - Canadian balance of payments: optimal paths compared with the equilibrium (trade-off experiment B where q33 = 105, q‘h4 = 105). . . . . . . . . Two-country model (Case B) - U.S. government expenditures: optimal paths compared with nominal path (trade-off experiment B where q33 = 105, q44 = 105). . . . . . . Page 131 135 135 136 136 138 138 139 140 141 142 143 145 Ive-country 7 ; Tao-country : mg‘a- ‘ expenditures path (trade— 1'33)! a o I 1 'UJ optiral gat‘r. ti; = 103). Figure 4.11(B) 4.12(B) Page Two-country model (Case B) - Canadian government expenditures: optimal paths compared with nominal path (trade—off experiment B where q3 = 105, q44 = 105). . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 Two-country model (Case B) - U.S. interest rate: optimal path (trade-off experiment C where q33 = (144'10) . 105, . . . . . . 146 xi . h. ‘I "I: ‘4 ‘I nl .“ Ono-R. ’\' new . hauibaa. “Hill 0? Egg .VF Age?“ 0‘\ J E'V‘Iu'xl L'ATROI. SOL 1):" 1 VD”; In“ 10": mafia) COX. I '" 1:32on r» at! I‘KOL PRL'E \‘Fv ”WI .: ' TA.‘O\' l lirreszlggg) Er)?" ““::O\S {U W‘w' '31. PFC} LIST OF APPENDICES Appendix Page A-l THE REDUCED FORM OF ONE-COUNTRY MODEL 0 o o o o o o o o o o 186 Ar2 THE DETERMINISTIC LINEAR QUADRATIC TRACKING PROBLEM: A SUMMARY OF CHOW'S AND PINDYCK'S APPROACHES. . . . . . . . 191 A-3 ON THE EQUIVALENCE OF PINDYCK'S AND CHOW'S OPTIMAL CONTROL SOLUTION WHEN THERE IS NO CONSTRAINT ON INSTRUMENT-VARIABLE MAGNITUDES. o o o o o o o o o o o o o o 198 Ar4 COMPUTATIONS OF THE OPTIMAL SOLUTION FOR THE ONE- COUNTRY CONTROL PROBLm O O O I O C O O O O C O O O O O O O 2 0 6 A25 EVALUATION OF THE WELFARE COST FOR THE ONE-COUNTRY CONTROL PROBLEM . . . . . . . . . . . . . . . . . . . . . . 212 A-6 THE REDUCED FORM OF THE TWO-COUNTRY MODEL (CASE A). . . . . 218 A‘7 COMPUTATION OF THE INVERSE OF A SYMMETRIC MATRIX (Ayres:1962) O O O C O O O O O O O O O O O I O O O O O O O O 231 A‘8 COMPUTATIONS OF OPTIMAL SOLUTION FOR THE TWO-COUNTRY CONTROL PROBLm (CASE A) O O O C O O O O O O O O O O O O O O 2 35 A~9 EVALUATION OF THE WELFARE COST FOR THE TWO-COUNTRY CONTROL PROBLm (CASE A) O O U C O C O C C O C I O O O O O O 2 46 A"lO THE REDUCED FORM OF THE TWO_COUNTRY MODEL (CASE B). . . . . 252 A‘11 COMPUTATIONS OF THE OPTIMAL SOLUTION FOR THE TWO-COUNTRY CONTROL PROBLEM (CASE B). o o o o o o o o o o o o o o o o o 266 A‘12 EVALUATION OF WELFARE COST FOR THE TWO-COUNTRY CONTROL PROBLEM (CASE B). o o o o o o o o o o o o o o o o o o o o o 280 ‘A‘13 SELECTED BIBLIOGRAPHY ON THE THEORY AND APPLICATION OF DIFFERENTIAL GAMES TO ECONOMICS . . . . . . . . . . . . . . 289 xii Toe problem of V . | merge: and ..eaoc his?" {Meade:l951 :trsiutec' the most 5272:, studies re amt with one e) Earallel to t‘. 25:16: of interna stern control .:. .C‘ interest in ":7“: in Spite 1223‘, L 3‘ tdeory to .ie: L... - ..jE INTRODUCTION The problem of internal—external balance was first raised by Tinbergen and Meade under the title of "Theory of Quantitative Economic Policy" (Meade:l951, Tinbergen:1952). Along this direction Mundell (1962) introduced the most controversial issue called "The Appropriate Use of Monetary and Fiscal Policies for Internal and External Stability." However, studies related to these problems are presented in a static context with one exception of Votey's work (1969). Parallel to the Meade-Tinbergen-Mundell formulation of the static Problem of internal-external balance, significant advances both by engineers in modern control theory and by econometricians in macroeconomic modeling led to interest in the problem of the dynamic economic policy-making. HOWever, in spite of numerous studies on the application of optimal cOntrol theory to economics, a survey of which is found in Kendrick's Unpublished paper (1972) as well as in Park's dissertation (1973), no studies have considered the problem of internal-external balance. The purpose of this study is twofold: first, to show that optimal ciontrol theory is also applicable to internal—external models; secondly, t0 introduce a new framework, called differential game theory, for SOlving the problem of internal—external balance under more sophisti- cated assumptions than those of Meade-Tinbergen—Mundell models, e.g., active responses from other countries, conflicting targets, and decen- tralized decision-making . 1 ictey's macro: 2:325 of optimal natal-external ‘: fzle‘iTla, 1970b. 3:75: (1972, 1971 straightforward ap; rial, and require: 7,2231 solution. Starter I will :5 acetone policy :5 :gtizal control Etcozrern of the 33:33:“. Chafi lli'nl . . ...; tEthnques 111:1 . '1 PLOblem Vi] 2 Votey's macrodynamic model will be borrowed to illustrate appli- cations of optimal control tools. To derive optimal policies for internal-external balance Chow's deterministic optimal control approach (Chow:1970a, 1970b, 1972a, 1972b, 1972c) is preferred to that of Pindyck (1972, 1973), because the former approach involves a more straightforward application, given the structural form of Votey's model, and requires only basic matrix manipulation to compute the optimal solution. Chapter I will present a survey of the literature on the theory of economic policy for internal-external balance and on the application of optimal control theory to problems of economic stabilization. The main concern of the chapter will be to show a gap between these two developments. Chapter II will deal with the application of optimal control techniques to the one-country model. The two-country optimal cOntrol problem will be treated in Chapter III for the case of common eXternal balance, and in Chapter IV for the case of linear dependent external balance. Chapter V will introduce a two-player deterministic multistage game with a linear and quadratic system for further research on its application to problems of internal—external balance, or more generally to problems of conflicts in economics. Chapter VI will Present the main conclusions of the study and suggest some 1‘ecounnendat ions . l ‘ LL Vi Ll. 3: the Theory Eternal-Ext. .___.__ Forty years a. -:.: a fatal blow :: f‘iL-ezplomen t raised through a: >22 hterrention. ~31" EIIEC’C s tend 1-5 equilibriu illiéd pIESSure 0 :zic internation Ins: ‘Qder fixed e Ike's 1. ~ “a. balance" .: : .\‘3;‘ a ‘ “Che gOvEmmE 3.1; .' M .are of itselj " Vie-rs are u . :‘a‘a 91:10th to ‘ <1 balame V. .«l‘L'ES of p01; 1 CHAPTER I SURVEY OF LITERATURE 1.1. On the Theory of Economic Policy for Internal-External Balance Forty years ago, John Maynard Keynes with his "General Theory" dealtra fatal DIOW'tO the classical notion that "internal balance", cu'full-employment equilibrium in the domestic economy, could be attained through an automatic adjustment mechanism with little govern- nmnt intervention. But the notion that the "automatic" price and income effects tend to restore "external balance" or balance-of- Payments equilibrium persisted until the middle 1950's. Under the cOmbined pressure of modern economic theory, and observation of the chronic international financial difficulties that plague the real wOrld under fixed exchange rates, economists have come to regard "external balance" as one of the specific economic objectives of deliberate governmental action, rather than as something that will take care of itself (von Neumann Whitman:1970). The pioneers in developing a formal body of analysis incorporating these views are Meade (1951) and Tinbergen (1952). They introduced a new approach to the problem of simultaneously achieving internal and external balance via quantitative economic policy, that is, of finding tile values of policy variables given some desired levels of real \‘7 2:2; and balance “Ezraiitional mul in addition t; :13, liabergen it more of a sol. ;:i::iple of equali :ezutoer of inde; fiai, all the poll; 1 targets achieve Defining exte: mtg that fiscas, 3:32.;ng the 38‘ 211‘: .' ‘ I ..ea noely at! title 1 “1. caSeS ‘ \ 4 income and balance of payments. This approach is just the inverse of the traditional multiplier analysis. In addition to the criteria on the effectiveness of policy instru- ments, Tinbergen in his work also formulated a rule on determining the existence of a solution to attain the joint balance, called the prhuflple of equality between the number of independent targets and thetnmmer of independent instruments. Once Tinbergen's rule is satis- fied, all the policy variables can be set at the necessary levels and all targets achieved simultaneously. Defining external balance in terms of the current account, and assuming that fiscal and monetary policies are equivalent methods of controlling the aggregate demand, Meade (1951) developed a theory of economic policy to cure internal and external disequilibrium. Two of Meade's four cases have potential conflicts (Table 1.1). This analysis has been widely accepted for a considerable period. Table 1.1. Cases of internal and external disequilibrium Disequilibrium Internal External Case Deflation Inflation Deficit Surplus Cures l x x Inflationist policy 2 x x Deflationist policy 3 x x Devaluation (conflict) 4 x x Appreciation of (conflict) exchange rate faint-l " 3‘! f"'.. twin"- I'm 4. L“ G'. ‘75. 7. In the 1960's :afzieve lnterna station. The ar. =21:ng rate lxe .7.: :rale and cap aside, proposed . tie: Classl-flcat -.‘c' be directed ~~~~~~ e ' a H 3.3.53» InpaLt . -o.. Ila-rial policy 1 “fiv'i- .-.. ...ese assun: E;:.:..' 1" ....on mnetar ::u.1 - ....ar policy ~= :;p:si:e would .32 min cri: :‘r;:‘~l968. Coo: {U u ‘ Wdell' SOlution The Sh . hr fundanes Tile Pro- instrU‘; 5 In the 1960's, the problem of using monetary and fiscal policy to achieve internal and external stability received considerable attention. The analysis assumes that a country chooses to keep its exchange rate fixed and chooses to avoid the use of direct controls over trade and capital movements. Mundell, in his classic 1962 article, proposed a new theory based on the Principle of Effective Market Classification (PEMC) which states that "each policy instrument should be directed towards that target on which it has relatively the greatest impact." Unlike the old theory of Meade, Mundell defined external balance in terms of the current account and short term capital flows. The latter assumed to be responsive to changes in interest rate differentials among countries. Mundell also divided financial policy into two separate policies: monetary and fiscal. Under these assumptions Mundell concluded that in a disequilibrium 8ituation "monetary policy ought to be aimed at external objectives and.fiscal policy at internal objectives", on the grounds that to do the opposite would worsen the disequilibrium situation. The main criticisms of Mundell's theory are (Yeager:l966, Votey:l969, Patrick:l968, Cooperzl969): (l) Mundell's proposal and proof are set forth as a short-run solution. (2) The analysis deals with one, very small country. (3) The suggested policy mix may thus temporarily palliate a fundamental external imbalance. (4) The proper assignment assumes full freedom in assigning instruments to variables. Therefore, so long as nations IL 'I- IBJ remain if. could not {5) line "corn; with the distance controlle- that con:- takers he However, linking j ficient ._ ’\ i V’ l'ne 31131 the spee: a prOPEr 51:63?“ to re C. 511::— s::} and Votev (I 43:; ..; . K. . ‘h «5“ I" A 50‘: C e! Istrk ' Pv-d ' i!‘ . .. A — — _ — 6 remain independent in their actions, some targets could not be reached. (5) The "comparative advantage" of instruments may vary with the environment, the level of targets, and the distance from the targets of the variables to be controlled. (6) The Mundellian model prescribes a set of policy responses that converges on equilibrium even when the policy makers have limited knowledge of the economic system. However, uncertainty about the analytical relationship linking instruments and targets can also lead to inef- ficient use of instruments. (7) The analysis is mainly static. No account is taken of the speed of adjustment of the system, the effect of a proper policy assignment on growth, or cyclicity. Attempts to remedy the shortcomings of Mundell's analysis under fixed exchange rate have been undertaken. Cooper (1969), Patrick (1968) and Votey (1969) extended Mundell's theory to a two-country Case with emphasis on the interdependence of the economies. Cooper's work (1969) is mainly concerned with the gains from Coordinating the instruments of economic policy both within and between nations. He found that as the economic interdependence i‘Ilcreases, the effectiveness of decentralized policy making a la MIundell will decrease, and the case for coordination of policy making, f0r‘directing all the policy instruments at all the targets, becomes “Hire compelling. The analytical framework used in his study is Mia: to that in 3:73:13 of ecc- ?*ttitk (196E :. ;:-:;:sa1for a twc 'iazze-of-pajmen: 5:: Country II vi :zgets. fie cont: izrlé dses not in: 21:25 introduce: I raise the possi Recent 1? , Pat z: a decentraliz ,.::a::en:l968, 1‘. .=.:~.:ents case 1 ‘- m’zeq. .‘ M L Wilcy in 7 suular to that introduced by Tinbergen (1952), involving targets and instruments of economic policy. Patrick (1968) studied the stability of Mundell's assignment tuoposal for a two-country model under various assumptions: common balance-of-payments, passive responses from Country II, active responses fumICountry II with international cooperation as well as conflicting targets. He concluded that the explicit inclusion of the rest of the world does not invalidate Mundell's theory; however, the other compli— cations introduced by the assumptions of active responses from Country II raise the possibility that Mundell's conclusions are inappropriate. Recently, Patrick (1973) reexamined the convergence of assignment for a decentralized system using McFadden's criteria for stability (McFadden:1968, 1969). He discovered that for the two targets and two instruments case the minimum information necessary to establish con- vergent policy in a centralized system is virtually identical to that necessary to establish a convergent decentralized system. Votey (1969) extended Mundell's analysis in two senses: first, Votey's model allows reactions from the rest of the world; and secondly, it is dynamic in that it adds a production function and allows for accumulation of the capital stock over time. Then, he studied the effects on stability, cyclic response, and growth from applying MMndellian policy, for the cases of a simple, open-economy, one-country model and Of a two-country model. The main conclusions of Votey's study are: (1) A prolonged solution based on Mundellian policy requires a higher degree of sensitivity of capital flows to interest rate differentials than Stein's (1973) results L .cqi .... ..r' “1’ 1’ ‘CF . fluid 1 within F) If (”er hiéher I Of all 1 leads tc Swath- m we“ establis a more f favorabl‘ (1) Eithin t? yclicit3 adoption 2: spite of ti" exit of Mundel1 szszizportant tc 1:115 external prc xiii? EUSI be Ina ' K s. . ke..¢t mg actic ., P \ bl ucscl e tool of of Mundell 8 would indicate if interest rates are to be kept within the range of past movements. (2) If there exists international cooperation, with higher rather than lower growth rates as the goal of all parties, the adoption of Mundellian policy leads to a choice of action which may stimulate growth. (3) International cooperation with respect to the establishment of interest rates can lead both to a more favorable balance of payments and more favorable rates of growth. (4) Within the confines of Votey's model it appears that cyclicity is not a problem associated with the adoption of Mundell's solution. In spite of the above results, Votey is not an advocate for the adoption of Mundellian policy. He recognized that monetary policy is a most important tool for achieving domestic goals and its abandonment to the external problem is too large a price to pay. Furthermore, the lags which must be accepted in the effectiveness of fiscal policy both in initiating action and in achieving results makes it unacceptable as the sole tool of domestic policy. To these two objections to the adoption of Mundellian policy, we would add a third one: It is not likely that the fluctuations in interest rates necessitated by foreign balance can be exactly offset in the domestic economy by government expenditure to the extent that the sectoral imbalances do not occur, at least without some very selective countermeasures. {‘7 "ofi ‘d‘lea’ ’ Another exter 1373). He incor; ecisians of suit: rifles a better tie SITJC‘ZUIE and :15 night influe: gzals of "full emy'. ta fixed-exchan; gelicies have dif: zeiiter countrieg 1': :1.” , v.5 vo“ the PrE\‘lk {a the AWL \k {0 Veeroecm \ in the 19601 4.43. This Was ......etrics and se aeen found ‘ 5:5: . . ‘5 8.0th n: mikes ' ) sagrt .._,_. “be;fi ‘E5. «3, d‘v’niwn: ' ‘ A as. ‘kfai. 3 LdelSs §;’r~‘,. "'3). 77:11 s Sectit 4011 V‘ 9 Another extension of Mundell is the recent work by Krainer (1973). He incorporated the production, investment, and financing decisions of multinational firms into a macroeconomic model. This puovides a better understanding of how factor endowments influence the structure and interrelationship of trade and investment and how due might influence the impact of monetary and fiscal policies on the goals of full employment and equilibrium in the balance of payments in a fixed-exchange-rate world. He concluded that monetary and fiscal policies have different effects in resource rich and resource poor creditor countries. However, his model is fundamentally static as are all the previous ones with the exception of Votey's. 1.2. On the Application of Optimal Control Theory to Macroeconomic Stabilization Policy In the 1960's optimal control theory found substantial applica- tions in economics because of increasing interest in dynamic decision- making. This was facilitated by the development of quantitative econometrics and of computers. Numerous applications of control theory have been found in macroeconomics as well as in microeconomic fields, such as growth models, planning models for sectoral allocation of resources, short-run economic stabilization models, consumer choice problems, dynamic models of investment and pricing by firms, portfolio analysis models, and pollution control problems (Kendrick:197l, Park:l973). This section surveys studies on optimal planning for economic Stabilization via optimal control techniques. In the earl exciting mat! in: this direc trier of staE :25 of the sys! retried that l elicies to mu11 siesired osci results are als. 9:357). H :iesezse that ‘ Elizsi'zie, they :at time atten ”‘37 °f Optima 10 In the early fifties Tustin (1953) noted that the problem of determining macroeconomic policies is a feedback control problem. Work along this direction was also done by Phillips (1954, 1957). He proposed a number of stabilization policies and considered the stability proper— ties of the system when these policies were implemented. In particular, he showed that the application of certain types of stabilization policies to multiplier-accelerator macroeconomic models could result in undesired oscillations and even instability in economic activity. These results are also found in other studies (Baumol:l961, Chow:l968, A11en:1967). However, Phillips' analysis was purely descriptive in the sense that while the alternative policies he considered were Plausible, they were not derived from any optimizing behavior. Since that time attention has shifted to more normative questions and to the Study of optimal stabilization policies. Van Eijk and Sandee (1959), Holt (1962, 1965), and Theil (1964) applied a more modern analysis to macroeconomic systems without correlating the analysis with the control system aspects of the problem. Their works have generally related to the derivation of linear decision rules of the type first derived by Simon (1956). These decision rules minimize the distance between actual and desired levels of the target variables, i.e., the social welfare function is quadratic in form. But neither Holt (1962) nor Theil (1964) used modern control theory to derive their decision rules. Recent advances in control theory have led to the development of new, more convenient techniques than the calculus of variations: Pontryagin's maximum principle (Pontryagin et al. :1962) and Bellman's :5; program | 1215 no: known t'rl :ia linear model tea: feedback 1;. :::':tain the sol; Eeterainisti; rials by Buchanar‘ Eyck (1972, 19.“ are function a Buchanan (195 ...’ f'. is: problem of 'uyu or mcrOeCC jaw: . who and fOIEigr Shit OptimiZatj ...t: and 1111165 tine If?" v- ' q ”M tnat Eros he n' in“: °f Stabi] ‘3 : ‘: ! area the Eton ‘313’35 . "3" (i973) , :3: 11 dynamic programming (Bellman:l957). As a result of these developments it is now known that with the assumption of quadratic utility functional and a linear model it is possible to obtain the optimal policy as a linear feedback law. This is a particularly convenient form in which to obtain the solution. Deterministic control theory has been applied to macroeconomic models by Buchanan (1968), Sakakibara (1969, 1970), Sengupta (1970), Pindyck (1972, 1973), and Turnovsky (1973). They all used a quadratic welfare function and linear deterministic model. Buchanan (1968) showed that modern control theory is applicable to the problem of economic policy determination for domestic stabili- zation of macroeconomic systems. Sakakibara (1969, 1970) integrated demand and foreign sectors to an ordinary growth model and applied dynamic optimization to evaluate actual economic policies (unemployment rates and investment - GNP ratio) for the United States and Japan. He found that growth policies have been too conservative while the movement of stabilization tools has been too erratic for 1952-1967. For Japan the growth policy in the 1960's was quite successful while that of the 1950's was too conservative. Both Sengupta (1970) and Turnovsky (1973) applied optimal control techniques to the stabiliza- tion of the deterministic Phillips' multiplier-accelerator model. Pindyck (1972, 1973) applied the deterministic control theory to study the optimal time path for the policy variables, using a linear econo— metric model of the U.S. economy. His analysis provides empirical 'measures of the trade-off between unemployment and inflation. However, the 3:231 control p 5:1:iou is quadr‘ Lastly, some at our knowledg' :ecooooetric no; u the structural azedeith it. To test: has been it frathe Optimal , strziiestic contro. “60313181 dew ie of a constan 12 However, the above studies are restrictive in three senses: the optimal control problem has been completely deterministic, the cost function is quadratic, and the econometric models used are linear. Recently, some attempts have been made to remedy these shortcomings. Our knowledge of the economy is incomplete; the coefficients of an econometric model are themselves random variables, and each equation in the structural form of the model has an implicit error term associ- ated with it. To cope with uncertainty, stochastic optimal control theory has been introduced. Chow has shown that there are two gains from the optimal stochastic control policy: the gain of the optimal stochastic control over the optimal deterministic control and the gain of the optimal deterministic control over the deterministic control rule of a constant growth rate for each policy variable (Chow:l972b). At the extreme, if the error terms are additive, uncorrelated normal random variables, the cost functional is quadratic and the system is linear, the principle of "certainty equivalence" (known as the "Sepa- ration Theorem" in the control literature) allows the stochastic control problem to be reduced to one that is essentially deterministic (Theil:1957, WOnham:1968, 1969, Sorensonzl968). The optimal control becomes a function of the expected value of the state vector, and if there is no measurement noise, the solution for the optimal control is the same as for the deterministic problem (Chow:l972b, 1972c). Paryani (1972) has applied the tools of stochastic control to derive an optimal control policy for the U.S. national economy. He found that the optimal control variables differ from the actual values of these variables during the period 1954-1963, suggesting the use of are flexible weal bala Harv. Instead . :eziersoo (l9 0 O . ‘ 951-331 CG. ( "I :¢~—‘:- ‘ “16;; d \ 112:. 31:93:33 7&4 "n :5». u .;7'-. ‘ “‘= a“: ‘1‘- In -- h_ L .- l- ‘ ~~1: 1:.‘V “ ‘ E‘ r ‘\ 13 more flexible control policies by the decision—makers to reach the internal balance. Instead of additive stochastic disturbances, Turnovsky and Henderson (1972, 1973) introduce stochastic parameters in a somewhat specialized dynamic context. Benjamin Friedman (1972) extends Theil's stochastic optimal con— trol approach to economic policy to the case where the welfare function may not be quadratic but is approximated by several quadratic segments. It is an attempt to solve dynamic optimization problems with more general cost functions. Stein and Infante (1973) have sought optimal stabilization policies which drive the quadratic cost of deviation monotonically to zero instead of minimizing the cost. Finally, the most serious restriction is that of a linear model. Most econometric models are at least quasi-linear in structure, but sometimes the more interesting aspects of their dynamic behavior arise from non-linearities. Recently, several studies on the application of optimal control theory to non-linear macroeconomic models have been conducted (Livesey:1971, Holbrook:1972, Norman:1972, Shupp:1972, Haurie and van Petersen:1973). In spite of these efforts, further research needs to be done on the development of stochastic, non-linear and non-quadratic optimal control theory and its application to macroeconomic stabilization problems. This survey of the literature shows that: (1) The pro interns static ' (2) Optimal the pro is take? targets and of one cou: imrefore, t: h.- ‘.. I flu -. \ 351 Van Can J" and mO'coun. 23: internal and t 14 (1) The problem of quantitative economic policy for internal and external balance has been basically static with the one exception of Votey's study. (2) Optimal control theory has been mainly applied to the problem of domestic stabilization. No account is taken of the regulation of balance-of-payments targets, of the interdependence between countries and of the repercussion effects when more than one country is considered. Therefore, the next three chapters will attempt to bridge this gap (Thai Van Can:l972) by applying optimal control tools to Votey's one- and two-country macrodynamic models to derive optimal policies for internal and external balance. 1D. =e ”.5 ....is C In- a N '- . .n‘ ol prc en...’ ~94»; O 0 "V‘ a CHAPTER II ONE-COUNTRY MODEL This chapter deals with a two-target and two-instrument case. Both fiscal and monetary authorities of Country I act to attain full employment and balance-of-payments equilibrium while Country II is assumed to be passive with respect to its targets of internal and external balance. To find the optimal policies for internal-external balance in Country I. First, Votey's one-country econometric model will be presented (Votey:l969). Secondly, the one-country optimal control problem will be formulated in terms of a linear-quadratic (L.Q.) system to find the optimal control solution which minimizes the quadratic cost function subject to the constraints of a linear dynamic system. Pindyck's and Chow's approaches (Pindyck:1973, Chow:l972a) will be used to put the reduced form of Votey's econometric model into the "state-space" system. Next, Chow's formulation will be used to derive the optimal solution both analytically and numeri- cally with reference to the United States. Finally, the optimal solu- tion will be appraised and amended. 2.1. Presentation of Votey's Model The main assumptions of the one-country model are: 15 (l) The ex; at the tre ate: The fo: the pa: Countrj exterm or cou: The Sh( interes COUECTI he model h, 15.”; ... Contro l6 (1) The export demand is given. It is assumed to grow at the same rate as it has in the past and is treated as an exogenous variable. (2) The foreign interest rate is given. This assumes the passive cooperation of Country II such that Country I may adjust its own rate to achieve the external balance without any foreign interference or countermeasures. (3) The short-term capital flows are sensitive to the interest rate differentials between the two countries. The model has five equations and five unknowns which lead to a determinant system. The variables are classified in three classes: (1) Endogenous variables: Blt: Balance of payments Clt: Consumption expenditures If£z Gross investment Ilt: Net investment Klt' Capital stock Mlt: Imports 0 ° Net capital outflow Y ' National income (2) Controlled exogenous variables: Glt: Government expenditures rlt: Rate of interest "H O ‘4. "I c ‘1: ~ 2 (3) Non-cor .ae equations are Y = ' 1t Clt+‘ 3 = 1t Xlt ‘ 33 K = n it I1t ‘ I I g n Y =: . 1t “10*: c 1t 110'” 12' 'J M :2 1t “10% . +5 . 12 . 3. 5 n ' I :3 l7 (3) Non-controlled exogenous variables: th: Labor force PM : Import price 1t Px : Export price 1t th: Foreign interest rate Tlt: Tax receipts X : Exports 1t P TTlt =- (l) : Terms of Trade P X It The equations are as follows: (1-1) (1-2) (1-3) (1-4) (E-l) (E-Z) (E-3) (Is-4) G Ylt= Clt+llt+Glt+Xlt-M1t National income identity B1t=X1t - M1t - 011: Balance of payments identity n Klt—Ilt + Klt-l Capital stock identity IG = In + 6 K Gross investment identit 1t 1t * 1t Y Y1t 3610+611K1t+612L1t Production function Clt=a10+all(Ylt-Tlt-5*K1t) Consumption function W810H11(Y1t lt 6*Klt) + 812 TT1t Import function u Ilt3Y10+YllYlt-l-Y12[q(6*+T)]lt-l +713K1t-1 Investment function ~t-i I :':5 first 1 :2 -533' :6 SEE: t 311' o... Y‘ : - h: LIL-ta; ‘K- ... ‘s ‘lEd “\: “A “‘11 b 2: ‘_. “‘d; gm L) é ,'._ b‘lu \ VI- «q? x 3‘; \‘S. 5 g . 19-. \.1 h. *L \ 18 (E-S) 01t=n10+n11(r2t-r1t) Net outflows of short term capital All the equations are linear functions, and none are greater than the first order of difference. Equation (E—l) - The output is a linear function of both capital and labor. It embodies the assumption of perfect substitutability between the two factors of production. Equation (E—Z) - Consumption expenditures are made a function of disposable income, using Klein's approach in the econometric model of the United Kingdom (Klein:l96l). Disposable income is GNP, less capital depreciation (6*Klt where 6* is the rate of replacement) and also less taxes (Tlt) which are determined by a linear function of the form: Tlt '3 110 + A 11 ch' Equation (E—3) - Imports are simply a function of disposable income and the ratio of foreign to domestic prices. Equation (E-4) - Net investment expenditures, that is, net addi- tion to the capital stock, are assumed to depend on money output, the c«'='»131ta1 stock, and the user cost of capital which prevail at the time the investment decision is made. In Votey's formulation, the user cost of capital has two principal components: the opportunity cost of funds tied in the capital, plus the cost of the actual capital consumed. This can be written in the form: q(6* + r), where q is the price of capita1 goods, 6* is the rate of replacement of capital stock, and r is the rate of interest, which represents the opportunity cost of funds. To simplify the model, it is assumed that relative prices Within the country do not change, in which case q = 1 over time. Equation :17: to inter: Stein's resul1 :agital moves: 52:5 as a dire results will t 21:1.‘n.i‘;1 .m-__ OH 1115:: R It is ass Elgezfiitures C 43% that j \- . '9 4" “MCE, that ‘ '7-L: _ . r4 7.. \s‘ “L‘Ketar‘, 19 Equation (E—S) - The capital transfers are assumed to be sensi- tive to interest rate differentials. Two results are available: Stein's results (or minimum results) dealing only with short term capital movements, and maximum results dealing with all capital trans- fers as a direct function of the existing differentials. Stein's results will be chosen for the analysis of short-run stabilization. 2.2. Qgtimal Control Problem Without Constraints on Instrument-Variable Magnitudes It is assumed that the Country I has two instruments: government expenditures G1': and the short-run interest rate rlt' It is also assumed that its goal is to attain simultaneously internal and external balance, that is, a situation of full employment without inflation combined with balance-of-payments equilibrium. The exchange rate throughout the study is assumed to be fixed; therefore, the decision- makers of Country I try to steer their GNP and balance of payments close to the targets by choosing the appropriate combination of monetary and budgetary measures (rit, Git). The external balance is represented by the balance-of-payments equilibrium, that 18,51t = 0 while the internal balance is determined by the production function: Y1t = 5 which gives the potential output. It is noted that the capital stock 10 + 611 K1: + 512 L1: (2'1) Klt Which is an endogenous variable in the reduced form of Votey's econometric model also figures in the production function for the determination of the potential output. To overcome this problem of "double entry" of Klt’ the following transformation in the variables °f ec{nation 2.1 is needed: g I. ‘ ‘Ifl V .1 Y1: " 611 K1:: = 510 + 612 L1: Let {I -- Y - 6 1t 1t 11 Km: be the output net of capital accumulation. Y1: = 510 + 512 L11: which determines the internal balance. 20 (2.2) (2.3) (2.4) To apply the optimal control theory, the one-country econometric model has been put into the reduced form (see Appendix A-l): = X + Yt yt_1 + B ut + C ut_1 D zt where g1. means "defined as" and d D [er’ Blt’ Klt yt no. t [Cu ' rlt] ' t [1’ TTlt’ T1t’ Xlt’ rzt] N I A11 0 A13 Q11 0 T A3 A21 0 A23 ‘ 3‘ Q21 ”11 ‘ C3 311 0 0.34 L0 0 0 A25 0 Y12 — p 0 A157 (2.5) F D11 D12 D13 Q11 0 D21 D22 D23 1’Qzl n11 c.0000 36 h. Given the structural form (2-5) the one-country optimal control Problem can be formulated in either Pindyck's or Chow's terminology. Both lead to the same result. However, Chow's method will be applied Then equation 2.2 becomes: 5?: {33:81.1 ‘1 -<:5 the L o" . u e ..- Vi. . , ¢ fl ~f1'”“h’ ~H' - JU: .... C .L l t J. an . U 2 n C D: t :5 ... I c x u “a. FL u l p. 5'. 5.5 o .t‘ .13 o. d: E a) u 5: §‘- . .4 H s. z “uh val. = «u c u.“ ...N “I sixul-hb ,W-_ opeulnnlo cut .5 ‘ . EEK Gus Var; \. ‘ "3 ‘3. b. 21 to derive the optimal solution first analytically then numerically, using the U.S. data for the period 1961-1970. 2.2.1. Formulation of the Problem Given the reduced form of Votey's one-country model to use Pindyck's approach the following transformation to equation 2.5. must be made : yt ._ B Lit—D zt = A(yt_1—B ut-l- D zt_1) + (AB + C)ut-l (2.6) + A D zt_1 to get Pindyck's "state-space" system (see Appendix A-2, equation A- 2.1a). In other words, the "state-space" system for the one-country optimal control problem is given by: P P P P P P = A - . xt xt_1 + B ut—l + C zt_1 (2 7) YYlt’ BBlt’ KK1t is the (3x1) state vector; P d d ' u = = -l O t ut_1 [Glt-l’ r1t_1] is the (2x1) control vector, P g d v 2t-1 zt—l' [l’TTlt-l’ Tlt—l’ Tlt-l’ xlt-l’ r212-1] is the (5‘1) exogenous variable vector; fl!‘ U.“ tame q uhfl na" 4...“: u v - b ..- ‘ "Q .Ls A11 0 “‘13.l A11Q11 A15 7 Pd-d —d—— —g . A. = A A21 0 A23 , B AB +-c - A21Q11 A25 , LYII 0 “34 Y11Q11 ‘le A _ d F A — A11D11+A13 a36 A11D12 A111313 11Q11 0 _.d __ ._ _ ,. , C ‘ AD = A21D11+A23 a36 A21D12 A21D13 A21Q11 0 LY11D11+334 a36 Y11D12 Y111313 Y11Q11 0 Id P are known matrices. Since R =0 under the assumption of no constraints on instrument-variable magnitudes, the cost functional to be minimized is: P N J=l P__.P,PP__P 2 2 [(xt-l xt-l) Q (xt-l xt-li] (2'8) t=1 where ' _P d ~ xt_1= yt_1-But_1-th_1 = [%Ylt-l’ BBlt-l’ KKlt-l] is the nominal or ideal state variable vector; q11 0 0 P Q = 0 q22 0 with q11 and q22 are weights attached to LC 0 O l the respective quadratic deviations d ~ " 2 -—- 2 (YYlt-l YYlt-l) and (BBlt-l'BBlt-l) ' Unlike Pindyck's approach, Chow's state-space system (see equation Ar2.17) does not require any transformation to the reduced form (2.5). In fact, in Chow's terminology the dynamic system of the one-country Optimal control problem is given by: : CC “ WAX +13 X: H V [w P I a: current ens: "Q- . ext “:6! ' t Lie {311) vectc fl! n I ‘A 23 C=AC x t c cc cc + . xt-l + B ut C 2t (2 9) 1 C d u d r ' where xt [yt tut] [Ylt’ Blt’ Klt Glt’ rlt] is the (5x1) vector I of current endogenous and controlled variables; 11: g-ut= [Glt’ r1t is the (2x1) vector of controlled variables, zt- zt- [1, TTlt’ Tlt’ xlt’ is the (5x1) vector of exogenous variables; - . P "" I— D A 1 c1 All 0 A13 0 A15 13? 1 Q11 c: 1 g o c: = A .--1--- A21 0 A23 0 A25 ’3 ---_ 'Q21 ‘ . Y11 0 8‘34 0 412 0 I o : o o o o o o I 1 I . o o o o o o L ’ _ L a. - _ L 1 I’11 D12 D13 Q11 0 D D21 D22 D23 1'Q21 ‘“11 C= “" 3 a36 0 0 0 0 are known matrices. o o o o o o L o o o o o Since there are no constraints on the policy-variable magnitudes, the weight matrix in the welfare cost function (see equation A—2.18 in Appendix A-2) has the form: 1'21:] ... . a i m .C s \\ L¥ ....s \A C “a; J... a: Q» ‘1 .. a.» .3 L1. : ‘ . x‘ x: QIL 1“ .«U. A.“ 9‘ ‘ ~ ..s FE; 5-. . e— g ; fl "1 = - ". L U “ D. - attrt . -“‘Q g .t. -c; ... .- ‘ 1 ...:b '5. . ‘\u 5 “ 24 I. ‘ I- — T qll o o o 0 PI Q . o o q22 o o o I QC: --..---‘=' o o o o o | o o o o o 1 o .0 o o o o o L | .. L .- instead of - | - -. ' T qll o o o 0 QP Q 0 O q22 0 0 0 QC: —--:---- 3 o o o o o o '11P o o o r o I 11 I L t . L0 o o o rzzd where q11 and q22 are weights attached to the respective quadratic deviations from the internal and external balance, i.e., weights 7' —- 2 lt-Ylt lt-Blt) . The weights r11 and r22 are attached to the respective quadratic deviations from the limits set attached to (T )2 and (B on fiscal and monetary variable magnitudes, i.e., weights attached to ..f )2 1t 1t ' Appendix A—3 shows the equivalence of these two approaches. How- —- 2 (Glt-Glt) and (r ever, this study will use Chow's approach to derive the optimal policy mix for internal and external balance for two reasons. First, Chow's state-space system formulation is more straightforward and simpler to apply, given the particular structural form of Votey's model, than Pindyck's formulation, which requires a transformation of variables ‘before applying his result (see equation A-2.16). Secondly, Chow's 'method just requires basic matrix manipulations even when constraints palicy-varl requires the C 4:;a1c'ix A-Z) Rhetder 3121?; 0911:. 25319.3. Boa :geiratic f0: hearse it a: :egative as 1 in .L a “3 real 5:: the adju flag this c :«5 pro-Me: ‘;"H' ' ‘ 29.4 «4 [I 5. Ch p0 :Egign 31" e v V‘. 's‘ 3“. 51.; ‘On f 4“-.. . hi4: ‘ «n th. I The Si ‘1;' . ‘5‘, ‘T‘i -:5 4'_ u .1 ‘ laalu :e 25 on policy-variable magnitudes are introduced, while Pindyck's method requires the determination of Riccati and tracking equations (see Appendix A-2) in the case of RP#O. Whether formulated in Pindyck's or Chow's frameworks, the one- country optimal control problem is typically a linear quadratic tracking problem. However, it is restrictive in two senses. First, the quadratic form of the welfare cost function is subject to criticism because it assumes that the deviations from either side of the targets-- negative as well as positive deviations--are of equal cost. However, in the real world there is no such symmetry in the sharing of the burden for the adjustment of the balance of payments equilibrium; for example, along this critic Friedman (1972, 1973) has made a contribution to this problem by introducing a "piece-wise quadratic criterion function" which divides the range of possible values for each endogenous variable and each policy variable into three regions: values within the middle region are assigned zero cost, but values within the two extreme regions are penalized quadratically but asymmetrically. Still, the criterion function remains quadratic, and further research needs to be done in this field. The second criticism is of the linear form of the econometric model, which is a very crude approximation of the real world. However, the disadvantage of a linear quadratic framework is compensated for by the computational advantages (Pindyck:1973). m' [in ..~ 1.: 3. :.:.2. Optimal and Ex: Cnov's 0;: After subs :26: Z 26 2.2.2. Optimal Solution for Internal and External Balance Chow's optimal solution is given by A-2.20 in Appendix A-2. After substitutions and computations (see Appendix A—4), it becomes: I. y c W . Eels u-Ev H— 1 u f: h ~ ‘ :I _ . .—-— ...”...Lm _.- 217 AaH.~V Aaua Ao~.~v yam ado i add «a e» A + on AH o oc' AOH 0° IHH luau“ 0' 00 «any ....I Has: H-umu - snows snows .-.“. alun 0N 3. an H H an ..t: H-u«u Anuwo Huumu ”name HIUHW 7% in as “a e .au as: a .... c H 7 Asia 33' A-$fiaae+ alas: as as use a «I manage: «Hanan: «so as. .Q an~¢+na '4 0—¥ OOOOOOOOOOOOOOOOOOOOOOOOOOOOO o ooooooooo coco-co... U 0H o H .‘H a: a -10 I I I ! L I L A I L 1960 61 62 63 64 65 66 67 68 69 70 Figure 2.4. U.S. balance of payments: optimal paths compared With the aCtua]. path (one-country model). boon... _: .IH n-flUIU-N - ~ A y. in u g . j .l l L hr . Q n u .x -\ a I N . w. dk~ .NI - INN KEEN»: «6 42 Optimal Path ------ Nominal Path 2.5-1 2.0-1 ‘__——_—“““-———- Run 3 Runs 5,7 ‘ Run 2 Run 5 1.5-1 .1 U a 1.0— 8 H m 0.. Run 1 0.5-‘ 1 0 _ l 1 1L_ 1961 62 63 64 65 66 67 68 K0 d RUHZO 00% Figure 2.5. U.S. interest rate: optimal path compared with the nominal rate (one-country model). 1 ‘4‘ ‘ ' ‘- Cfalthflc apchcy 1t 11 ‘I‘ ...e 9011' target var 9.5.} N. W. . figz" ,7 . . fl_‘ . a Y'~ “ '. \ in“ ‘2‘. -‘~ ‘1 7‘ ‘a “I ¥. .4“. w 43 "traditional" or multiplier approach (Tinbergenzl954). In other words, a policy instrument generally is more effective the smaller the change in the policy variable required to bring about a given change in the target variable when all other targets are held constant, e.g., the smaller'—-;L==-—, the more effective G* . Within the confines of 6G* la? 1': 1t 1t Votey's model, the matrix — ~ _ Row Y1t Blt 1 0.3810 0 2 0.0028 0.216 of equation 2.13 shows that the fiscal policy or government expendi- ture is the most effective instrument to regulate employment since it is governed only by the target §1t labor force. The optimal path for the fiscal instrument is derived for which, in turn, is determined by the the period 1961-1970. However, as Figure 2.1 shows, the optimal path is far below the actual or historical path. This means that during the period in question, fiscal policy was overused by the U.S. policy- makers and, as a result, the target of internal balance was overshot. Figure 2.3 shows that the actual or historical path of the GNP fluctu- ates above the nominal path T1 employment without inflation. Therefore, it is not surprising that t’ which represents the situation of full during the last decade a troublesome inflationary spiral has threatened the U.S. economy. This is because of an excess demand created by a boom in government expenditures. 4?“? .«c‘a Unlike I :c 359? both "EffectiVene’ is are effe- i: :egmat'm ‘33-‘51 sat at :e a situati sf external 3;;ation 2.1 Samant- az' extema :em in: re - um baud-.3 CC ‘7'“T‘ a weed {1.11: ‘ all tfiE 2m.” :5; . ‘ts ‘ Y V‘l ‘- {é‘l‘H‘lI-‘I 44 Unlike the fiscal policy Glt’ the optimal monetary policy rit has to obey both internal and external balance. But when using the "effectiveness" criterion described previously, the monetary policy is more effective in maintaining balance-of-payments equilibrium than in regulating full employment. Therefore, once the fiscal policy has been set at the optimal level, a level which brings the economy close to a situation of full employment without inflation, the maintenance of external balance can be taken care of only by the monetary policy. Equation 2.13 permits the computation of optimal path for the monetary instrument. It is found that the simultaneous achievement of internal and external balance would require high negative values in the short- term interest rate (Figure 2.2). Therefore, a limit or a so-called boundary condition (Tinbergen:l967) has to be set on this policy- variable magnitude to indicate that negative interest rates resulting from the application of optimal control theory to the U.S. economy are technically impossible. Boundary conditions and their violation by the optimal policy mix, the so-called inconsistency of the optimal policy, require the problem of quantitative economic policy to be reformulated (Tinbergen:l967). Within the framework of optimal control theory it can be done by penalizing the deviations of the control variables from their nominal values, which represents the limits on acceptable instrument variable magnitudes. For the one-country problem, the newly defined matrix Qc of the cost function, including the boundary conditions, is: be normal \ :5ch are . A sensi ”« the con m does m 2353?: the 5333on Of ; Eil‘en in Ta .é'zle 2.3 -.. LL. . A) .... . / [j ‘ / r ' - r I P . I Q ’ 0 q11 0 0 0 0 c -..- -- -.. = Q = ' 0 q22 0 0 0 Lo I RP o o o o o ' a 0 0 0 r11 0 0 0 0 0 r22 L .1 The nominal values for the monetary and fiscal instruments to be tracked are given in Table 2.1. A sensitivity analysis of the weighting factors on components of only the control vector is performed. Since the optimal control solu- tion does not depend on the weights q11 and q22, it is assumed that through the performance of seven experiments, both are equal to l, for reasons of simplicity. The cost function for all the experiments is given in Table 2.3, and the results are shown in Figures 2-1 to 2.5. Table 2.3. One-country model: penalty weights attached to the deviations of state and control variables from their nominal values Rn“ q11 q22 q33 r11 r22 0 1 1 o o o 1 1 1 o 0 1o5 2 1 1 o 0 5x105 3 1 1 o 0 106, 4 1 1 o 105 1o5 5 1 1 0 5x105 5x105 6 1 1 0 1o5 106 7 1 1 0 5x105 106 ,4 .9... .O‘L 1. t’ 1. 5351.25 Since the negatii assigned 11 :zstrzm nt Eelance. the Siiula Pfifie paid attained, am is CI 5331115 '1‘ 3.311 in i'n A ““3433; y d d I. r} 6.18 EK‘ ‘6 46 Results Since only the optimal monetary policy is not feasible because of the negative value of interest rates, the first three experiments are assigned increasing penalty costs for deviations of the monetary instrument from the nominal value, fixed at a rate of 2.15 percent, while government spending is free to fluctuate to achieve the internal balance. Figure 2.5 shows that the higher the penalty cost, the closer the simulated optimal path for r is to its nominal path. But the It price paid is that the balance-of-payments equilibrium is no longer attained, and within the context of Votey's model, the more the interest rate is constrained to the limit of 2.15 percent, the higher the surplus in the balance of payments is (Figure 2.4). In summary, the joint internal and external balance cannot be achieved for the U.S. economy during the period 1961-1970 once the deviations of the monetary policy from the nominal path are penalized. Since the interest rate is the most effective instrument with respect to the external balance target, the latter has to be dropped. Unlike the optimal monetary policy, which in the first three experiments gets closer and closer to its nominal value, the optimal path for the fiscal policy diverges strongly from the nominal path (Figure 2.1). This means that a feasibility trade-off exists between the two components of the control vector. However, even if the degree of freedom given to the fiscal tool allows internal balance, federal budget constraints as well as political constraints could prevent a large change in government spending which might be required for feasi— bility and stability in the monetary instrument. In the agitudes : :ta-ie-oii b ;er:ent, I .L,l to . . .4: rezant 47 In the last four experiments, limits on both policy—variable magnitudes are introduced. Figures 2.1 and 2.5 show the feasibility trade-off between the two instruments. The tracking of government expenditure to the nominal G1t is sensitive only to the increasing value of the penalty costs on deviations of monetary policy from 2.15 percent, while the tracking of the nominal interest rate depends on the weighting factors for both instruments. It is noted that for the same penalty costs r22, the effectiveness of the interest rate in tracking};t diminishes with introduction of limits on government expenditure magnitudes. Furthermore, neither the internal nor the external balance target is achieved (Figures 2.3 and 2.4). In fact, within the confines of Votey's model, the U.S. economy must be in a situation of deflation combined with a balance of payments surplus for the monetary and fiscal policies to be feasible. Does this mean that the initial conditions for the one-country optimal control problem are far away from.the "ideal" conditions or the arbitrary set of nominal values for monetary and fiscal policies to be tracked need to be more inflationist, or simply the policy for internal and external balance is not feasible within the optimal control framework due to political, social and ethical constraints established in any given economy? No definitive answer can be given. Conclusion Under the assumption of constant price, fixed exchange rate and passiveness on the part of the second country, i.e., fixed foreign interest rate, it is found for the case of one-country that: Fw a- W (1) Th an pr pe: a t in gm fe (3) If.) (1) (2) (3) (4) 48 The solution for simultaneously attaining internal and external balance can always be determined via the optimal control techniques if Tinbergen's principle on equality between the number of inde- pendent instruments and the number of independent targets is met, and if there are no constraints on instrument-variable magnitudes. There is no guarantee that this optimal policy mix will be feasible in a given economy. Policy inconsistency in Tinbergen's sense (Tinbergen:l967) arises in the U.S. economy; that is, the optimal monetary policies to achieve the overall balance violate the boundary conditions or limits on interest rate variable magnitudes that, for practical or political con- siderations, have to be set. One of the targets has to be dropped, and within the confines of Votey's dynamic model the choice of that target rests on Mundell's "division of labor" principle. For the U.S. case, the tracking for external balance has to be dropped at the expense of the tracking for the nominal interest rate fixed at 2.15 percent. If constraints on all policy-variable magnitudes are included in the optimal control problem, neither the internal nor the external balance is achieved 49 and a trade-off exists between attainment of the overall balance and feasibility of monetary and fiscal policies to be carried out toward the given targets. ‘1- [.7 CHAPTER III TWO-COUNTRY MODEL CASE A: PASSIVE RESPONSES AND COMMON EXTERNAL BALANCE This chapter deals with a three-target and three-instrument problem: the fiscal authority of Country II acts to attain full employment in II while the fiscal and monetary authorities of Country I strive for both internal full employment and common external balance. To find the optimal policies for internal and external balance in both countries the procedure used in the one- country model case will be repeated. After the presentation of Votey's two-country model (Votey:1969), the two-country optimal con- trol problem will be formulated into Pindyck's framework as well as that of Chow. Then for reasons similar to the previous case, Chow's approach will be chosen to derive the optimal solution both analy- tically and numerically with reference to the United States and Canada. Finally, the optimal solution will be appraised and amended. 3.1. Presentation of Votey's Model The two-country model is obtained by adding to the one-country model a second set of behavioral equations for consumption, invest- ment, imports, production function and identities defining capital stock and national income with Country II subscripts. Votey's 50 Im- f . l:- E - ”.le ~..o-eount (l) I L. fi ‘5 3 . T "1‘. t ‘5 I . .\ ‘ I. .‘ . “’5 In, «a ~k . Q I‘ ‘ ‘ is k 51 two-country model is considered under the following assumptions: III , (1) (2) (3) (4) The foreign rate of interest is given. This assumes Country II's passive cooperation such that Country I may adjust its own rate to achieve external balance without foreign interference or countermeasures. Stein's results dealing with short term interest rates will be considered to represent the degree of sensi- tivity of capital flows to interest rate differentials. Unlike the one-country model, the exports of Country I will be composed primarily of imports of Country II and will therefore be a dependent variable in the system. The value of imports of Country I from Country II is assumed to be fixed or exogenous to the system. The variables are: i = 1,2 Common balance of payments Consumption expenditures in Country 1 Government expenditures of Country 1 Gross investment expenditures in Country 1 Net investment expenditures in Country 1 Investment earnings of Country I from abroad Capital stock of Country 1 Labor force in Country 1 Total imports demand of Country 1 Imports of Country III from I and II Imports of Country I from Country III Net short-term capital outflows of Country I ,,: Rate it P! : ImpcI .I it: Tax; “it: Capi'I B: it I >< I 52 rit: Rate of interest in Country 1 P : Import price of Country i M it P : Export price of Country 1 Kit T : Tax receipts of Country 1 TR ' Capital transfer of Country II xit' Total exports of Country 1 Yit: National income of Country 1 The equations are as follows: C (1‘1) Yit Cit + I1t + Git + (M2t+M3t+IElt)—Mlt n (1‘2) K1: ' 11: + Klt—l G n (I-3) I1t = I1t + 6* Klt (1‘4) Blt = (M2t+M3t+IElt) ' Mir-Git (1’5) Y2: = CZt+12t + G2t+(Mlt-MI:I) ' ”2: (1'6) KZt = Igt + K2t-1 (I-7) Igt = Igt + 6* K2t (E’l) Y1: = 610 + 611 Klt + 612 L1: (E‘Z) Clt a 0‘10 + a11(Ylt-T1t ' 6* Klt) P _ = - _ .3i (E 3) M11: 810 + 811(Y1t T1: 5* Klt) + 812(Px)lt (E—a) In = 1t Y10 + Y11 Ylt-l-le q(6='c+'r)1t-1+I13 Klt-l . O’. ._.._.-...__.__.—r—. "n.— I. I; v = 3 I-II ‘2: ‘ .4) C? 3 (I ' a (3‘ \_, La: | {1) Tue comments 01 iegter. Equation Eaz-z'n its facto Equation its disposable iifferent from epital deprec teamed to filial transf 53 (E's) 011: _ n10 + r‘11 (th'rlt) (3‘6) th = 520+521 K2t+622 L2:: “‘3'” C2t ' o‘20”‘21 (th’th'Tth'5* KZt) P _ _ .. _ .24. (E43) M2: .. 320+321 (Y2t T2t TRth 5* RR) + 822 (Px)2t n 03-9) Izt = v20+¥21 YZt-l-YZZ (“5*”) 2t-1+Y23 KZt-l The comments on (E—l) to (E-S) have been presented in the preceding chapter. Equation E-6 - The output of Country II is a linear function of both its factors of production: KZt’ L2t' Equation E—7 - Consumption expenditures of II are a function of its disposable income. Here the disposable income definition is different from that of Country I and it is represented by GNP less caPital depreciation (6* K where 6* is the rate of replacement and 2t is assumed to be the same for both countries), less taxes and less capital transfers of Country II (TRZt is assumed to be exogenous). Equation E-8 - Net investment expenditures in II are assumed to depend on money supply, the capital stock and the user cost of capital in II. The user cost of capital in II has the same definition as that in Country I, with the assumption q = 1 over time. Identity I-4 - It represents the common external balance between the two countries . 3.2. Optimal ( on PoliC\ Under the gassive with re to seek its bal are three indet F5! rent in both cc ._ ; agents equili instruments are and II coupled I‘M" ' 'I ‘.._ 1.1K {nutty 11's it The exterr Din internal b; respective pro: 'I‘Jére the subs. ...... 54 3.2. Optimal Control Problem without Constraints on Policy Variable Magnitudes Under the assumption of common external balance, Country II is passive with respect to its balance of payments, allowing Country I to seek its balance-of-payments target unimpeded. Therefore, there are three independent targets to achieve simultaneously: full employ- ment in both countries and maintenance of Country I's balance-of- payments equilibrium. To attain these targets, three independent instruments are considered: government expenditures in Countries I and II coupled with a short-term interest rate in Country I, while Country II's interest rate is regarded as fixed or given. The external balance, as defined previously, is Elt = 0 while the two internal balances for Country I and II are determined by their respective production functions: Yit = 610+6fl Kit-+612 Lit (3.1) where the subscript 1 stands for country 'i (i = 1,2). As before, to overcome the problem of "double entry" of K in the reduced form, it ecluation 3.1 is transformed to: Yit-611 Kit = 610+612 Lit ; 1 = 1,2 (3.2) Let {lit = Yit-6il Kit' Then equation 3.2 becomes: In = 6 +612 Lit ; 1 = 1,2 (3.3) 10 Which determines the internal balance for the respective country. To aPply the optimal control theory, Votey's two-country model has been formulated into the following reduced form (Appendix A—6): . :7 +311 4"; I t 3yt-l t :‘tere E‘Iht'er’B 5: = [G t’GZt’r ’- M l’IElt’ul 'f L 0121.. '——_I H A11 A12 A31 32 2:23-01 Problem Imw “hated into ""Qaches are Red for the :1, . ass”Eamon 1 = — +_ — +- yt Ayt-l But+Cut_1 th [Ylt’YZt’Bit’Klt’KZt’Glt’GZt’rl where Q ~ yt d “t ‘ [G -t’GZt’rlt] d III zt - [l’IElt’Mlt , I- A11 A12 0 A21. A22 0 3': A31. A32 0 r11 0 o 0 721 0 In 11 D12 D13 D 21 D22 D23 3': 1) 31 D32 D33 a46 O 0 Lase 0 0 TE 14 24 A34 45 14 24 34 55 15 25 >- >- o. Gil ll 35 O 55 15 25 35 "“11 .] Q11 Q12 Q21 Q22 0 0 1 Q31 Q32 "11 0 O ’Y22 l7 16 27 26 37 36 17 27 37 U. 0| 17 27 37 lt’ Tlt’ r2t’ r2t-1’ TTZt’ T2t’ TRZt’ M3t] (3.4) The“: given the structural form of equation 3.4, the two-country optimal Control problem under the assumption of common external balance will be formulated into Pindyck's as well as Chow's framework. Since the two approaches are equivalent (Appendix A-3), Chow's method will be con- sidered for the derivation of the two-country optimal solution under th e assumption of no constraints on policy-variable magnitudes. 3' 4145—“ ' 11., 1 :1I Mid 31.2.1. Fomu] For the t E‘aéyck‘s fran P P P r=A x + I t-l . Pd tater =v- t “t state vector; "Q. 111'? 42'le t l,lE 131) current atrices: A11 1 A 1 P 21 P31 P '11 I o .7” 7 ’61:] r.- t M 56 3.2.1. Formulation of the Problem For the two-country optimal control problem formulated into Pindyck's framework (Appendix A—2.l), the "state-space" system is: PPP PP PP zt_ x = A xt_l + B ut_1 + c (3.5) l V Pd — — (1 ~ ~ where xt = yt-But-th [YYlt’ YYZt’ BBlt’ KKlt’ KKZt] is the (5x1) P c_l d . . state vector, ut - ut - [Glt’ G2t’ rlt] 18 the (3x1) control vector, I P III d and zt — [l’IElt’Mlt , TTlt’ Tlt’ r2t, TTZt’ TZt’ TRZt’ M3t] is the (11x1) current exogenous variable vector. AP, BP and CP are known matrices : r- a A11 A12 0 A14 A15 P A21 A22 0 A24 A25 A 3““ A31 A32 0 A34 A35 Yll 0 0 a45 0 O YZl O 0 a55 _ - IFA11Q11+A12Q21 A21Q12+A12Q22 A16 7 :BPafi A21(211‘5“22Qz1 A21Q12+A22Q22 A26 +c= A31‘211J'A32Qz1 A31(2124‘A32sz A36 Y11Q11 Y11Q12 I21921 Y21Q22 157 “noxoouu on on neuoo> manofiuo> ououm Hocfiaoo Aaxmv mnu ma um I a o m Narnnanh~ «Head» m~o-<+maaam< e~a-<+eanfla< n~o-<+maaa~< a~o-<+eaaa~< n~q~a<+nHaHH< e~a~H<+eHaHH< mNaHN» oHaHH» mNGNn<+anHm< w~n-<+anH~< mNo~H<+wHaHH< MNQHN» naaHH» nNn~N<+MHnH~< mNa-<+MHoHN< n~n~H<+nHQHH< HIuNMM .aluamm .HIuN r» .HIUHNV I HquoIHIuamIHIuh I Hqu when: h. h. U I. .l v ml Hlu Hun a-» Aauu Hun N NI x x: I.I A fil m Vmc . ml. mxv W H mm NNQHN» ANQH~» oNQHN» sanda» NHQHH» oa NNQNM<+NHQHM< NMGNM<+NHQHM< ONQNM<+cHGHn< NNONN<+NHGHN< NNQNN<+NHQHN< ONGNN<+0HOHN< NNQNH<+NHQHH< NNDNH<+NHQHH< cNGNH<+OHDHH< NNoHN» cmunmo+a~naa» ~HQHH» weane¢+HHnHH» -a~m<+~anam< anunm<+oanam<+ama~m<+aanan< Immnm ~NQ-<+~HQH~< enam~<+oaae~<+a~n-<+Hana~< NNDNH¢+NHDHH< enund¢+o¢u¢H \I-a‘nic c~au .IH‘I- hl I. II! «V In... N V I..." Am— .421! 71 Huom.oa ow.o hm.o~ «Hoo.a Hano.o qqco.o oo.~om mmoo.a He.o 5N.e Ao~¢avoa mmmH.m oo.e qq.¢~ m¢oo.a @Noo.o ammo.o on.oa~ moNo.H ow.m nm.m Amemav¢ mama.“ ma.“ m¢.o~ oomo.H mamo.o e~oo.o Hm.mo~ mmflo.a mm.m -.o Anomavm Noom.o -.o Hm.ma mwmo.a eaqo.o mono.o mm.m- owNo.H o¢.¢ mm.m Anemavm omoo.o mo.n mm.oa oaqowH qmqo.o omeo.o mm.m- «moo.H ~H.¢ mm.m Aoomave m-.o hm.¢ a~.qa maflo.a ~mmo.o emqo.o oo.omH oooo.a ~m.m mu.m Amomavm «qm¢.m ma.q m~.~a oooo.a w~mo.o ~wmo.o no.¢ha Noam.o mm.~ nm.¢ Aqemavq ommq.m mw.m wH.HH oooo.a ammo.o ammo.o o~.moa oooo.H «m.N mH.q Amoaavm ma~m.¢ Nfi.m mm.oa aomo.a ammo.o Hamo.o mo.HmH Hoao.a mq.~ mo.¢ Amomavm Hm¢m.q ¢¢.m hm.m moqo.a mmmo.o ¢a~o.o . mo.qqa oooo.a -.~ mm.m Aflooava um: away “my guys anumu um“ pay “Has “Hz “HmH Aummwy E 32$ 22-33 $850 25 .ms - 833.5 2.8 mwoxm mcu Mom mmDHm> HNUHHOumfim .N.m «News ”—15 4’1 7‘3" ' ‘ll' 4““ I" F "W' _' ‘L’ 33.2» firis aL1eS. Llap‘er II ( 72 . I (1) Compute [Gi(l), 63(1), ri(1)] from equation 3.10 using the initial conditions given by the historical data for t!1(0),‘Y2(0), 31(0), K1(O), 01(0), 02(0), r1(01] and the exogenous variables of period 1 given in Table 3.2. (2) Compute [?*<1). ?2(1). Bf(l). Kf(l). K§(1). 93(1). G3<1>. I ... ~ rf(1i] from equation 3.11. NOW'[Y*(1), Y§(1), 33(1), I * * * * * _ K1(1), K2(1), 61(1), 62(1), rl(1)] can be used in equa I tion 3.10 to compute [Gf(2), G30), rf(2)] , which can be used in equation 3.11 to compute [§1(2), §§(2), I Bi(2), Kin), K§(2), Gi(2), 63(2), rf(2)] , and so on. Continue the process until all of the control vectors I [Gf(t), c§(c), ri(ti] , t = 1,..., 10, and all the state variable vectors [§f(t), §§(t), Bf(t), Ki(t), I K§(t), Gf(t), G§(t), ri(t)] , t = l,..., 9, have been computed. flChe results are presented in graphical form (Figures 3.1 to 3.6) With tzime on the horizontal axis. First, based on the following matrix (equation 3. 7) — .— ~ ~ — R°w Y1: Y2: Blt 1 0.3810 -0.0977 0 2 -0.1315 0.3632 0 3 0.0028“ -0.0021 0.0216 some remarks will be made on the effectiveness of policy-instrument v ariables. Using the same concept of effectiveness as described in Ch aptel’ II (section 2.3), it is noted that the fiscal policy of each 51"."; A. than 3.4 U3“- “w b1. 1 1 LnnH ) (3 7 \T‘fl \‘\ b l l, I [anti ) (3 [ 73 500"' 1 Optimal Path ++-I Actual Path 400"‘ ------ Nominal Path ' ’3 a o H __ .4 300 :1 .0 3 Figure 3.1. Two-country model (Case A) - U.S. Government expenditures: the optimal path compared with actual and nominal paths. 10"' M'- ........ 0 l 1961 A U) G d O ‘H v--l H 2 ~10 —4 Optimal Path \u—v +—H Actual Path — r - ----- Nominal Path .20 § I Figure 3.2. Two-country model (Case A) - Canadian government expenditures: the optimal path compared with actual and nominal paths. :71 ‘UL ‘- 1 1 f .1.— '-:'. :1 fl 1’0 rcen t; 3H 74 Optimal Path -I—H Actual Path ' " ' ' ' ' Nominal Path 20.0"- d 60.0 “ ...-q u G '40.0— 8 s... Q) Dd 1. ~20.0—J Figure 3.3. Two—country model (Case A) - U.S. interest rate: the Optimal path compared with actual and nominal paths. ‘0 -_ w ‘. . 1.147 ... LIP; . n.1,}...t, H .M . ...., Amt-thw ~H H3 WV 1"! “I J.” 75 Optimal Path 4—H Actual Path """" Nominal Path 600'- 500-‘> 400'- ”a? 300 —-J 1 8 H r-i v-i 'H _ .0 1’5 200-- _ 'uns 11,12 Run 100- - Runs 8,9 Run 10 0 J l l I J L l J .1961 62 63 64 65 66 67 68 69 70 d . ~100 '— qu 3Figure 3.4. Two—country model (Case A) - United States: optimal 1’ netof capital stock compared with the actual and potential GNP. m w- :1: ‘I 1 .nv.‘ NH ‘ as u...» 76 120"‘ 90" Optimal Path +H Actual Path - ----- Nominal Path Run 7 ‘ /////Runs 8,9 60.- ’//,4’a¢/////// /\ Runs 4,5,6 30— ..J Runs 1,2,3 .._—’5 .----. ----- L ----1 ----- 1 ----- \1. ,31961 62 63 64 65 66 6 68 69 70 S -°H ..o 3 -30— ‘60.... ~90 q ~120 ..I Figure 3.5. Two-country model (Case A) - Canada: optimal G:1}> \ net of capital stock compared with the actual and potential GNP. ...J {cw E 3.. . nfiwww‘ I A 7.:an HH #3 T \l, v 5. WV 1 II. ‘I'.‘ a V SEEM 1 Fa I A 77 Optimal Path “W Actual Path """ Nominal Path 10"‘ m a H D.“ T fl u a CD 'un 7 o—r ........................................ ... ...... Runs 8,9 q ‘ Run 10 ‘10 Runs 11,12 ‘ E I o '1-4 .,.1 .n 3 —30--' Run 4 fi Runs 5,6 .... NH 0 '0 ~40 Run 1 _ Runs 2,3 ~50 1 1 1 1 1 1 1 1 1 l 1960 61 62 63 64 65 66 67 68 69 70 Figure 3.6. Two-country model (Case A) - U.S. balance of payments: optimal paths compared with the actual and nominal paths. country is than that own fisca; other com interdepe: country's Country I varying d1 balance t} 05 by the reduced d1 Short, the ES econom Xext, eptimal cc RS. and ( Micy in Pat' H, and 78 country is not only more effective in achieving its own internal balance than that of the other country, but also that a negative change in its own fiscal policy is required to bring about a given change in the other country's internal target variable. This means that the economic interdependence of the two countries lowers the effectiveness of each country's fiscal instrument in reaching its own internal balance. Country I's monetary policy depends on all three targets, but with “varying degree. It is more effective in achieving the common external balance than the internal balances which have already been taken care of by the fiscal policies. Furthermore, its overall effectiveness is reduced due to the existence of Country II's internal target. In short, the effectiveness of each country's policy making will decline as economic interdependence increases. Next, an appraisal of the results obtained by the techniques of (aptimal control will be conducted to see if they are feasible in the [1.8. and Canadian economies. Figure 3.1 shows that the optimal fiscal puslicy in the United States diverges from both its actual and nominal Imath, and that the economic interdependence prevailing in the two- CNDuntry model requires much higher U.S. government expenditures to I13ach the situation of full employment than in the one-country case. F'urthermore, the GNP resulting from using the optimal fiscal policy iii exactly on the internal balance path (Figure 3.5). Again it is noted that the actual path for the GNP diverges upward from the full employment situation. In other words, for the last decade U.S. ecOnomy dealt with inflation created by an excess-demand. 4.3 '5‘." .‘- J i "'_I‘ J . 13“"? For C States, tt t'te U.S. 1 fiscal p02 first hali tent exper diverges f acre stabl deviation therefore, Once . attaining out by the mitten 1 .egayme‘ 121!th ‘ A “Earl‘s; q 79 For Canada, which is a small country compared with the United States, the attainment of its own internal balance, when impeded by the U.S. pursuit of domestic stabilization, required a tightening of fiscal policy with a high accumulation of government savings during the first half of the planning period, then a steady increase of govern- ment expenditures starting at 1967 (Figure 3.2). The optimal path also diverges from the nominal and actual paths, the growth of which is more stable. Figure 3.5 shows excess demand of an amount equal to the deviation of the actual GNP from the situation of internal balance; therefore, inflation occurs in Canada too. Once the U.S. government expenditures are set at the level for attaining the internal balance, the common external balance is carried out by the U.S. monetary policy, defined as the U.S. shorteterm interest rate. Unlike the one-country case in which the value is negative, the optimal interest rate starts with a very high and even unrealistic level around 82 percent, then decreases over time with negative values for the three last periods. In short, the actual situation in the United States and Canada, compared with the given targets, is as follows: both countries are facing inflation combined with a surplus in the U.S. balance of pay- ments, which is termed as a situation of potential conflict in policy goals (Johnson:1966). To change this conflict situation to an overall balance by using traditional policies rather than appreciation of the exchange rate would require, within the context of optimal control theory, high government expenditures and a negative interest rate in the United States, while government savings would be forced in Canada. ‘ ‘ ‘1 1. q .‘ Nm’flfl All thre for eith boundary been vio theory t nenalty that is, Similar t “fighting 'e': Whe lie C031: : we r981111 80 All three of these optimal policies are declared to be inadmissible for either social, political, or practical reasons. Therefore, the boundary conditions or limits on policy-variable magnitudes which have been violated are to become active. Within the framework of control theory this can be done by introducing into the cost function the penalty costs for deviations of the policy variables from their limits, that is, by changing the diagonal elements of the matrix QC as follows: — I '- 1 7 : d qll 0 0 0 0 0 0 0 , 0 q22 0 0 0 0 0 0 P 1 Q ‘ 0 0 0 q33 0 0 0 0 0 0°= "--_.--” = 0 0 0 0 0 0 0 0 ’ 0 0 0 0 0 0 0 0 ' P 0 . R 0 0 0 0 0 r11 0 0 v 0 0 0 0 0 0 r22 0 | L | L0 0 0 0 0 0 0 r33 I - do Similar to the one-country case, only a sensitivity analysis of the weighting factors on the components of the control vector is performed, 1'8 . , when the weights for the endogenous variables remain unchanged. The cost function for all the experiments is given in Table 3.3 and the results are shown in Figures 3.7, 3.8 and 3.9. ..L'.‘ n“ i r F ,‘ "n ; t r‘ - f e“ . w Table 3.3. ' Runs lest 1 1 Test 2 4 Yes: 3 l 81 Tablxa 3.3. Two-country model (Case A): penalty weights attached to the deviations of state and control variables from their nominal values Runs q11 q22 q33 r11 r22 r33 Test. 1. 1 1 1 1 0 0 105 5 2 1 1 1 0 0 5x10 3 1 1 1 0 0 106 5 5 Test. 2. 4 1 1 1 0 10 5 105 1 1 1 0 5x10 10 5 5 1 1 1 0 5x10 5x10 1 1 1 0 105 5 5x10 6 1 1 1 0 5x10 102 1 1 1 0 105 106 1 1 1 0 5 106 105 7 1 1 1 10 5 0 105 1 1 1 5x10 0 105 1 1 1 10% 0 10 5 8 1 1 1 10 5 0 5x10 1 1 1 5x10 0 5x10 1 1 1 log 0 5x10 9 1 1 1 10 0 103 1 1 1 5x105 0 106 1 1 1 106 0 10 TeSt 3 10 1 1 1 105 5 105 5 10; 1 1 1 5x10 5x10 105 1 1 1 105 5 5x10 105 1 1 1 5x10 10; 10 5 11 1 1 1 105 5 105 5x105 l l 1 5x10 10 5 5x105 1 1 1 105 5 5x105 5x105 1 1 1 5x10 5x10 5x10 12 1 1 1 105 5 102 10g 1 1 1 5x10 10 5 106 1 1 1 105 5 5x105 106 1 l 1 5x10 5x10 106 1 1 1 106 106 10 82 Figure 3.7. Two-country model (Case A) - U.S. Government expendi- é“? tures: optimal paths compared with nominal path. Figure 3.8. Two-country model (Case A) - Canadian government expenditures: optimal paths compared with nominal path. oi l on — ($ hi 1 1101114) 1960 to_ so 1 l d (5 1. r .11 1mm ) I 83 300"" Optimal Path """ Nominal Path Run 1 Runs 2,3 — RUHZI Run 5 Run 6 —”(-D\ 200 g ”-4 H H *H ..D 8 100_> i 1 1 I 1 1 I 1 l J l 1960 61 62 63 64 65 66 67 68 69 70 Figure 3.7 Run 7 Optimal Path Runs 8,9 80 '''' Nominal Path 60 4O 2() ($ billions) 10,11,12 l I L ‘40 Runs 2 , 3 Figure 3.8 ..................... flRuns 4,5,6, M d "Sinai. regent: kl"- Ai I. e a ...-nun: U Go ...vru Noam .11., ~ 84 —"——‘ Optimal Path Nominal Path Run 1 4.0.——: Run I 3.6 3.2 'E a) o a... a) 13.. 2 ' 8 Run 10 Rum? Ron 2 ——“’—‘ Run! 2.4 L 2.0 l T I I I I 1961 62 63 64 65 66 67 68 69 Figure 3.9. Two-country model (Case A) - U.S. interest rate: 01) t 111131 paths comp ared with nominal path. Three assigned f: to all the For t instrument observed t closer it instrumen Upward an inadmissi is a tree In fact, Prevails large de: in both . Situatio. andc 2t' The instTune tither c (1) 85 Three tests of sensitivity are performed with penalty weights assigned first to one instrument, then to two instruments, and finally to all the three instruments. For the first test (runs 1,2,3) in which only Country I's monetary instrument is constrained to the limit fixed at 2.15 percent, it is observed that the more weight given to the interest rate rlt the closer it gets to the nominal path, while the two other unconstrained instruments Glt and G2 t diverge strongly from their boundary conditions upward and downward, respectively. Therefore, they are declared inadmissible for practical and political reasons. Furthermore, there is a tracking trade-off between endogenous variables and instruments. In fact, because a division of labor in achieving the joint balance Prevails in Votey's model when rlt is constrained within limits, a large deficit occurs in the U.S. balance of payments, while the GNP in both the United States and Canada keeps tracking the full-employment situation at the expense of non-feasibility of fiscal policies Glt and G 2t' The second test concerns penalty weights given to two of the instI‘uments--Country I's interest rate and government spending of either country. It is found that: (1) If both Country I's instruments are constrained by the boundary conditions, then its interest rate will track the nominal path more closely than if limits are set on either Country I's interest rate alone or on the pair of Country II's fiscal instrument and Country I‘s interest rate. In short, limits set on ~r...¢.u. . I .I . III p Pi. Fir three 1] (2) 86 G reinforce the feasibility of Country I's 1t monetary policy, while limits on G t do the 2 opposite. Since two boundary conditions are active, it is expected that at least two endogenous variables would not be on target exactly because a track- ing trade-off between targets and instruments occurs. Because of the particular structural form of Votey's model, the choice of targets to be dropped is based on Mundell's "division of labor" principle. That is, the target on which the constraint policy instrument is the most effective has to be changed numerically or given up, at the expense of instruments tracking for limits. IFinally, the third test in which weights are assigned to all the three instruments is performed. It is found that: (l) (2) Each of the three instruments is tracking its limits. However, giving increasing weight to r11 leads to a closer tracking of Country I's monetary policy on its limit, while the tracking for nominal government expenditures in both countries remains the same, whatever the weight number assigned to them. Compared to the second test (runs 7,8,9), the tracking for the fixed interest rate is less "'7 ‘———-A J. at"? It (3) 87 effective. This is due to the additional boun- dary condition set on Country II's government expenditures. Similar to the first two tests, the tracking of all endogenous variables 1 and B t is lost. lt’YZt 1 The deviations from their respective targets are larger than those obtained in tests 1 and 2. It can be concluded that: (1) (2) (3) As economic interdependence increases, the effect- iveness and impact of each country's policy on its own target will decline. If Tinbergen's principle on equality between the number of targets and the number of instruments is met, it is always possible to steer the endogen- ous variables to be controlled exactly to targets by using the quadratic welfare function (equation A-2.l6) and assigning positive weights only to the deviations of the variables selected. However, nothing guarantees the feasibility of optimal policies to achieve the internal and external balance. When the boundary conditions become active, a tracking trade-off between endogenous variables and instruments occurs. van 5.1m ' I:. CASE 1 CHAPTER IV TWO-COUNTRY MODEL CASE B: PASSIVE RESPONSES AND LINEAR DEPENDENT EXTERNAL BALANCE Unlike the preceding chapter, this one deals with a four-target and three-instrument case: the fiscal authority in each country will act to attain its own internal balance while the monetary authority in Country I strives not only for its own external balance but also for that of Country II under the assumptions of passive responses from Country II and non—common but non-conflicting balance-of-payments tiargets. Since the number of targets is greater than the number of inStruments, Tinbergen's rule is no longer satisfied. Therefore, it is exPected that the optimal results for internal and external balance Will differ from those obtained in the two preceding cases. To show this. the same analysis procedure will be repeated. First, Votey's two“country model with modifications on the foreign sector will be pre- sented. Second, the optimal control problem will be formulated to derive the optimal solution analytically and numerically with reference to the United States and Canada. Finally, the economic evaluation of t he Optimal policy mix will be made as well as its amendment when b oundary conditions become active. 88 u —.~_n a"! {1 T1“ "' ‘ 4.1. P_r_e_s Vote: fied by 1’! adding an paments. land ll countries follOViné (1) 89 4wll. Presentation of Votey's Model with Modifications on the Foreign Sector Votey's two-country model presented in Chapter III will be modi- fieni by redefining all the variables in the foreign sector and by adding an identity for the determination of Country II's balance of payments. This is done because Votey ignores the trade of Countries I zarid.II with the rest of the world by assuming the total exports of I (3E1) equal the total imports of II (I) and by using a single equation-- Ccniritry I's balance of payments--as the common external balance of both countries. Now consider the "modified" two-country model under the following assumptions: (1) Country II's interest rate is given, that is, Country II is assumed to be in the position of passive cooperation with Country I for dealing with the external balance. (2) Stein's results for short-term interest rates represent the degree of sensitivity of capital flows to interest rate differentials. (3) Unlike Case A, the total exports and imports of Countries I and II will include the trade with the third country which represents the rest of the world. :FIJe variables of Votey's econometric model with modifications on the foreign sector are: f°r 1 == 1,2 :3 ita Balance of payments in Country 1 C a iut Consumption expenditures of Country 1 IEIII 1:1]: _ 9O Gross investment expenditures of Country 1 Net investment expenditures of Country i Investment earnings of Country I from abroad Capital stock of Country 1 Labor force of Country 1 Total imports demand of Country i Imports of Country 1 from Country III Net short term capital outflows of Country 1 Rate of interest in Country 1 Export price in Country 1 Import price in Country 1 Tax receipts of Country 1 Transfer payments of Country II P P (_yqit = Terms of trade of Country 1 x Total exports of Country 1 Exports of Country 1 to Country III National income of Country 1 The equations are as follows: (I—l) (1-2) (1—3) (1—4) (:[~{5) Ylt a alt + lit + Glt + x1: ’ M1: Bit = x11: ' M1t ' Olt K1t - II111; + K1t-1 lit a Ilt + 6* K1t x1t = x161 + (M2t _ M261) + IElt (1‘6) Y2t = C2:; + Igt + GZt + x2t _ M21: (1'7) 321: = X2t - M2t + 011: (1‘8) K2:; = Itzlt + K2t-1 (1-9) Igt = 131: + 6*K2t (Jr—10> X2. = Xi? + (M1: - Mi?) “3"” Y1: = 510 + 511 K1: + 512 L1: (ES-'2) C11; = 0‘10 + 0‘11 (Ylt - T1t - 6*Klt) (E‘3) Mlt = 810 + 811 (Y1: " T1: ' 6*K1t) + BlZTTlt (E40 1:11: ... "’10 + YllYlt-l ’ Y12 “521+” 1t-1 + Y131(1t-1 (E—S) 011: = n10 + n11 (th - rlt) 03"” th g 520 + 621K2t + 22th (E—7) C2t = 0120 + 0121 (Y2t - T2t - TR2t - 6*K2t) (IE-8) M2t = 820 + 821 (YZt - TZt - TRZt - 6*K2t) + BZZTTZt (E79) I211; = Y20 + Y21Y2t-l ' Y22 (5* + r) 2t-1 + Y23KZt--1 The comments Chap ter II. on equations (E—l) to (E—S) have been presented in Equation E-6 - The output of Country II is a linear function of bot}, its factors of production: capital stock (KZt) and labor force (th). a 37' fl. If?“ vw‘u-v Equ disposal: H is $1 is repre rate of less ta: to be e: Eq' disposa H (112 E1 a58uned cost of H has ‘l‘lc 92 Equation E—7 - Consumption expenditures are made a function of disposable income. But the disposable income definition for Country II is slightly different from that of Country I (Votey:1969), and it is represented by GNP less capital depreciation (6*K2t where 6* is the rate of replacement and is assumed to be the same for both countries) assumed less taxes and also less transfer payments of Country II (TRZt to be exogenous). Equation E—8 - Import demand of II is simply a function of disposable income and ratio of foreign to domestic prices in Country II (TTZt)° Equation E—9 - Net investment expenditures in Country II are assumed to depend on money output, the capital stock and the user cost of capital in Country II. The user cost of capital in Country II has the same definition as that of Country I with assumption q = 1 over time. Identity I-Z - The identity for the determination of the balance of payments in Country I is different from that of Votey. The same def inition as that of the one-country model is used, but Xlt and Mlt redefined to take into account the trade between Countries I and II and the rest of the world (called Country III). Identity I-5 - The total exports of Country I are equal to the exports of Country I to Country III, plus exports of Country I to Country II (Xi: = MZt - Mg?) plus investment earnings of Country I f r0111 abroad. Similarly, total imports of Country I are equal to imports of C olitxtry I from Country III (FIJI-:1) plus imports of Country I from Co II III untry II (M1t X2t - x2t ). h‘ lder of-payner (KR) le: capital 1 net shor lde ,7". of Count Country H'W' fl...“ f3"" '0 are equa .I-n . 4!? 1L 1 L": izports “ The [In Irfi 1 “mm 93 Identity I-7 - Added to Votey's model is the Country II balance- «of-payments identity. It is equal to total exports of Country II ()(Zt) less total imports of Country II (MZt) less net short-term ceapital outflows of Country II (O2t = -01t), which is equivalent to zieet short-term capital inflow from Country I. Identity I-lO — Total exports of Country II are equal to exports ()1? Country II to Country III (xgil) plus exports of Country II to (anuntry I (x;t = M1t - Miil). Similarly, total imports of Country II III zalre equal to imports of Country II from Country III (M2t ) plus I III imports of Country II from Country I (M2t xlt - X1t ). Then substituting th and M2t into th - X2t - M2t + O1t results in: III III III III 13 = - - - 2t x21: + M1: M1: X2: X1: + X11: + Olt _ III III III, III _ _ _ - (x11: +x2t) (Mlt +M2t) (xlt M1t 011:) (31' clefining X : total exports of Country III = MIII + MIII 3t 1t 2t , = 111 III M3t' total imports of Country III X1t + X2t res ults in: BZt a -x3t + M3t - Blt Therefore, is linearly dependent on Blt' th 4- :2 - Optimal Control Problem Without Con- straints on Policy-Variable Magnitudes Unlike Case A in Chapter III, the assumption of common balance of fiants is no longer held when the third country block is introduced .JIII-._. to rep to ach balanc- brium .' linear Countrj instrur govern: interes term in the as 5 94 to represent the rest of the world. Therefore, there are four targets to achieve simultaneously, but only three are independent. Internal balance in both countries and Country I's balance—of-payments equili- brium are independent, while Country II's balance of payments is a linear function of that of Country I instead of being equal to Country I's balance of payments with the opposite sign. As for the instruments, they are in the number of three as in Case A, i.e., government expenditures in both countries and Country I's short-term interest rate are used to achieve the joint balance, while the short- term interest rate of Country II is considered as fixed or given under the assumption of passive responses from Country II. The definition of external and internal balance is similar to that used in Case A (section 3.2) and the reduced form of Votey's modified two-country model given by Appendix A-lO: yt = A yt—l + But + Cut_l + th (4.1) where d ' y = t [Ylt’ th’ Blt’ th’ Klt’ K2t’ Glt’ G2t’ rlt] ' (1 L1 = '5 [Glt’ G2t’ r11:] :1 III III 2 : t [1’ IElt’ x1: ’ Mn; ’ TTlt’ Tlt’ r2:9 r2t—1’ Tth’ th I III III TRZt’ x2t ’ MZt] r u: Mair run: ... A44. :33... .II. ,3 I! . its“. . : ||+| 95 Hmo NMa: chn A527 Hmou HHOI mm mm mm ma NNO NAG mm mm mm ma u Io cm «m cm «a HA AH 0 CI 0 «mo Hmou Nmo- Hmo «No Hmo NHO HHO o «mm mm mm NH o o o 0 33:. so- Nmo Hmo+a mac- Hmc NHOI HAG 4.. com o o mmm qm< mm ‘¢ .¢ ‘< <3 ‘¢ £17m - r A “ “it‘d a Next "mod: where is {h is th 96 Next, the optimal control problem for the "modified" two-country model will be formulated into Pindyck's and Chow's framework but only the latter will be used to derive the optimal solution. 4.2.1. Formulation of the Problem Expressed in Pindyck's terminology, the "state-space" for the "modified" two-country optimal control problem is: P = AP X? + BP up + CP zP xt t-l t-l (4'2) 1 P d —- - d ~ ~ where xt — yt-But - th -[YY1t, YYZt’ BBlt’ BBZt’ KKlt’ KK2;] d -1 ut—l [Flt—1’ GZt-l’ rlt-I] n '11 "Q: is the (6x1) state vector; u III III 1t , M , TT 1t P d is the (3x1) control vector, zt [l, IElt’ X lt’ Tlt’ r2t, TR X M 2t , 2t is the (12x1) vector of current exogenous 2t’ ' III III Tth’ th’ J variable; zP_ is the (12x1) vector of lagged exogenous variable; and t l P P A , B and CP are known matrices. Matrix AP = X Row II YY BB BB KK KK 1t 2t 1t 2t 1t 2t 1 A11 A12 0 0 A13 A14 2 A21 A22 0 0 A23 A24 3 A31 ‘A32 0 0 A33 "A34 4 'A31 A32 0 0 "A33 A34 5 yll o o o ass 0 6 0 0 0 O a 66 Row Glt—l A11Q11+A12Q21 A21Q11~+A22Q21 A31Q11'A32Q21 ”A31Q11+A32Q21 Y11Q11 Y21Q21 97 P _ _ Matrix B = AB + C GZt-l A11Q12+A12Q22 A21Q12+A22Q22 A31Q12‘A32Q22 "A31Q12+A32Q22 Y11le Y21Q22 r1t-1 15 25 35 35 'le 9&3 HNOHN»- HHcHH»- AH~a~m¢+HHon Aa~o~m n~aa~r «NnHN» nanaa» nHoHH» «Hana» «NOHN»- HNOHN» HNOHN» Nuo~m<+uanamos¢-o~ oa<~a> -»¢~gatls from the internal and external balance in both countries, i.e., weights attached to (§lt-?lt)2’ 2t-§2t)2’ 2 ) 3 and (th-th)9 (I (B lthlt ~—\ (4 ) r—J ¢- f x.) I. mp; prob 103 respectively, while r11, r22 and r33 are weights attached to the quadratic deviations from the limits set on both fiscal policies and Country I's monetary policy, i.e., weights attached to (GltIEIt)2’ 2 2 ) (G 1t_rlt) ° Zt-G2t and (r 4.2.2. Optimal Solution for Internal and External Balance Using Chow's result (equation Ar2.20) for the optimal control (Appendix A-2), the optimal solution for the "modified" two-country problem is computed (Appendix A-ll): ].()13 ad ha an 0 m“ ... U d M u N [no u H In 0 N ’1 |0 u H “V 3.3 wa H-u~. an one- ado- nae- “no- and- one- was- «me. fine- «no- fine. ”no- one: o~<- “Na. “Na- and- n~N 2 J (u*) = 3% 33 44 2 (M3 -X3 ) (4.8) +q t=1 t t q33 44 The results for the two-country optimal control problem under the assumption of linear dependent balance of payments differ from those obtained in Case A of Chapter III. From equation 4.4 it is noted that only the variables I* (i = 1,2) are on the targets exactly, while the it variables Bit (1 = 1,2) deviate from their equilibrium. How much Country I's balance of payments Bit deviates from its equilibrium depends on the welfare weight q33 assigned to it, and Country III's balance of trade. Similarly, the deviation of Country II's balance of payments depends on the weight q44 assigned to it and on Country III's balance of trade. Unlike Case A, the optimal welfare cost is no longer (equal to zero, but it depends on the penalty costs that each country aissigns to the deviation of its balance of payments from the equili- torium as well as Country III's trade balance. In summary, the concluding remarks for Case B are: (1) Similar to Case A, the economic interdependence between countries is reflected in the policy inter- dependence; that is, no policy can be considered in isolation. Again within the confines of Votey's model, there is one peculiarity to be noted. Both fiscal policies GIt and Cat affect 109 the full-employment situation in both countries, while Country I's monetary policy is directed at all four targets (internal and external balance in both countries) and the relative effectiveness of these three instruments is measured by the matrix: Row Y1t Y2t 1t B2t 1 "22 312 o 0 Q Q 2 321 "11 o 0 Q Q 3 Q22‘231J'QzIQ32 '(Q12Q31+Q11Q32) q33 "44 ”11" ”nQ ”11(q33+q44) "11(q33+q44) which will be evaluated in the next section by sub- stituting Votey's numerical values for the estimated coefficients into the two-country model. (2) Along with Chow (1972c), it is concluded here that if the number of variables to be controlled is larger than the number of instruments, the variables will not reach the targets exactly, and their deviations from the targets will depend on the welfare weights in QC assigned to them. However, due to this study's formulation of the optimal control problem and the particular structural form of Votey's model which has been constructed to investigate the effects of Mundellian policy assignment on the (3) (4) 110 stability of the system, the conclusion holds only for the external situation where there is only one instrument, i.e., Country I's interest rate to deal with two external targets or balance-of-payments equilibrium in both countries. As for the internal situation of both countries where the number of variables to be controlled (the GNP of both coun- tries fiit (i = 1,2) is equal to the number of instruments (the fiscal policies of both countries Git [i = 1,2]), Tinbergen's rule is met and the former variables are exactly on target. Unlike Case A in Chapter III, the optimal policies are no longer unique in attaining the joint balance in both countries. In Case B there is a family of optimal policies, depending on the welfare weights assigned to the deviations of both balance of payments from the equilibrium, because there are not enough instruments to attain the targets fixed by policy-makers. Similar to Case A, nothing guarantees the feasi- bility of the optimal policy mix in a given economy. Therefore, in addition to a trade-off between the two external targets due to an insuf- ficient orchestration of instruments to secure the simultaneous attainment of external balance in both countries, there may exist another trade-off 111 between attainment of targets and consistency of policies. 4.3. U.S. and Canada thimal Policies for Internal and External Balance: Appraisal and Amendment of the thimal Solution Similar to Case A in Chapter III, Votey's numerical values for the structural coefficients and the historical data for the United States, Canada and the European Economic Community (EEC), which represents the third country in our "modified" two-country model, will be used to derive the U.S. and Canadian optimal policy mix for the 1960's. Since there is a trade-off between the U.S. and Canadian balance-of-payments targets, a sensitivity analysis of the weighting factors on the components of the target vector will be performed.v Then the result of each run of the test will be appraised to determine if the policies (flotained by the techniques of optimal control are feasible. In other anrds, if the optimal policy is in accordance with the boundary condi- tix:n.that the instrument variable cannot surpass certain numerical values for practical or political reasons, the boundary condition does not interfere. However, if the values found for the unknown instruments Vixalate the boundary condition, the optimal solution has to be perfected and the boundary condition becomes active by redefining the weight maltlfiix Qc to include the penalty cost assigned to deviations of instru- ments from their limits. IFirst, using Votey's numerical values for the estimated structural constants (Votey:1969), the matrices of coefficients for equation 4.4 anej t11e dynamic system of equation 4.3 are computed: 112 . a u S u S flap 0N." UN: e-euu UN“ UHF 01—: U S U a HAHN e r l 3 en 3 new 3 new 3 2.. lflljeeeed ale-Fleas... :86 :86 o :26 e 28.? 336- .IwPICeeeod- TIN-133.c- 386- $36 2... 3.. 3.. new 886 88.7 33.0 326 o 28.2: o 226 32.8 88; 886 98°... 22.3- + 884 886 :86 :3... o 82.3 o 236 Sende- 886 88.7 88.7 :26? .1 name. new «as “as new 332.. 3:2: 1 :- fio o o o o All-«elveeenoé- Tile-«H323. 38.? 386 3.: 3» o o o e e o o 33... 32.? + 0 fl e~ e o o o o o e :86- 386 J Ten Tum. e-uuo To? Tar Tem- e-e 6“. an 326- o o 88.? 386 o o 88...- o awe 7e . mm 33.. o o 28... 28.0- o o 216- Sooo .. 6H0 ‘6“ s 0° r .mL £263 o o 33.? 286- o o 28¢- 28 21113 AOH.¢V e-eeu en ue uH HHH emu emu emu _ 1 _lIII-flfllllllllllllilllll. o o o o o o o o o O o o o o o o o o o o mth.OI wmom.o NemH.o Nme.o mHNn.°l NemH.o NeoH.OI NQQH.OI nhco.Hl Hmn0.n Hnno.Nl Hnn0.Nl Nmmw.Nl MQNN.O NQNN.OI NO~N.OI fl. 4 H o o o H O o o H o o o o o O ohN.o¢l NQaH.OI nth.O an~.0¢ NQQH.O nth.OI o Hnmo.n nnnco.d o ~e~6.o ~nee.~ I. l admo.mem1 o somm.no honm.no I mOHQ.HnoI H¢o~.N~NI mn~.o¢ a-.oc1 o o ~w-.o1 Nmn~.o cheo.dl waam.al N.Hoc I o mnwn.w~a I c nmmm.w~H o mnno.nm¢ I o sous.noNHI c nnna.cdl nmnm.e~ o~on.onl annm.om o c o o o o o o o o nmom.ol ~¢md.cl unno.ml nwnm.01 o o o nmoa.a o wHoo. Homo.01 ~NHO. Homo.c «sac. ammc.o1 ocoo. cowo.o anHo. o o mme.o mHNn.o nmvc.H Nnmw.N o o o o H o o o o1 o o o o 9 COD 0 c mHNn.OI nHN~.o nuco.H «mow.u o o moofl.o o euno.o1 <~n0.o wnnc.o wm~H.o o o ammo.mH ao~d.m oNHm.wN o~am.m~l NNHQ.NOH wmoo.aNH o c NNOO.OI nsoc.cl nnoo.o w~oc.ou nhoo.01 114 Given these two equations, the optimal policy mixes (GIt’ th, rit) for the United States and Canada over 10 periods running from 1961 to 1970 can be derived with the following initial conditions: 383.37 21(0) 12(0) 31(0) = 4.17 20.06 32(0) = 1.24 K1(0) = 508.69 K2(0) = 49.33 G1(0) = 94.90 G2(0) 8 6.97 0.0215 rl(0) where (0) refers to 1960. All these values are historical. The nominal values for state variables are given in Table 4.1. The nominal value for state variables is similar to that in Table 3.1, to which is added the value zero for Country II's balance-of-payments equilibrium. The historical values for exogenous variables are given in Table 4.2. The U.S.-Canadian optimal policies for internal and external balance are obtained by the following steps, which require only some basic matrix manipulations: (1) Compute [Gf(1), G§(l), r1(l)]' from equation 4.9 using the initial conditions: [Y'l(0), 12(0), 31(0). 32(0). K1<0). K2. c1<0). 02(0). r1(oi]' and the exogenous variables of period 1 given by Table 4.2. 115 mHNo.o mmam.oa mq.oca mmmm.cc mem©.mmo o o menm.N qum.Nmm AONmHVOH mHNo.o nomm.m ¢O.mma cecm.¢c ocwm.meo o o mmmw.m nowm.¢mm Amoaava mHNo.o Hmmm.m mm.mNH n¢w¢.No mmmm.¢¢o o o NmHm.N NNNm.on Amomavw mHNo.o NNNH.m ow.¢NH omoo.oo N¢N0.mmo o o Nmmn.N memm.oam Anomavn mHNo.o emaw.w mo.o~a mmom.wn NNoc.noo o o mmme.~ qwmm.eom Accmavo mHNo.o Nowe.w n¢.mHH moma.nm moan.mmm o o wcmm.~ Hmom.¢mN Amwaavm mHNo.o oqma.w mo.HHH MHNm.mm memm.-m o o m~©¢.N mmmw.mm~ Aqomave mHNo.o coew.m 05.00H Naom.mm Nmmw.nmn o o Nnam.N “wom.wmm Amomva mHNo.o wwmm.n mo.NOH qum.~m Nao©.mmm o o mmcm.N mmow.mw~ Amcmavu mHNo.o wmem.n m.wm maom.om noma.mwm o o mch.N NoeN.mwN afloaflva uHm. uwm. ufiM. uwm. ufim. ufil ufil umw uHm Aumowv l .... to.“ Hum ohmHIHoaH "meadow can moumum woufica I mmHnMHHm>omuMum mnu How monam> Hmnfiaoz .H.e oHAMH I .-...!\ J h. I I.DU~A~.H‘I.~ EN) 51NQUAVI-s.n A, V31 .- H ~95 ..-,.- V N pt.‘ ll‘ I a a. . 11‘ {4‘ U‘ C > i ‘ ~0-‘E‘N‘ ? V ~ .\ i - ,. K1 . N N 1‘ d u U 0 1 a 6 I r..~ 1.7 .1 (’2‘? .ah r47. 1 .5 “— 116 55.0 mH.H ow.o mm.o~ «Hoo.a ammo.o eeeo.o oo.~om mmoo.a Ho.o ~e.w m~.o Aenmavoa mn.o an.o oo.o ec.e~ meoo.a emoo.o ammo.o o~.omN wo~o.a ow.m mm.o mm.m Amomavm Ho.o Hm.o ma.n mm.o~ oomo.a mmmo.o «Noo.o Hm.mo~ mwao.a mw.m ea.o NN.o Awomavw mm.o «0.0 N~.o Hm.wH mwmo.a ooeo.o mmmo.o mm.wNN owNo.H oe.e no.m aw.m Anoaavn Hm.o oo.o mo.m mm.oH omeo.a emeo.o cmeo.o mm.mHN emoo.a NH.¢ cm.m nm.m Aoomavc ne.o mm.o mm.m m~.eH mec.H ~wmo.o emeo.o oo.oma oooo.H mm.m om.m m~.m Anomavm mm.o ~m.o ma.e mn.NH oooo.H mmmo.o Nwmo.o mo.¢na moma.o mm.~ m~.m nw.e Aeomave mm.o me.c mw.m mH.HH oooo.H ammo.o wnmo.o em.woa oooo.H ~m.~ mm.e mH.e Anomavm Hm.o me.o -.m mm.OH mono.a mmmo.o ammo.o mo.~mH Hoao.a me.~ om.e mo.e Amomavm Nm.o m¢.o ee.m nm.m woeo.H mNmo.o mamo.o mc.¢ea oooo.H NN.N nm.m mm.m haemava HM: mex “New ewe ”as. Tom“ “we sea :5 Hum: Human new: WWW“ OmmH-Homa "cam now memnmo ..m.D I moHomHum> moooomoxo man How mooam> Hmofiuoumfim .~.e wanes 117 ~ ~ I * * * * * * * (2) Compute [21(1), Y2<1>. Klu). 13(1). (11(1). czm. 5(1)] from equation 4.10. Now [§i(1), 23(1), Bf(1), 33(1), Kf(1), K3(1), I Gi(l), G§(l), rf(li] can be used in equation 4.9 to compute I ~ ~ [Gi(2), G§(2), ri(2)] which can be used to compute [Yf(2), Y§(2), I * * * * * * * B1(2), B2(2), K1(2), K2(2), 61(2), 62(2), r1(2i] and so on. Continue I the process until all the control vectors [Gi(t), G§(t), ri(t)] , t = 1,..., 10 and all the state variable vectors Yi(t), Y§(t), I BI(t)’ B§(t), KI(t)’ K§(t), Gf(t), G§(t), rf(t)] , t = 1,..., 9, have been computed. Unlike the first two cases, the optimal control solution for achieving the overall balance is no longer unique, but is a function of the weights q33 and q44 attached to the deviations of the U.S. and Canadian balances—of-payments from their equilibriums. To study the effects of the trade-off between the two external targets on the optimal policy mix, three experiments will be performed. Table 4.3. Two-country model (Case B): penalty weights attached to the deviations of U.S. and Canadian balance of payments from the equilibrium Trade-Off Experiment q33 944 A 10" 10"5 B 105 105 5 -4 118 First, observations will be made on the relative effectiveness of the impact of three instruments on the internal and external targets. Based on the following matrix (equation 4.9) — — ~ ~ — — Row Ylt Y21: Blt BZt 1 0.3810 -o.0977 o o 2 -o.1315 0.3632 0 o q q 3 0.0028 -o.0021 0.0216(—§§;7——) -0.0216(-éfi:a—-) q33 q44 q33 44 it is noted that the policy interdependency prevails, but in a strict sense within the confines of Votey's model. That is, both fiscal policies affect the internal balance in both countries, but with a positive impact on its own target and a negative impact on the other country's target, while the U.S. monetary policy affects the overall balance positively with respect to the U.S. internal and external targets and negatively with respect to Canada's joint balance. Further- more, the impact of the U.S. monetary policy is greater on the U.S. external target than on the U.S. internal target. Therefore, due to the particular structural form of Votey's model, Mundellian policy assignment to internal and external balance is inherent in the two- country model. It is also noted that the effectiveness of the impact of U.S. monetary policy on the U.S. and Canadian balance of payments depends on the welfare weights q33 and q44: the greater the weights, the greater the impact. Next is an appraisal of optimal policies for internal and external balance in the United States and Canada. The results are presented in graphical form (Figures 4.1 to 4.5) with time on the horizontal axis. J | 119 400... Optimal Path 4 -l-—-u—a Actual Path ' °°°°° Nominal Path 300 — Experiment A r\ Experiment C ~ 2 .2 Experiment B ...; I: 200 -4 .o 8 J l l I l l l 1 I l 0 1960 61 62 63 64 65 66 67 68 69 70 Figure 4.1. Two-country model (Case B) - Effects of trade-off between the two external targets on the U.S. government expenditures: optimal paths compared with actual and nominal paths. 40 -- . Experiment B 1; Experiment A S :H 30— 2 Experiment C E 33 - ee~ m..r‘i.--.§.---fi§ ........... x 0._. - Optimal Path Actual Path -20-—- ""*'* """" Nominal Path J I I l l J L l I I 68 69 70 1960 61 62 63 64 65 66 67 Figure 4.2. Two-country model (Case B) — Effects of trade-off between the two external targets on the Canadian government expendi- tures: optimal paths compared with actual and nominal paths. 120 Optimal Path Actual Path ------ Nominal Path 50-- 40'- ‘ Experiment A .1 4..) 30 --- 8 Experiment C 8 o o- - Ex-eriment B 20 " 1960 61 62 63 64 65 66 67 68 69 70 . Figure 4.3. Two-country model (Case B) — Effects of trade-off between the two external targets on the U.S. interest rate: optimal paths compared with actual and nominal paths. J I 4“. n1.-- .._ h-.?a. E.‘ ‘1.- .....- ILn. ~ nuxn-e1x \ 2.~ ~..\. deficit surplus deficit 121 Optimal IPath *‘H Actual Path 1 """ Nominal Path 6— 5 J 4'-‘ r\ m 8 ...; 3 .- :1 ...; ..D (D- 2__‘ V, ’/////r 1_ ‘ xperiment A Experiment B Experiment C o—IIIbO-oo out-ere. 1960 61 62 63 64 65 66 67 68 69 70 ‘1- ‘ Figure 4.4. Two-country model (Case B) - Effects of trade-off between the two external targets on the U.S. balance of payments: optimal paths compared with actual and nominal paths. 4— Experiment C $ billions) 1 ll ' xperiment B perimfnt A 960 61 62 63 64 V 66 67 68 69 7o 'i—H Actual Path ----- Nominal Path Optimal Path Figure 4.5. Two-country model (Case B) - Effects of trade-off between the two external targets on the Canadian balance of payments: optimal paths compared with actual and nominal paths. 122 The following remarks on the performance of optimal policies can be made: (1) All three optimal policies are sensitive to the trade-off between the two balance-of-payments tar— gets. This trade-off is shown by Figures 4.4 and 4.5. The more the deviation of the balance of payments from its equilibrium is penalized, the closer the external balance. In fact, in experi- ment A where the U.S. balance of payments diverges strongly from equilibrium, the Canadian balance of payments is exactly on the target, and vice versa in experiment C. However, if equal penalty costs are assigned to both deviations (experiment B), both balance of payments diverge equally from the equilibrium. Therefore, the external balances in both countries can never be attained simul- taneously. A trade-off exists between them. (2) The actual paths of fiscal and monetary policies deviate from their optimal ones. This implies that the joint balance is neither attained in the United States nor in Canada during the 1960's. In fact, the United States was faced with inflation and balance-of-payments surplus as was Canada, with the exception of 1965 when Canada had a deficit. (3) When using the optimal fiscal policies, the U.S. and Canadian economies are found to be on the 123 targets exactly no matter what the trade—off between the balance of payment targets. On the contrary, the U.S. optimal monetary policy by itself cannot bring the U.S. as well as the Canadian balance of payments to the equilibrium. This requires additional constraints: heavy penalty costs have to be assigned to deviations from the external balance. (4) It is found that the optimal policies for internal and external balance are inconsistent, i.e., inadmissible for political, social, and technical reasons. In other words, they violate the boun- dary conditions which have been set up to limit the instrument-magnitudes. Therefore, these additional constraints become active by reformulat- ing the cost functional Jc (equation A—2.18) in the two-country optimal control problem (Case B) with the assumption RP 4 0. Next, the amendment of optimal solution generated by the intro- duction of boundary conditions will be examined. The newly defined weight matrix Qc under the assumption RP # O is: 124 Row Ylt Y2t Blt B2t K1t K21: G11: 621; rlt 1 q11 0 0 0 0 0 0 0 0 2 o c122 o o o 0 o o o 3 0 0 q33 0 0 0 0 0 0 4 0 0 0 q44 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 ILjr 6 0 0 0 0 0 0 0 0 0 H 7 0 0 0 0 0 O r11 0 0 8 0 0 0 0 0 0 0 r22 0 r—‘ 9 0 0 0 0 0 0 0 0 r 33 A sensitivity analysis of the weighting factor on components of the control vector is performed. Unlike Case A in Chapter III because of a trade-off between the two countries' external targets, the sensi— tivity analysis will be done for each trade-off experiment with the cost functional given in Table 4.4. Three sensitivity tests will be performed for each trade—off experiment: in test 1, only the deviations of U.S. monetary policy from its constant limit set at 2.15 percent are penalized; in test 2, the deviations of two out of three instruments policies--the U.S. monetary policy coupled with the fiscal policy in either country--from their nominal values are penalized; and in test 3, all three instruments are tracked to their limits. 125 Table 4.4. Two-country model (Case B): penalty weights attached to the deviations of control variables from their nominal values Runs r11 r22 r33 Test 1 1 o o 1 5 2 o o 10 5 3 o 0 5x10 4 o 0 106 Test 2 5 0 l 5 102 6 o 10 5 106 0 5x10 106 o 106 106 7 1 5 o 106 8 1o 5 o 106 5x10 0 106 106 o 10 Test 3 9 l l l 10 105 5 105 5 105 5 11 5x10 5x10 5x 0 12 1o6 106 105 13 1 1 106 14 1 1 5 105 15 1 6 10 105 16 10 5 1 5 105 17 5x10 10 5 105 105 5x10 10 18 1 6 106 102 19 105 - 1 6 106 20 106 105 106 ‘ 10 10 10 600"‘ 500"‘ 400 'f" ($ billions) 126 Optimal Path +H Actual Path Nominal Path 1 l l J l l l l l l O 1960 61 62 63 64 65 66 67 68 69 70 Figure 4.6(A). Two-country model (Case B) - United States: .optimal GNP trajectories compared with the desired and actual GNP (trade-off experiment A where q33 = 10"4 = 5 , q44 10 ). 127 80 “J Run 5 60 Run 7 Run '1 film 3 u: Run “1 40 ‘- q — R 13 20 , 11:: z ‘ Pan 9 -31": E . . o 4:” l .2 69 70 — F4 '1 B U)- -20 -‘ _40 __ Ran ‘ - Run 15' Optimal Path -60 “ * i ’ Actual Path _ """ Nominal Path Run 1! -80 ‘— Figure 4.7(A). Two—country model (Case B) — Canada: optimal GNP trajectories compared with the desired and actual GNP (trade-off experiment A where q33 = 10'4, q44 = 10 ). 128 Optimal Path '-7 ------ Nominal Path ’8 8 -H -H F4 F4 -H .n d 1’3 Run 1,...,20 _r ----------------- o ................................ I I r l T I F I 7| T 1961 62 63 64 65 66 67 68 69 70 Figure 4.8(A). Two-country model (Case B) - U.S. balance of payments: optimal paths com ared with the equilibrium (trade-off experiment A where q33 = 10' , q44 = 105). Optimal Path '1 d A U) {3 -—..S H ... 'H E "‘ <0- V d1-ocoerVVV—Vjvv-T‘}.v‘j‘j‘——-tuwur- I T I I | I I 'l l I 1960 61 62 63 64 65 66 67 68 69 70 Figure 4.9(A). Two-country model (Case B) - Canadian balance of payments: optimal paths compared with the equilibrium (trade-off experiment A where q33 = 10‘ , q44 = 105). 129 Figure 4.10(A). Two-country model (Case B) - U.S. government expenditures: optimal paths com ared with nominal path (trade- off experiment A where q33 = 10' , q44 = 105). Figure 4.11(A). Two-country model (Case B) - Canadian govern- ment expenditures: optimal paths compared with nominal path (trade— off experiment A where q33 = 10'4, q44 = 105). Optimal Path """ Nominal Path 130 Run 1: 300"' ,\ Ru": 0) 5 at, + :3 Run? 2 / Rum 0,53 14, u '0 Run 7 (l)- 200"' \I + . . . o 100"4 l l 1961 69 70 Figure 4.lO(A) _ n"? M 3171' M3 Cunt -20 '- ($ billions) -----:-~ RI” 5,15 18 v " --- -' Optimal Path ------ Nominal Path Figure 4.11(A) 10" I .- ..h. ' ...»: has“! to‘fllzrl... 4N‘fi 131 50"1 Optimal Path - —----- Nominal Path 40"‘ 3O " Run 15 N <3 I 1 Percent 13 C a p Run; ‘,19 Run). 10 " an” Run! Run! .1 "—_?' ' Rub O I l I l l 1 I l l I Figure 4.12(A). Two-country model (Case B) - U.S. interest rate: optimal paths compared with nominal path (trade-off experiment A where q33 = 10'4, q44 i 105). 132 (i) Trade-off experiment A [Figures 4.6(A) to 4.12(A)] In test 1 (runs 1 to 4) it is observed that the tracking for the U.S. nominal interest rate gets closer the greater the value given to the welfare weights r Because of the policy interdependency 33' (equation 4.8) the U.S. and Canadian fiscal instruments are also track- ing their nominal paths. However, the tracking for U.S. nominal government expenditures is positively correlated with that of the U.S. nominal interest rate, while there is a negative correlation between the latter tracking and that of the Canadian nominal government spending. Furthermore, it is found that the optimal path of U.S. balance of payments on which the U.S. monetary policy has the greatest impact remains insensitive to the tracking for the U.S. nominal interest rate; and only for the three other endogenous variables, ~ Ylt’ §2t’ and B2t is there a trade-off between attainment of targets and feasibility of the U.S. monetary policy. In test 2 (runs 5 to 8) it is found that the additional boundary condition reinforces the tracking for both the nominal U.S. interest rate and Canadian government expenditures, but only if a very low weight is attached to deviations of U.S. government spending from their nominal path (r22) compared to that of U.S. monetary policy (r44). Otherwise, there would be deviations from both nominal paths and the amended rlt’ GZt optimal control solution could be explosive. Similar to test 1, it generates a trade-off between attainment of three other targets, U.S. internal balance and Canada's overall balance, and the tracking for the nominal values of two instruments--U.S. fiscal and monetary policies or on Canada's fiscal policy and U.S. monetary policy. 133 The results of test 2 are also found in test 3 in which the limits are set upon all three instruments. (ii) Trade-off experiment C [Figures 4.6(C) to 4.12(C)] This is the opposite case of trade-off experiment A, in which a high penalty cost is assigned to deviations of U.S. balance of payments from its equilibrium. The same results are obtained, with one exception. Since the external balance trade-off between the United States and Canada has been reversed (the United States attach more importance in welfare weights to the performance of its external balance than Canada does), it is found that the tracking for limits on policy-magnitude leaves the optimal path of Canada's balance of payments unchanged [Figure 4.9(C)]. However, the tracking affects the attainment of external balance in the United States by using the inconsistent or nonfeasible optimal policies. Furthermore, trade-off experiment A assigned heavy penalty costs to deviations of Canada's balance of payments from its external balance, resulting in a surplus of Canada's balance of payments at the expense of tracking for limits on policy-magnitudes. Unlike experiment A, it is found in the trade- off experiment C that the tracking for the nominal policies brings a deficit in the U.S. balance of payments. (iii) Trade-off experiment B [Figures 4.6(B) to 4.12(B)] When both countries give equal importance to the achievement of their external balance, a trade-off occurs between attainment of the four targets--internal and external balance in the United States and Canada--and the feasibility of all three policies. In other words, to 134 Figure 4.6(C). Two—country model (Case B) - United States optimal GNP trajectories compared with the desired GNP (trade-off experiment C where q33 = 105, q44 = 10'4). Figure 4.7(C). Two-country model (Case B) - Canada: optimal GNP trajectories compared with the desired GNP (trade-off experiment C where q33 = 105, q44 = 10'4). Optimal Path 4 "" " Nominal Path 135 ’3 300‘“ g ...I H PI ‘H ‘ .o 3 200'-" Figure 4.6(C) . 1 60 -# Run‘ Run! .1 all!!! Role Run? 40... J A Bold} U) a .o _ ....4 20 —- 2‘. Run: ...; ..D {O- - V Rafi - ___ '__ v___"‘Rm1 ' """"""""" X I 0 r I T I T I 1 I 1961 62 63 64 65 6 8 69 70 q Optimal Path -20 -- Nominal Path q -4o 1 Run it Figure 4.7(C) 136 Optimal Path 4 ° """ Nominal Path 0 .--. 1+ ”a? ‘ a o H e4 ...; H 1"“ a ‘3 q 2 I I l l l l I I l T 1960 61 62 63 64 65 66 67 68 69 7O . Figure 4.8(C). Two-country model (Case B) - U.S. balance of payments: optimal paths compared with the equilibrium (trade-off experiment C where q33 = 105, q44 = 10'“). z—d —— Optimal Path '°"" Nominal Path Runs 1,...,20 ($ billions) 1961 62 63 64 65 66 67 68 69 70 Figure 4.9(C). Two-country model (Case B) - Canadian balance of payments:' optimal path compared with the equilibrium (trade-off experiment C where q33 = 105, q44 = 10‘4). 137 Figure 4.lO(C). Two-country model (Case B) — U.S. government expenditures: optimal paths compared with nominal path (trade-off experiment C where q33 = 105, q44 = 10‘“). Figure 4.11(C). Two-country model (Case B) - Canadian government expenditures: optimal paths com ared with nominal path (trade-off experiment C where q33 = 10 , q44 - 10'4). Optimal Path 138 J ------ Nomi a1 Path RIM" 300 Rant. 3 ‘Ilfl' s 2““: " :1 «113 6." E-SI RUM 7!“ ._J 200 3 Run, 4,5 100 - 1961 70 Figure 4.10(C) In: 1 Optimal Path ”an" In" ------ Nominal Path / “a“ Run: 20 A ’8 - I: o H .... d 10 __i ..D ____-::.-=RII$‘,‘I$,W {D- 0 I I I J 1961 69 70 -10 “i Figure 4.11(C) -20 '- 139 Optimal Path ----- Nominal Path 50 " 40'-1 Percent 20 I I l I l I I -I l l 1961 62 63 64 65 66 67 68 69 70 Figure 4.12(C). Two-country model (Case B) - U.S. interest rate: optimal paths compared with nominal path (trade-off experi- ment C where q33 = 105, (144 = 10'4). HIJ 4!in . .vfiute." . ...6. If... 1* 140 Optimal Path - ..... Nominal Path ’3 zoo—J 8 'H e :3 \ H “II"' 33 \\\\ . t \ \- 100"' RM” "'32, 20 Rm ’,n Rania," 0 \ I I I ’I’ I I I I I I 1960 61 62 63 -64 65 66 67 68 69 70 Figure 4.6(B). Two-country model (Case B) - United States: optimal GNP trajectories compared with the desired GNP (trade-off experiment B where q33 = 105, q44 = 105). 141 Optimal Path Nominal Path 80 "'" a... s . Run 16,19 \V A ‘ RHI‘ . ‘ 7! "r' \___.. -—— RflxzugIG I' v 1 \ A :1 It.“ -1 _Y:____ ,1_r_i_:r:.m...t."~ 0 - _ 1 13 a '1 .3 .._I v-I 'I-I .D -40 -# (0 -80 -4 —120 —4 Rum Io,u,I2,I1,zo —160 I l L l L, J, 11 CL L_ 1961 62 63 64 65 66 67 68 69 70 Figure 4.7(B). B where q33 = 105 = 105). : Q44 Two-country model (Case B) - Canada: GNP trajectories compared with the desired GNP (trade-off experiment optimal 142 Optimal Path ------- Nominal Path 40 - d 30I- _ . I! I + R::|1 20fi- s I, . .J g / Run“ PI 2 c / 10"‘f3 // Rumba I RUMW,'71 to + d , Rhwlyfl 0 — ooooooooooooooooooooooo - m -10 --i 5:5 If alt .. Stain," Run‘ . “has '20 "" Runs 3,9,3' 1 1 I l I 1 I l l g 1961 62 63 64 65 66 67 68 69 70 payments: experiment B where q33 = 10 Figure 4.8(B). optimal paths co Two-country model (Case B) - U.S. balance of gpared with the equilibrium (trade-off a Q44 = 105)- 143 Optimal Path Nominal Path Run ’0 33:2 20"II Run!- Rhnb “In"! J hall Raul! lO«- 4 / RUIS'Jq O—P---‘ -" -..-C-OO-OOOOCOO 0- O -.--boo-onoouo ’5? a o _ '1-4 j “\ Ramon," .... kahuna.» ..D -1o-I 3 q [R I4 _20 '_1 an I ‘ Runsl,FI -3O " l J I I I, I J, 11 I -40 I 1961 62 63 64 65 66 67 68 69 70 ’ Figure 4.9(B). Two-country model (Case B) - Canadian balance of payments: optimal paths compared with the equilibrium (trade-off experiment B where Q33 8 105, q44 = 105). Figure 4.lO(B). 144 Two-country model (Case B) - U.S. government expenditures: optimal paths compared with nominal path (trade-off experiment B where q33 = 105 Figure 4.11(B). ment expenditures: off experiment B where q33 = 105 = 5 , q44 10 ). Two-country model (Case B) - Canadian govern- optimal paths compared with nominal path (trade- : 5 , q44 lO ). Optimal Path ..... Nominal Path 145 300'- Run I ,/ RumL‘nS .. 1; Run 9 Run 1: '3 Runs guns," .2 .4 200 —> E <1} \\ " ‘ e/ “I' Run: 830,», 11,“, I1, n, 2.0 1 1 I I I l l I I I 1961 62 63 64 65 66 67 68 69 70 Figure 4.10(B) 8° '1 I ‘ Optimal Path - ~ " '°-' Nominal Path ‘ “an? 60 — Rants,” 1 40 '1 g H :1 Rhn' ‘ :3 Run! 8 \ 20 "1 3 HmnISII .' 2 /\ I am am I1. If, n I! to ..‘.‘1" " h"' //3,)9I: \\ I“. A / O R /' ‘\\\\///r ' l J l l l l I I__ —20 l 1960 61 62 63 64 65 66 67 68 69 70 Figure 4.11(B) 146 Optimal Path - ----- Nominal Path 50‘- 4 ‘1 40— ‘- ‘ 30"1 ‘é _‘ 8 Paul 3.. o n. 20'-‘ q lO-# d M" R |I LTAJh‘L a“ -----.---z----T-----' ----- Runsiflflmm'hflo“ fifi‘~“““-=====::J§E‘“5” 0 / 1 l 1 I l 1 1 I l I: 1961 62 63 64 65 66 67 68 69 70 Figure 4.12(B). Two-country model (Case B) - U.S. interest rate: optimal ath compared with nominal path (trade-off experiment C where q33 = 10 , q44 = 105). 147 track the two nominal fiscal policies (4 percent growth per annum) and the nominal U.S. interest rate (2.15 percent), both the United States and Canada must give up their targets of internal and external balance--def1ation coupled with balance-of-payments deficit in the United States and inflation paired with balance-of-payments surplus in Canada. Under the assumptions of fixed exchange rate, passive responses from the second country and linear dependent balance of payments, it is concluded: (1) Within the optimal control framework, if the number of variables to be controlled is larger than the number of instruments, the variables will not reach the targets exactly, and their deviations from the targets will depend on the welfare weights in Qc assigned to them (Chow:l972c). As a result, the solution for simultaneously achieving internal and external balance in both countries is no longer uniquely determined. Within the confines of Votey's model where an inherent Mundellian policy assignment prevails for internal and external balance, only the variables of balance of payments (B BZt) are not on target exactly since there lt’ is only one instrument, Country I's interest rate, to hit both external balances. Therefore, the solution obtained by the optimal control techniques is found to be a function of trade-off between the (2) (3) 148 two countries' balance of payments or, more specifically, a function of welfare weights q33 and q44 assigned to their deviations from equilibrium. Furthermore, unlike the two preced- ing cases in which the optimal cost 3c is zero, the trade balance of the third country with Countries I and II as well as the welfare weights q33 and q44 determine the optimal cost 3c (equation 4.8). The larger the welfare weight q33 (q44), the closer the variables B1 (BZt) reach the equili- t brium targets. A trade-off exists between the two external targets. For one country to hit its external balance the other one must consent to attach only a small importance to its own external balance. Otherwise, if both attach equal importance to the attainment of their own external balances, neither will reach the targets and both will face a balance-of-payments disequilibrium. When boundary conditions become active, a second trade-off occurs between attainment of targets and feasibility or consistency of policy-instruments, in addition to the trade-off between the external targets. CHAPTER V TWO-COUNTRY MODEL CASE C: CONFLICT OF INTERESTS AND GAME THEORETICAL APPROACH This chapter introduces a new framework for analyzing the problem of internal and external balance under the following assumptions: (1) Economic interdependence between two countries, (2) Active responses from the second country, (3) Conflicting internal as well as external targets pursued by both countries but with the possibility of international cooperation. The case of conflict of interests can be resolved by presenting a systematic and concise introduction to a two-player or controller multistage non-zero—sum game with linear quadratic system and perfect information will be presented. Its application to policy mix for internal and external balance will be left to future research, because more realistic assumptions will imply a more complicated model than that of Votey. According to Y. C. Ho (Ho:l970), Differential Games (DG) is nothing but an extension of optimal control theory within the context of "Generalized Control Theory" (GCT) which incorporates all optimi- zation problems possessing the three main ingredients: the criterion 149 150 function, the controller (or player), and the information set available to the controller. As in the optimal control problem, the differential game problems consist of finding strategies which minimize the Performance criterion subject to the constraints described by the state system, with the difference that there is more than one controller. Furthermore, in the non—zero-sum (NZS) game the players' interests are not in direct conflict. Therefore, contrary to the zero sum (ZS) game it is no longer possible to have a unique definition of optimality (Starr and Ho:l963a, 1963b). In this chapter, the noncooperative (or so—called Nash) equilibrium strategies will be determined first, then the inferiority of Nash equilibrium strategies will be proved, which leads to the search for a noninferior solution in particular the so-called Pareto Optimum Strategy. 5.1. Formulation of the Problem The two-player NZS game is defined by: (l) A linear, time-invariant, dynamic system, so—called "kinematic" equation (Isaacs:l965): x = Axt + B u +IB u v+ICz (5.1a) t+1 1 1t 2 2t t x - xt = Fxt +'B u + B u + C2 (5.1b) 1 1t 2 2t t 1.. t+l where ern is the state variable, ultErl is player (Country) I's control variable, uzeErZ is player (Country) II's control variable, zeES is the exogenous noncontrolled variable, r—‘u. 151 A = I+F is a nxn matrix, B1 is a nxr matrix, 1 B2 is a nxr2 matrix, C is a nxs matrix, and all the matrices are assumed to be time-invariant and known. (2) Quadratic performance criteria: 1 N T _. T J1 = 5':=0{[xt7it] Qllxt-xt] + u1t Rllult T (5.2a) + u2t R12u2t} J2 g %-§=o {[xt‘iilTQ2[xt‘i£] + ultTR21ult T (5.2b) + “2c R22u2t} where J1 and J are players (Countries) I and 2 II's performance criteria to be minimized; Q1 Q2, R11, R12, R21 and R22 are assumed to be symmetric; and R are positive definite R11 22 matrices; and Q1 and Q2 are non-singular matrices. Therefore, the problem consists of finding a pair of strategies (61,62), called Nash equilibrium or non-cooperative equilibrium strategies such that: 152 5.2. Determination of a Nash Equilibrium Set (filial; 5.2.1. Definition Following Starr and Ho (1963a), the strategy set a = (61,..., fik} is called a Nash equilibrium strategy set if for i = 1,2,...k: ~ ~ ~ < ~ ~ ~ Ji(ul,...,u1,...,uk) Ji(u1""’ui"°"uk) for all admissible u that is, for controls which belong to a Euclidian 1, space. 5.2.2. Necessary Conditions for a Nash Equilibrium Solution Starr and Ho (1963a, 1963b) derive the necessary conditions for a Nash equilibrium in the continuous case while Haurie's work (1970, 1971) is in connection with the discrete case. Applying Haurie's results, a Nash equilibrium strategy pair for the linear quadratic NZS game problem is obtained by solving the following equations: 631t+1 .. _ 5n . xt _ 0 (5.3a) 1t .. “it 538 l 5 2t+1 i = O (5.3b) u2t t l12:: agelt+l 5592131 it+1 " "t g “S'— = T“: Fxt+Blfilt+BZfi2t (5'4) 1t+1 2t+1 T A _ X = _ 6% 1t+l+ 63Et+1'S 6u2 2t (5.5a) 1c+1 1t Ts—x 6x t t 12t+1 - AZt = - 2t+1 + 6§t+1 6x1t (5.5b) xt 1t with the boundary conditions: 2(0) = RC (5.6) 11m) = Q1[2(N) - am] (5.73) 51201) = Q2[i'c(N) - 5200] (5.71:) 2 where 3'81: +1 and 3'6 2 t +1 are the respective hamiltonians for the two players. 5.2.3. Determination of the Solution Along the optimal path, the Hamiltonians for the two players are defined as follows (Starr and Ho:l963a, Haurie:l970, 1971): __1_[ __ T .. __ .. T .. g£1t+l _ 2 (it Kt) Q1(xt xt) + u1t Rllult+fi2t (5.8a) R12u2t] + A1t+1 Fxc + Blult+B2fi2t+Czt] 38 =-1-(i-i)TQ(fi-i)+fi TRii-h‘i 2c+12 tt2tt 1t211t2t (5.8b) ~T ~ ~ ~ R22fi2t] + A2t+1 [Fxt + B1"1‘.;""32“21:+Ct] where X are co-state variables. lt+1’ A2t+1 154 Then, solving equations 5.4a and 5.4b yields ~ _ -l T ~ “1t ‘ R11 B1 A1t+1 (5°98) ~ _ _ -1 T ~ “2: " R22 B2 A2t+1 (5'9b) Therefore 3(1t+1 and 2t+1 are at their minimum with respect to u1 and u2 since R11 and R22 are positive definite. Since the performance criteria are assumed to be quadratic, the closed-loop Nash equilibrium strategies of the form fi = S (it) + s can be derived (Starr and Ho: it :1 J 1963a). Similar to the linear quadratic optimal control problem, it is known that (Lee et aZ.:l972, Athans and Falb:1966): A1t+1 = Klt+1 it+1 + k1t+1 (5°103) k (5.10b) A2t+1 K2t+1 itt+1 + 2t+1 Substituting equations 5.10a and 5.lOb into equations 5.9a and 5.9b: - _ —1 T ~ -1 T “1t ’ R11 Bl K1c+1 xn+1 R11 B1 klt+1 (5.11a) d - ' D1t+1 xt+1+ d1t+l - _ _ -1 T 1 _ -1 T “2: ‘ R22 B2 K2t+1 xt+1 R22 B2 k2t+1 (5.11b) d D2t+1 xt+1 + d2t+1 Again, substituting equations 5.lla and 5.1lb into the "kinematic" equation 5.1b: 155 i = Fit + B D t+1-xt 1 1t+1 it1+1 + B2D2t+l 5tn+1 + Bldlt-I-l (5.12) + B2d2t+1 + Czt 1 d . Since [Gt+J- = [I-B1D1t+l-B2D2t+l] exists for all t, t {O,...,N—l}, then: .. -1 .. 1 -l xt+1 [CHI] Ax: + (Gt-I- Bldlt+1 + (Gab -1 g B2"21:+1 +(Gt+]) CZt ’ A- -1 ~ -1 Q4 ”(GtH) AXt + gt+1 + Gt+l CZ: I + F (5.13) Substituting equation 5.13 into equations 5.11a and 5.11b: CI l -1 1t " H1t+1 it + h1t—I-1 + Dlt+l (Gag CZ: (5’1“) CI ll .. -1 2t H2t+l xt + thH + D2t+1(Gt+9 Czt (5.141)) where “Q -1 H1t+1 Dlt+1 (Gm) A "D: 1c+1 D1t+1 gt+l + dlt+1 no. -1 H2t+l D2t+1(Gt+1) A [ID- 2t+1 D2t+1 gt+1 + d2t+1 Solving the canonical equations 5.5a and 5.5b, and rearranging the terms: 156 = [Q +11T R H 1]}? +A [G- A +HT R h It 1 2t+1 12 2t+ t c+1 1t+1 2t+1 12 2t+1 T 1 (5.15a) + H2t+lR12D2 t+1Gt+1Czt-Q1xt >J2 >’I l x +A T[G-11]T~ T T t+ 1 +H 2t ' [Q2+H1t+1R 21 H1t+1]t 2t+1 1t+1+H1t+1R21Dlt+1 (5.15b) G'lc t—xQ t+1Cszt Substituting equations 5.10a and 5.10b into equations 5.15a and 5.15b: K 3t+k T 1t t 1:: =T[Q1+HZt+1R12H2t+l+A (Gt- lTKl) lt- 1G t+1A] tilfr K +k + HT R (5.16s) i tT+A (G 1t+1gt+1 1t+l 2t+1 12 h +AT (G HT 2t+1 “if 2t+1 R12 D21:+1 t+1) Kn+1] _1 _ Gt+1 CZI: - Q1 xt T TK K2tit+k2t ' [Q2+H1t+1R21H1t+1+AT(Gt-1)K2t+1Gt-1A] 1 T -1 T T “c” (Gt-1) K2t+lgt+1+k2t+l + Hlt+l (5'16” -1 T _ Rh21 1t+1+[let+1R 21D t+l+TA (Gt+1)K K2131161131 CZ: Q2 ’5: Then equating the coefficient yields: -1 T -1 _ T Klt ‘ Q1+H 2t+1 R12H2t+1+A (Gt+1) K1t+1Gt+l A (5'173) = T -1 T -1 K2t Q2+Hlt+l R21H1t+1+AT(Gt+1) K2t+1Gt+1 A (5'1”) 157 T -1 T klt = A (Gt+1yTEFlt+lgt+l+klt+1] + H2t+1 R12 h2t+l (5.18a) + [H2131 R12D2t+1+AT(G;il)TK1t+1J G31 C21: _ Q1 it k2: = AT(G;11)T[K2t+lgt+l+k2t+1] + H§t+l R21 hlt+1 ( ) 5.18b + [H1t+l R21 D1t+1+ATK2t+1] G31 CZ: ‘ Q2 it And K1N = Q1 (5.19a) K2N = Q2 (5.19b) k1N = Q1 ET (5.20a) k2N = Q2 ifi (5.20b) Summary of the solution The Nash equilibrium strategy pair (31, flz) is found by the follow- ing steps: (1) Solve the Riccati equations 5.17a and 5.17b with their boundary conditions 5.19s and 5.19b backwards in time to get values for Kt’ t = 1,...,N. (2) Solve the tracking equations 5.18a and 5.18b with their boundary conditions 5.20s and 5.20b backwards in time to get values for kt’ t = 1,...,N. (3) Compute the Nash equilibrium strategy set u(O) = {fi1(0), fi2(0)} from equations 5.14a, 5.14b: [IO- D "O- D "D- D no. D "G- no. IICL no. -1 g (G131) " d = H2 t+l xt -R B 158 .. _-1 ' Hlt+1 xt +[h1t+1+D1t+1Gt+1 at] (5.14a) ~ + [h (5.14b) -1 2t+l+D2t+1Gt+1 czt] —1 lt+1 (Gm) A -1 2t+1th+)\.) A lt+1 gt+1 + d1t+1 2t+1 gt+1 + d2t+1 T 11 1 K1t+1 K21;+1 -R B k 11 1 lt+1 k -l T 2 2t+1 [I‘BID1t+1 ' 1321231]-l _ -1 81+1 {Gab [B1d1t+l + BZdZt-l-l] using i(O ) = i Then compute x(l) from equation 5.4. Now i(1) can 0. be used in equation 5.14s and 5.14b to compute u(2) é {fi1(2), fi2(2)} which can process until all of the u g {a the it, t = be used in equation 5.4 to compute 1(2), etc. Continue this t It, fiZt}’ t a 0’ 1,000,N-1 and all Of 1,...,N, have been computed. r4 159 ~ ~ (4) The costs J1, J2 can be computed from equations 5.2a and 5.2b. 5.2.4. Principle of Optimality in the NZS Game The strategy set u is optimal in the Nash sense; that is, the equilibrium solution is secured against any attempt by one player unilaterally to alter his strategy. If every player is using his Nash control, and if a given player plays non-Nash optimally, he will do no better. The conditions of Section 5.2.2 are necessary for the Nash equilibrium strategy set to exist, but these solutions are not pro- tected against cheating and thus are unstable in a noncooperative sense. In other words, it is possible for the "prisoner's dilemma" to occur and the optimality is non-unique in the NZS differential game (Starr and Ho:l963a, 1963b). Therefore, there are solutions other than Nash's, such as noninferior. solutions and minimax solutions. This study is concerned with the noninferior solutions, and in particular with the Pareto optimal strategy set. The noninferior solution has the property that it is not dominated by any other solu- tion point in its neighborhood. In other words, any deviation from the noninferior solution cannot result in simultaneous improvement of all Ji's (i = 1,...,k). First, the sufficient conditions for the Nash equilibrium to be inferior are derived to check if Nash strategies are inferior. If they are, the Pareto optimal solution, which is noninferior and therefore superior to the Nash solution, will be determined. 160 5.3. On the Inferiority of Nash Equilibrium for a Two Player Multistage Game 5.3.1. The Basic Sufficient Condition for Inferiority EEri, r Consider k players with respective controls ui 1 given integer and respective criteria: J1: Er+R 1 = 1,...,k. where r = r1 + r2 + ... + rk. Definition 5.3.1: The control utEr is inferior if there exists ueEr such that: Ji(u) s Ji(fi) for all 1 {1,...,k} Jj(u) < Jj(fi) for some j {1,...,k} Lemma 5.3.1: (Rekasius & Schmitendorf:197l) If each functional is C1 in a neighborhood of u., then a suf- ficient condition for u to be inferior is that there exists a vector k h E such that: h < 0 (5.21) and r[A] = rIAE h] (5.22) where A is the following er matrix: 1 ~“IS :11 d o A = 3 6 3ek ~ ('5 u 161 and “1’ . . . ’mk are the Hamiltonians: 19:29 i (x,ui, Ai’ t) i = 1,...,k. 5.3.2. Application to Two-Player L.Q. Multistage Game Consider the linear system: xt+1 = Axt + Blult + B2u2t + Czt and the quadratic criteria 1 N T T J1 = 2 t=0 [Kt-xt] Q1363] + “1c R11‘111: T + “2c R12 “2t __1 J. T _ T T J2 - 2 =0 [xt xt] Q2 [xt StI: + u11: RZlult; + u2t R22 u2t ' Construct the matrix A: where 1 “lt+1 = 2 [xt A1t+ 53? l 6u1 a}? 6u 2 1 all? 6u2 6392 Gu l 2 Ti+uQ1 t It R11mm 2tT Ru12 “2c] 1_[Fit + 131 1 +B2fi2t + Czt] 162 _1 * T 2:;TEX szt + “1:: R21 “1: “1 TR“22 ‘11:] + 2t+1 ‘38 2t+1= [Fxt + Blu1t+B2u2t + Czt]2 Since G 3 (El, fiz) are a Nash equilibrium, the following conditions are necessarily satisfied: 1r 1 2 ul 0 L 1 where: 6gPlt+l ‘ fiT R + BT i 6n 2t 12 2 -1t+l 2t ———6382t+1 == fiT R + 13T i Gult 1t 21 1 2t+1 According to the previous results (Section 5.2.3), a Nash equilibrium is given by: .. _ -1 T ~ “1c‘ ‘R11 B1 Alt+1 .. _ _ -1 T ~ “2: ' R22 32 "2c+1 163 therefore, it is rather unlikely that 52'? 5'38 1 2 6u2 6u1 Thus A is non-singular and has a rank 2. And rankI:AE h] = rank A = 2 for h < 0. By Lemma 5.3.1, fl is inferior; that is, the Nash solution of a two-player NZS game will usually be inferior. 5.4. Determination of the Pareto-Optimal Set Nash equilibrium strategy set (fil, fiz) has been shown to be inferior and it is assumed that the two players agree to cooperate. Therefore, the two-player multistage NZS game is reduced to a vector- valued criterion optimal control problem in which the decision maker is a team of two players, and several optimality criteria are relevant to the players, although their relative importance is not obvious. Different approaches to the vector-valued criterion optimization problem.have been developed. Basile and Vincent (1970a); Vincent and Leitman (1970b); Leitman, Rocklin and Vincent (1972); and Stalford (1972) used a geometric approach based on the concepts of convex cost cones to derive the Pareto optimal set for both static and continuous games. Schmitendorf et a2. (1971, 1972, 1973) also derived the neces- sary conditions for Pareto optimal solutions for static and continuous games. This approach does not require any local convexity assumptions like the method of Vincent et aZ., but is based instead on the rank conditions of the cost matrix. Above all, the most popular technique is the scalarization technique, where the vector performance index problem is converted into a family of scalar index problems by forming an auxiliary scalar index as a function of the vector index and a r‘é- 164 vector of weighting parameters (Zadeh:1963, Klinger:l964, Da Cunha and Polak:1967, Starr and Ho:l963a, 1963b, Reid and Citron:l97l, Haurie: 1970, 1971, 1973). Haurie's approach and results (1970) will be applied to the two-player multistage NZS game with LQ system, perfect information and complete cooperation. 5.4.1. Definition A set of control functions u g (0 62,...,fik) is said to be 1’ Pareto optimal if for each set of control functions u g (ul, u2,...,uk) there exists: either Ji(u) = Ji(fi) for all i {1,...,k} or at least for one i {1,...,k}: Ji(u) > J1(u) 5.4.2. Scalarization Procedure The Pareto optimal set 6 could be obtained by minimizing the following scalar performance criterion J = 111.11 + 11sz “1 + “2 = 1 ul>0.u2>0 provided that the set of cost vectors (J J2) generated by all the 1’ admissible controls is convex (Starr and Ho:l963a). Therefore, the problem consists of finding a set of controls u(u , x,t) g u = (fil, 62) 3 such that: 165 ”1‘11 (6) + 11sz (G) < “1J1 (u) + 11sz (11). 5.4.3. Determination of the Solution The problem of finding the Pareto optimal set a is formulated as follows: minimize the performance criterion d 1 N T T J = “1.11 + 11sz = 2 i=0 {[xt-xt] Q[xt-Xt] + ut Rut} (5.23) given the dynamic system xt+l - xt = Fxt + But + Czt (5.24) where "1Q1 0 Q: L 0 u2Q2 J F '7 u1R11+“2R12 0 R: O “1R21+“2R22 D. B = [31, 32] u 3 [ul, uz] To get the necessary conditions for the Pareto optimal solution, the Hamiltonian is constructed: 166 _ 1_ —- T 4. T “T . Mal - 2 {[xt-xt] Q[xt xt'] + ut Rut} + At+1.[Fxt _+ But + Czt] (5.25) Similar to the linear quadratic optimal control problem, it is known that (Lee et aZ.:l972, Athans and Falb:1966): A gK At+1 t+1 xt+1 + kt-.+ 1 Then the minimization of the Hamiltonian is written as follows: 53.9 t+l_ . T. = Sut — Rut + B lt+1 0 and the canonical equations are: 619 t+1 _ = __——_—-= + xt+1 xt 52 Fxt + But Czt t+1 i —i =.6_¥£:11=_Qfi_f _FTX t+1 t dxt t t t+1 with the split boundary conditions: x(0) = x0 AN = QbLN-KN) From equation 5.27: _ -1 T" “c"R B At+1 Substituting equation 5.26 into equation 5.32: A _ _ -l T _ -1 T xt: - R B Kt+1 itt+1 R B kt+1 Substituting equation 5.33 into equation 5.28: A A _ A _lT xt+1 xt - Fxt BR B K A '1 T t+1 "t+1"BR B kt+1 + CZ: (5.26) (5.27) (5.28) (5.29) (5.30) (5.31) (5.32) (5.33) (5.34) 167 Rearranging the terms, equation 5.34 yields: A _ -1 A -1 -l T _1 xt+1 — W Axt - W BR B kt+1 + W Czt (5.35) where: A = I + F g -l T W - I + BR B Kt+1 -1 g -1 T -l W - I + BR B Kt+1 Substituting equation 5.35 into equation 5.33: 1 T 1 1 . - -1 . - — T -l -l T ut - -R B Kt+1 [w Axt-W BR B kt+1 + w Czt] -R B kt+1 (5.36) Now re-examine W-l. Using the matrix identity (Pindyck:1973), W— becomes: -1 _ T -1 T w - I—B[R+B Kt+1BJ B Kt+1 (5.37) Substituting equation 5.37 into equation 5.36: A _ -1 T T -1 T . -1 T ut - -R B Kt+1[I B(R+B Kt+1B) B Kt+IJAxt + R B Kt+1 (5.38) T -1 T -1 T -1 T T -1 [I-B(R+B Kt+lB) B Kt+1] BR B kt+l-R B Kt+1 [I—B(R+B Kt+lB) T -1 T B Kt+1]Czt-R B kt+1 And after rearranging: t R {1 B Kt+1BLR+B Kt+lBJ } B Kt+1Axt + R {I-B Kt+1B (5.39) T —1 T -1 T -1 T —1 T [R+B Kt+1BJ } B Kt+lBR B kt+1 - R B kt+1-R { I—B Kt+lB T -l T [R+B Kt+1BJ } B Kt+1 Czt. 168 Now, using the identity: I-x (T+X)'l = Y(Y+X)-l (5.40) Letting X = BTKt+ B and Y = R, simplify equation 5.39 to: l ’1 BTK Ait + [R+BTKt+lB] 1 t+lBJ t+1 '1 BTK BR- . _ T T ut [n+3 K t+1 B kt+1 1 T T -1 T B kt+l - [R+B Kt+1BJ B K Cz - R t+1 t Next, solve the second canonical equation 5.29: A Substituting equation 5.26 into equation 5.42: A A _ A T = _ T Ktxt+kt Q[xt xt] + A Kt+1xt+l+A kt+1 After rearranging the terms: T A A Kt+1xt+1 Substituting equation 5.35 into equation 5.44: T {- A Kt+1 W A A T Czt} + th - Ktxt — -A kt+l 1 l -l T -1 BR B kt+ + W Axt-w 1 (5.45) + kt + Qit. Rearranging the terms: (5.41) A T A A T A A = -_ A = —— t Q[,xt xt] + (I+F) t+1 Q[xt xt] + A lt+1 (5.42) (5.43) . A _ _ T + th-Ktxt - A kt+l + kt + Qii. (5.44) (5.46) 169 Equating the coefficients of the left-hand side and the right—hand side yields: Kt = Q + ATRt+lw'1A (5.47) kt = -.4T1 B Q yt - (B 'Q B ) BP'QP 3' * ut: (BP,QPBP)-l 31>,st “t Then u: is given by: “4.13) [(BR,Qpo)-1 BP.QPEJU: .. _(BP.QPBP)-l 3?.pr YZ-l + [(BP'QPBP)'1 BP'QPX B - I] u§_1 + (BP'QPBP)-1 BP'QPy't P P P -1 BP - (B 'Q B ) P ' ._ Q D zt. 204 (A_3.14) u: a: [(BP'QPBP)-l BP'QP'B'J-l { ”(BP'QPBPy-l BPIQPX y:_1 + [(BP'QPBP)'1 BP'QPB B - I] 112—1 + (BP'QPBP)_1 BP'QP'y't P P P -1 - (B 'Q B) BP'QPD zt} . Let us examine the following expression: (A—3.15> (BP'QPBPY'1 BP'QPB E (BP'QPBP)-l BP'QP(QP>"1 (B'>’1 (B' QPB) And the inverse of expression A-3.15 is: (A—3.16) [(BP'QPBPf1 BP'Q“B']‘1 = (B'QPB)‘1 B'QP(QP)‘1(BP'>'1 (BP'QPBP) Substituting A-3.16 into equation A—3.14: P P -1 P -1 P (A-3.17) u: = -(§'QP§)‘1 pQ (Q ) (B ) (B .QPBP) (BP p p -1 'QB) BP'QPK yg_1 + (B'QPBY1 B'QP(QP>"1 (BP')'1 (BP'QPBP) (BP'QPBP>‘1 BP'QPK B up1 - (B'QPB>'1 B'QP(Q1’>‘1 (BP')'1 (BP'QPBP) “221 + (B'QPB)_1 B'QP(QP)'1 (BP'>'1 P PP-l P,P (BP'QPBP) (B 'Q B ) B Q 5'": - (B'QPB)'1 B'QP (QP)-1 (BP')-l (BP'QPBP) (BP'QPBP)-l BP'QPD zt Rearranging the terms, equation A-3.l7 becomes: (A—3.l8) u: = -(B'QPB)‘1 B'QPB y§_1 + (B'QPBY'1 B'QP(B B - BP) u§_1 + (B'QPBY'1 B'QP 7t - (B'QPB)'1 B'QPB zt 205 Substituting BP = A:B + E into equation A-3.l8: _ _ _ _ p... _ .. _ (A-3.19) u: = -(B'QPB) 1 B'Q A y§_1 - (B'QPB) 1 B'QPE u§_1 P yt - (B'QPB) B'QPB zt Q.E.D. + (B'QPB)'1 B'Q Therefore, Pindyck's optimal solution expressed in terms of the endogenous, exogenous-controlled and noncontrolled-variables is equivalent to that of Chow (equation A-3.10) even if the formulation of the state-space system is different. This equivalence verifies the property of unique- ness of the optimal control; that is, different formulations or approaches always lead to the same unique solution. APPENDIX A-4 COMPUTATIONS OF THE OPTIMAL SOLUTION FOR THE ONE-COUNTRY CONTROL PROBLEM Chow's optimal solution is given by equation A—2.20 (Appendix A—2): (”Z-2°) “’E = -'1 B“'Q°A°x:fl + o.houo> uo nOHusuHuunnu Hound noncoon n.0Id aOHuqavo cosh o o o m-c H o o o c o o o o o o o H o o o o HHaHHc I H-uo.onH-Hu nuo.umvun 217 Since q11 O O 0 O 0 q22 0 O O c Q = 0 0 0 O O 0 0 0 O 0 0 0 O 0 O h n- Then _ c*;_c . c c*;_c = (A 5.7) (xt: xt) Q (xt Xt) 0 a Therefore “ _ l_ c*;_c , c c*__c = J — 2 (xt xt) Q (xt xt) 0 Q.E.D. I'I'MZ =1 APPENDIX Ar6 THE REDUCED FORM OF THE TWO-COUNTRY MODEL (CASE A) Under the assumption of common external balance, the variables of Votey's econometric model are classified as follows: (1) Endogenous variables: Bit’ Cit’ lit’ lit, Kit’ Mit’ Olt’ Yit’ i = 1,2. (2) Controlled Exogenous Variables: Glt’ GZt’ rlt. (3) Noncontrolled Exogenous Variables: IElt’ Lit’ PMit’ Pxit’ r2t, Tit’ TRZt’ xit’ M3t’ MEI’ 1 a 1’2 And the structural model is constituted of: 9 equations: (E‘l) Y1: = 510 + 511 Klt + 612 L1: (E’Z) Clt = “10 + “11 Y1: ' “11 Tlt ' “11 5* K1t + TTlt (E'3) M1: = 810 + 811 Y1: ' B11 Tlt " B11 5* Klt + 812 TTlt + 3 wk 15- P It (E‘4) 1?: g Y1o + Y11 Ylt-l " Y12 5* ‘712 r1t-1 + Y13 Klt-l (E'S) 01: = n1o + n11 r2t ' n11 r1: 218 (“’6’ th = “20 + “21 KZt + “22 L2: (“‘7) C2t = “20 + “21 Y2: + “21 th “21 Tth ' “21 “* ch (“'8) M2t = “20 + “21 th ’ “21 T2t;" “21 Tth " “21 5* “2: + 8 TI - TT =(3!) ' 22 2t ’ 2t Px 2t (“‘9) Igt = Y20 + Y21 YZt-l Y22 “* “ Y22 r2t—1 + Y23 K21-1 7 Identities: (1‘1) Ylt = Clt + lit + Glt + (MZt + M3t + IElt) - Mlt (1'2) K1t = 11: + K1t-1 (1-3) lit = lit + 6* Klt (1‘4) B1t = (M2t + M3t + IElt) ' Mlt - Olt (1’5) th = C2t + Igt + G2t + (Mlt M1:1) ' 2t (1'6) K2t = Igt + K2t-1 (1-7) Igt = Igt + 5* K2t Substituting {(I-3), (E-Z), (E-3), (E-4)} into (I-l); {(1-7), (E-7), (E—3), (E-8), (E—9)} into (1-5); {E-8, E-3, E—S} into (1-4); (E-4) into (I-2);(E-9) into (1-6) and regrouping the terms, we get the following system of equations: 220 _ — — - * * (I 1“) (1 “11 + 811)Y1t “21 Yzc “ (1+“11 + “11) “1: + “21 “ K2: 3 Y11 Y1t-1 + Y13 Klt-l + Glt ’ Y12 r1t-1 B + B + y (“10 — 10 20 5*) + IE - 8 TT 1o ' Y12 1t 12 1t + (“11 ‘ “11) T1: + “22 Tth ‘ “21 th + “21 T“2t + M3t .. .- — * —* _ (I 5“) “11 Y1: + (1 “21 + “21) Y2: + “11 “ Klt “ (1 “21 + “21) KZt = Y21 Y2t-l + Y23 KZt-l + “2t ' Y22 r2t-1 +(a +3 —8 +y —y 5*)-Mm+e TT 20 10 20 20 22 1t 12 1t ‘ “11 Tlt ' “22 Tth + (“21 ' “21) T2: + (“21 ' “21) TRZt - _ — * * (I 4“) “11 Y1: “21 Yzc + Blt “11 “ K1: +'“21 “ K2t 3 n11 r1: ‘ n11 th + (“20 ' “10 ' "10) + IE1: ‘ “12 TTlt 11 T1: + “22 Tth ‘ “21 th ' “21 Tth + “3: _ = — —- * (I 2“) K Y11 Y1c-1 + (1 + Y13) K1t-1 Y12 r1t-1 + (YIO Y12 “ ) (I-6a) K Y _ _ * 2t Y21 2t-l + (1 + Y23) K2t-1 Y22 th-l + (720 Y22 “ ) Then doing the same transformation of variables as for the case of the one-country model: 221 Y1t = “10 + “11 K1t + “12 L11 1 = 1’2 “r (1'7“) Y1t ' “11 “1: = “10 + “12 Lit let (I-7b) Yit = Yit - 511 Kit = 610 + 612 Lit then (I-8) Y = 1 t + 5 it 1 11 Kit (1'9) Y1t-1 = Y1t-1 + “11 Kit-1 substituting (I-8) and (I-9) into the above system of equations: ~ ~ _ _ — _ _ *— (I 1“) (1 “ 11) Ylt “21 Y2: (1 “11 + “11) (“ “11) “1: 11 + B + B (5* - 6 21 21) “2: = Y11 Ylt-l + (Y11 “11 + Y13) Klt-l .. ._ _ * +- + Glt Y12 r1t-1 + (“10 “10 + “20 + Y1o Y12 “ ) IE1 - 8 TT t + (B 12 1 ) T1t + 8 TT - B 11 ' “11 22 2t 21 T2: 21 ~ (I-Sb) -Bll Y1t _ ~ * _ + (1 “21 + “21) Y2: + “11(“ “11) K1: - (1-5-2 + 321) (5* - 5 5 1 21) K2t = Y21 Y2t-1 + (Y21 + B _ 8 II _ _ * ... + G2t Y22 r2t-1 + (“20 + 10 20 + Y20 Y22 “ ) Mlt + “12 TTlt ' “ ~ ~ (I-4b) 811 Y1t - B 21 th + Blt ‘ “ 11 = n11 r1: ‘ n11 r2t + (“20 ‘ “1o ' ”10) + IE1: ' “12 TTlt + 8 TT TR + M 11 Tlt + “22 2t ' “21 T2: " “21 2t 3t - 8 TR 21 + Y23) K 11 Tlt ' “22 Tth + (“21 ‘ “21) T2: + (“21 + *_ *— 11 (6 6 ) K1t + 821 (6 621“ K 1: 2t +'M3t Zt-l I “21) TR2c 2t 222 (I‘Zb) Klt = Y11 §1t—1 + (1 + Y13 + Y11 “11) K1t-1 ' Y12 r1t—1 + (710 ' Y12 “*) (I-6b) K2t = y21 §2t_1 + (1 + Y23 + Y21 621) K211“1 - Y22 r2t_1 + (Yzo ' Y22 “*) And in matrix notation, the above system becomes: 223 Huu~u aw um uH HHH -N_‘ HIuNM HIuHM Hind ..u H NV L H H» o o o c Ham «an- Adm- Nam has «as. mas o c men 0 o mNu o o as. H~> HN» on. can can cm was as» AH» 0". H o o A «ms «nu. «Na. «Na- «an nae HNm- NN HNm- Ham Adm- AH 361$ 224 where “11 = 1 ' “11 + “11 “22 = 1 ' “21 + “21 “13 = "(1 ‘ “11 + “11)(“11’“*) “23 = “11 (6* ' “11) “14 = “21 (“* ' “21) “24 = (1 ‘ “21 + “21)(“*‘“21) “15 = Y11 “11 + Y13 “25 = Y21 “21 + Y23 “16 = “10'“1o+“2o + Y10 ‘ Y12“* “26 = “10+“1o'720+720‘ 22“* 11% “17 = “11 ‘ “11 “27 3 “21 ' “21 “33 = “11 (“* ' “11) “45 = 1 + Y13 + Y11 “11 “34 = “21 (“* ’ “21) “46 a Y1o ' Y12 “* “36 = “20 ‘ “10 n10 “55 = 1 + Y23 +'Y21 “11 “56 ‘ Y20 "722 “* Then multiplying both sides of the matrix equation by: _ _ -1 .q “11 '“21 0 “13 “14 ““11 Q12 0 Q14 Q15 ’“11 “22 0 “23 ‘“24 Q21 Q22 0 Q24 Q25 “11 "“21 1 ‘“33 “34 = Q31 Q32 1 Q34 Q35 0 o o 1 o o o o 1 o o o o o 1 o o o o 1 L 4 _ A 225 where IQI = “11“22 ‘ “11“21 Q = 822 Q g -(813822-+ “23 “21) 11, 1Yfl' 14 lQl = 811 Q = ’(“11“23+ “13 “11) Q21 Tifl' 24 lQI B (B - a ) _ 11 21 22 g 1__ _ Q31 ‘ lql Q34 Q[“11(“23“21 “22“33’. ' “11“21(“ 33 " 23) “13“ 11 (“21 ‘ “228 = 821 Q a “24“21 “14“22 “12 W 15 M Q = 811 Q = “11“24 ' “14“11 22 W 25 M B (a -B ) 32 _ 21 11 11 = 1 _ _ Q _ lQl Q35 Wt “11(“22“34 “24“21) ’ “11“21(“ 24 “34) “14“ 11 (“21 ' “221] We get: 226 ....-- HLHH Huumo HIuH HH»- on on 0H Vonn»- z w 2 game m “NH. ~ 2 UNHH. m 2, H-u~u . . o “NH O uH a an: 3.2. _2 uH 1 M mN HHHx W a uHuH m mH H . a r ._ ... _ o o w 4 o o T: _ 1 HH Hm ..UNU C O w NH uHo . o o r t o NHo mm um um 5N NH NH HMO HNO HHO o o o m nm N mm H 5H H - H awe HuuHM HuuHm ...u H NV “.7.un u. _l de HH»- mm mm mmu mm nN nH «m «N «H m« «m «N «H mm mm MH Nm NN NH Hm» NM 5 E emu” o6» Hm H~ HH HH» Hm 5 E “N am “He pH 1» .uH u» 227 11 Y11(Q11 + Q14) 3 “14 “15“11 + “45“14 3 16 'Y123Q11+Q14) 21 Y1133321 + “24) 3 “24 “15“21 + “45“24 3 “26 = '7123Q21+Q24) o A = o A 31 Y1131131 + “34) 3 34 “15“31 + “45“34 3 36 'Y123Q31+Q34) 12 Y21(le + “15) 3 “15 “25“12 + “55“15 3 “17 ‘Y223Q12+Q15) 22 Y21(sz + “25) 3 “25 a25sz + “55“25 3 “27 ‘Y223Q22+Q25> 32 Y31(Q32 + “35) 3 “35 = “25“32 + “55“35 3 “37 = "Y223Q32+Q35) 11 “16“11 + “26“12 + “46“14 + “56“15 3 D12 = Q11 3 13 = Q12 D21 = “16“21 + “26“22 + “46“24 + “56“25 3 D22 = Q21 3 D23 ' Q22 D31 = “16“31 + a26Q32 + “36 + “46“34 + “56“35 3 D32 ' 1 + Q31 D33 _ Q32 D14 = ’“123Q11 ‘ “12) D15 = “17“11 ' “11“12 D24 = '“123Q21 ’ Q22) D25 = “17“21 ' “11“22 D34 = '“123Q31 ‘ “32‘1) D35 = “17“31 ' “11(Q32 " 1) D16 = “22(Q11 ‘ “12) D17 = ’“21Q11 + “27“12 D26 = “22(Q21 ‘ “22) D27 = '“21Q21 + “27“22 36 82233331 ' Q32 + 1) D37 = "“2131 + “31) + “27“32 228 D18 = Q11 D28 = Q21 D33 = 1 + Q31 Therefore, the structural form of the two—country econometric model is: (A—6.3) yt_1 = Kyt_l + Eut + Eut_l + 5“at And using Votey's estimated structural coefficients: alO = 52.57 810 = 14.5131 YIO = 31.707 610 = 0 all = 0.75 811 = 0.131499 Yll = -0.0027 611 = 0.2051 alZ = N.S 812 = -28.3566 le = 461.7 612 = 4.02 aZO = 31.26 820 = -12.53 713 = 0.0624 6* = 0.0489 a21 = 0.7345 821 = 0.09769 720 = 1.5812 620 = 0 822 = N.S. 721 = 0.1668 621 = .36345 n10 = -0.6915 722 = 349.8519 522 = .35525 n11 = 46.279 723 = 0.04291 we compute the numerical values for the matrices X; E. E and B} Row Row Row Y1t-1 -0.0073 -0.0028 0.0008 -0.0027 0 G1t 2.8933 1.0476 -0.2781 0 0 lt-l ~ YZt-l 0.1298 0.4538 0.0324 0 0.1668 G2t 0.7782 3.0351 0.1942 0 0 Zt-l 229 Matrix X B K lt-l lt-l 0 0.0131 0 0.0648 0 -0.0524 0 1.0618 0 0 Matrix3§ 1t Matrix3E rlt-l -1263.7107 - 483.6579 143.7013 - 461.7 0 KZt-l 0.0806 -0.0329 0.0201 0 1.1035 o 3 2 momm.mol mum.e«l mOHm.HmaI H«oN.NNNI o mHmw.m«mI o o o~«o.HI o mem.HI UNH m. NHHUNZ mumm.o mHom.oml mmwo.le wmmH.m mooc.mml HwH«.NoH m«no.m~H APPENDIX A-7 COMPUTATION OF THE INVERSE OF A SYMMETRIC MATRIX (Ayreszl962) Let us compute the inverse of the following matrix by partitioning: (BC'QCBC) 3 B First, partition (BC'QCBC) into: D b b b31 11 21 “21 “22 “ b31 b32 “ 32 33J c. c g 3“ QC“ 3 “11 “21 “31 “21 “22 “32 Lb31 “32 “33 where r11 = “11 “21 3 F12 g “21 “22 g 31 32 231 31121 g [“31’ “32] 31122 = [“33] .19. .1115! .1. 232 Now 1 b22 '“21 “11.1 = b b -b2 3 11 22 21 -b21 611 p 1 . 1D c- F r -1P = 1 “22 "“21 “31 = '“22 “31 ' “21 “32 11 12 2 2 b b -b b b - b 11 22 21 —b21 611 J b32 11 22 21 3b - u '“21 “31 + “11 “32 2 b11 “22 ’ “21 1 33b b b b -1 g = r22 — r21 r11 r12, =l:b33] — [631, 63%] 22 31 21 22 “11“22 ' “21 ‘“21“31 + “11“32 2 “11“22 ' “21 ' H = b _ Lb31 (“22“31 ' “21“32) + “32 (’“21“31 + “11“32) 33 2 “11 “22 ' b21 b 52 - 2b b b + b b2 _ b _ 22 31 21 31 32 11 32 33 2 “11“22 ' “31 b b b - b2 b - b b2 - b b2 + 2b b b = 11 22 33 31 33 11 32 22 31 21 31 32 2 “11“22 ' “21 2 -1 “11“22 ‘ “21 and E = 2 2 2 (“11“22 ' “21) “33 ' “11“32 ‘ “22“31 + 2“21“31“32 2E323 HHHHAHHn 1 HHnHHe HHAHHA 1 HHHHHe HHnnHHa 1 anHHe H”; 1 HHHHHp HM; 1 HHAHHn HnnHHa + HHAHHA- HHAHHA 1 HnnHH g H”; 1 HHAHHH HnnHHp 1 HnaHna HHAHHAHHHH + HmnHHa 1 HmaHHa 1 nna AH“; 1 HHaHan 1 H nnaHHn 1 HHaHnn H”; 1 HHaHHa HHnnHHa 1 HnnHHAV HHnaHHn 1 HnnHHav Hm HH HH Hn HH m an H H HH HH HH HH fl a a pH + a a 1 H nHHn 1 a A Hg 1 Ha pm \ H a 1 HHnHHnV a a HHn 1 HHnHHn H H H H + H 1 HH< n H HHA- HHn H HAH n_HH.. 1 HHAHHe 1 fl J W HM» 1 HHAHHa HnaHHp + HnnHHn1 HH HH HH HH HH a a- HH HH HH 1 n 1 a a HHpHnnHHaH + HMAHHA 1 HMAHHa 1 nnn AHm; 1 HHHHHe H» a a + H H H 1 HHH.H HM He H ”HHH.H HM1ev+ HM1L 1 HH< HH HH HH 1 HHnHHa 1 HnnHHn _ HHp1 H a . H5 1 a a r 1 r :23 HH HHHHHHHHHH + HMHHHA 1 HMAHHH 1 Han HHMH 1 HHn no m _<_ 1. mums: pr 1 HHpHHn . HHHHHH 1 HHnHHn anHHn 1 HHHHHA fi1 15 1 . we _ - 1 -.111 1 1 1 . HH< HH< _ N E m l HIAomoOva _ H HH HH . Hm mm HH < < _ HHHHHnF 1 HHnHHn . HHHHHH 1 HmnHHn Hp 1 a a 1 . 1 I .1 eq< 4. B Hm Hm HH Hm HH Hm HH 2 HH HH HH HHn 1 HHaHHa a a pH + Ha a 1 Ha a 1 a A Ha 1 a He 1 u 1 HH< N H HI Hm Hm HH Hm HH Hm HH Hm HH HH HH n a. pH + a a 1 a n 1 a A n 1 a He HHHHHH 1 HHAHHH . HHnHHn 1 HHHHHH H H H1 H 1 HH< 1 HH< - fil J HHaHHn 1 HHnHHn HHHHHHHHAH + HMHHHH 1 HMHHHH 1 «an AHMH 1 HHHHHe HH HH HH Hm HH Hm HH H 1 1 H1u A e H1ev1 1 < a a 1 a a APPENDIX A-8 COMPUTATIONS OF OPTIMAL SOLUTION FOR THE TWO-COUNTRY CONTROL PROBLEM (CASE A) The optimal solution is given by equation A-2.20 (Appendix Ar2): * — * — (A—2.20) u: = -(B°'Q°B°) 1 BC'QCAcx:_1 + (BC'QCBC) 1 BC'QCE: _ (BC'QCBC)-1 Bc,QcCczc c* ~ ~ ' = * * it * * * * xt [élt’ Y2t’ Blt’ K2t’ G1t’ GZt’ 1.It] i“ = 2 § 3' ET 67 E’ E’ 3 g 5 + 5 L 5 t 1t’ 2t’ 1t’ 1t’ 1t’ 2t’ 1t 10 12 1t’ 20 — _ _ _ ' + “22 LZt’ 0’ Klt’ Glt’ GZt’ r1t] C* - 3* c* -w 3 “t I [ 1t’ 2t’ r1t] c III zt [1’ IElt’ Mlt ’ TTlt’ Tlt’ r2t’ th-l’ TTZt’ T2t’ TRZt’ M31:] 235 236 -11 1- “15 “ “ “16 Q11 Q12 “ “25 “ “ “26 1 Q21 Q22 “ “35 “ “ “36 Q31 Q32 n11 o o o ylz ,8“= o 0 0 .1155 o o o o o o o o o o 1 o o o o o o o 1 o o o o o o o 1 “15 “ A17 “16 “17 “17 “18 “25 “ “27 “26 “27 “27 “28 “35 ‘Y11 “37 “36 “37 “37 “38 o o o o o o o o o yzz o o o o o o o o o o o o o o o o o o o o o o o o o .J o o o o o .1 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o 2137 3:26 Hmannu HnoHHu o HHaHHc HHGHH.v 1 a HnaHchno HHoHchnu o HHOHHu HHcH:v T GOO GOO HnaHchnv fi$$+HV§u+J%# HnaHnannc + HHaHHaHHu + HHoHHaHHa w 1 H o o o H a o o H o o o c o a HH: Hna Hno a o o ~No HNO 9 o 1 o HHc HHyl o o o o o o o o o o o o o o o o o o c c o o o o o o o a o o c o o o o o o o o o o o o c o o o o o mac o o o c o o o HHc OOOOOO HHOanHHv + HHaHHaHHu + HHoHHaHHu Hmonnu + chHH_v + HMaHHu o o 5.3V 6 o o o Hmannu HHcHHe HHaHH~v lane 3 o c Hnannu HHoHHu HHoHHu o u .u H o o o o HH: o o o H c o c Hno HHa HHa 1ua.un o o H o o Hna HHo HHa "gush-50 2E38 HH 1HH HH A N A A Hm HH 1HHHHHH A HH 1HH HH A A A NMAHN HHOHnonnu+HHoHHoHHu+HHoHH0HHc 1 HH 1HH HH A A A HHn1HH HH 2 a HH 1HH HH A N A A an; HHMH 1 HH 2:26 1 mm H c a Hmonnu + HMoHHc + HmcHHe 1 HHQ v Hm «nu HH HHu HH HV 1 HH He + Ho + Ho 8 n oqua aHHpV “AHAMUH waHSOHHow oAu cu von AUHAS HIAumuo.unv waHusmaou mo undue onu mume sud vaaoaa< .Aumoo.umv udHOnvm 0AA waHusqaou mo unuumcH AmomHuomumdv waHaOHuHuuua mo vOAuua oAu NA voaHuqu mH Aunoo.unv mo oouo>nH «Au can xHuuul oHuuanhu u mH Aumuc.umv uuAu moon 0: 239 Substituting the bij's and bii's (i = 1,2,3; 3 = 1,2,3) by their corresponding values, we get: 2 2 2 g _ 2 (“11“22‘“21) “33'“11“32‘“22“31+2“21“31“32 ' q11q22q33n11 (“11“22 “12“21) 2 g 2 2 2 2 “22“33 ' “32 ” q11q33n11 Q12 + q22q33”11 Q22 2 g 2 2 2 2 “11“33 ‘ “31 ‘ q11“33”11Q11 + q22“33”11Q21 b b - b2 3 (Q Q -Q Q >2 (Q Q -Q Q )2 11 22 21 q11‘122 11 22 12 21 +q113333 11 32 12 31 +c1c1 31 32 21 33 q33“11 q11 11 12 22 21 22 d “22“31’“21“32 g q11“33”11“12(“12°31'Q11Q32)+“22“33"11Q22(“22“31"Q21Q32) g “11“32‘“21“31 ‘ q11“33”11“11‘ H c HHc .HHoHH01HnoHHo HI HH0 o HHo1 an: “Huh “He “Has H1uH1 UNH UHF UH PH aha- U uH r H 1 H o ,HHc1 HHoHHo1HHoHHo H 1W1 HNOO 1mw1 NN0 HH 341 HH<1 HH QHCI 5H4! _ HnIOHHou no voaHuov on. 241 «H41 241 HH<1 2.- =4- 2.- 2.- » nuomuau OAH uuaHm Hun umo UN A‘1Q0‘v ac .2115 HH HHnoHH01HmoHHoe o c 22 I“?! an I: 8H 8H HH 73“— HH uH (4 UH UH HHH MH UN I: H: Flu-I HHHHoHHO1HmoHHoVoHe1HHHoHHo1anHHoVonH AoNoHH0 + OHoHN0IV HoHHHHo 1 OHHHH8 -3“... on<1 H H|0 H|0 +Hn an NN NH <3 fiH1umu H1umo H1umu Him”H HLmUH H1umn H1umw Er 241° $1 .HH? 2? HH 31 .HH? HH HH <1 o <1 <1 241° $1 .HH? .HHHr HH? 1! omol c o nmcl on1 0 o nNol F 0H01 0 0 nHol "nuaoqu « . 01¢ H5326... HH : Hn<1 AHHaHHHTHHOHH8 a HH» 0 HHHHHo 111mr11. HHHHHo1 «~61 o Hno1 Hno1 «H81 o HHo1 HHH1 «H61 o HHo1 HHo1 EU“ I h uH *H an an “mo l 3615 no Homeowdnuunuu van 8333095 .Houwa APPENDIX A-9 EVALUATION OF THE WELFARE COST FOR THE TWO-COUNTRY CONTROL PROBLEM (CASE A) The welfare cost is determined by: ,. T C )3 * *' * * (A-5.1) J =%- (xC -‘“) c C ‘1“ x Q (x - x ) t=l t t t t where (111-5.4) x: = [1-8°(8°'Q°8“)'18“'Q“]A° 11::11- ESC(BC'QCBc)-1Bc'ch)—c: + [I-BC (Bc,Qch)-1Bc ,Qc] Ccz: (A-5.5) x:*- i: = [I-BC(BC'QCBC)-1BC'QC]Acx::1+[Bc(Bc'Qch)-1Bc'Qc-I]E: + [I-BC (8° ' QCBc ) _ch ' QC]Ccz: . Let us compute for the two-country control problem the following matrices: 246 247' HH: HH HHnaHHa 1 HmaHHavo a HH AHmoHHo1HmcHHcvo c HI H aHn0NN01Nm0HN0V HH HHnoHHa 1 HmoHHovo c oHHc H H 1w1 HHo1 1M~1 HHa c o o o H H o o o H o o o H c o o o o o HHc Hno Hna o HHo HHo o HHa HH0L I o0.omH1Aono0.umvon 8 4 2 COCO ono1 o o mmo1 oHc1 o c an1 oHo1 o o «H61 o o o nan HH»1 o o o o o o o o o o o o o o o HH HH H1 H 8mm1 o Hmo1 Hmm1 L HH¢1 o HHH1 HHH1 «Ho1 o HHo1 HHH1 c o HH» new a o HH» o c o o o o o o o o o o HH AHmoHHHTHmcHHSo “1 o n oHHUOHuHH1Homoo.omvom1H_ 1 Huo.umH1Humuc.omvom1H_ 24$? HHHaHHo1HHoHHaywmmmr 1w: HHo 1m: HHa1 o o o o c HH<1 H mna 1 8m HH<1 o HHH 1 8H HH<1 o nHH 1 4H HH>1 o o o o o o o o o o o o o o HHHoHHa1HmoHHoV as. 1 H 1M~1 HHo1 1mw. HHa o o <1 <1 <1 nN<1 NN<1 HN<1 nH - Substituting {(1-4), (E-Z), (E-3), (E-4)} into (I-la); {(1-9), (E-7), (E-8), (E-9)} into (I-6a); {(E-3), (E-5)} into (I-Za); {(E—8, (E-5)} into (I-7a); (E-4) into (I-3) and (E—9) into (I-8) and regrouping the terms, we get the following reduced form system: 31’1“) 31’“11 + “11) Ylt ' “21 Y2: ' “*31‘ “11 + “11) K1: 3 “21 “* “21 = Y11 Ylt-l + Y13 K11:-1 + Glt " Y12 r11:-1 + 3“10 “ “10 + “20 + Y10 ' Y12 “*3 ' “12 ““11 + 1E1: + (“11 " “11) “1: + X1EI + “22 T12: ' “21 T21 ’ “21 ““21 ‘ “3:1 255 .— _ - 1- * — * 31 6“) “11 Y1: + 31 “21 + “21) th “ 31 “21 + “21) “21 + “11 “ “1: = Y21 Y21-1 + Y23 KZt-l 1 “21 ‘ Y22 r21—1 +(9 +8 -8 +1 -Oy 6*)+8 TT -8 T -M111 20 20 20 . 20 22 12 1c 11 11: 1t ‘ 111 ’ “22 T121 3 3“21 ' “211 121 + 3“21 ' “21) ““21 + “21 _ — — * * = _ 31 2“) “11 Y1: ,“21 Y2: + Blt “11 “ “11 + “21 “ K2: n11 r11 n11 th + (B - B - n > - 8 11 + B T + IE + xIII 20 10 1o 12 1t 11 1t 11 11 111 + “22 “T21 ' “21 121 ' “21 ““2: ' M21 _ 111 _ 111 111 _ 111 31'7“) Blt 3 “21 ’ “11 M1: 3 X2: “2: _ = -- * — 31 3“) K11 Y11 Y11-1 + 31+Y13) Klt-l + 3Y1o Y12 “ 3 Y12 r11;-1 _ = — * — 31 8“) K21 Y21 Y21-1 + 31 + 723) K21;--1 + 3320 Y22 “ ) Y22 r21:-1 To get rid of the problem of double-counting of K t’ we make the follow- 1 ing transformation of variables: 1.1. II (I-11) Yit = Yit + 6 = 6 6 1,2 11 Kit 10 + 12 Lit 1 (1-12) Y = i + 5 5 +15 it-l it-l 11 Kit-1 = 10 12 Lit-1 1’2 Substituting (I-11) and (I-12) for i = 1,2 into the above reduced form: ~ ~ 31'1“) “11 Y1: ’ “21 Y2: + “13 Klt + “14 K2t = Y11 Ylt-l + “15 Klt‘l TT + IE T + XIII + “11 ’ Y12 r11-1 + “17 ' “12 1t 1t + “18 1t 1t _ 3 TR _ MIII + “ TT ' “ 21 2: 2: 22 21 21 T21 (I-6c) (I-2c) (I—7c) (I-3c) (I-8c) where: 11 13 a) I 14 m I 15 3 CD I 17 3 18 3 — 1-a + B a 3 “213“* ' “ 256 ~ 11 Ylt 3 “22 “ '“ 21 3 “23 K1: 3 “24 “21 3 Y21 th-1 3 “26 KZt-l + 8 TT - B T - MIII - 8 TT 3 G 12 1c 11 11 1t 22 2: 21 ' Y22 r21-1 3 “27 111 3 “28 T2: 3 “28 TR21 3 “21 ~ 11 Y11: 3 “21 Y2: 3 B B 11 3 “33 “11 3 “34 K2: n11 r11 ' n11 r21 III + IE X + 822 TT 3 “ ' “12 1T1: 3 “11 T1: 3 1t 1t 37 3 “ 2: 21 12: 111 3 “21 ““2: 3 M2: 111 111 111 111 = -— + .. Blt 3 “21 x1: M11 “21 “2: ~ 1t 3 Y11 Y11;-1 3 “55 K11-1 ‘ Y12 r11-1 3 “57 2: ‘ Y21 Y21—1 3 “66 K21—1 ‘ Y22 r21-1 3 “67 1-a + B 11 11 22 21 21 (l-a 6*) a B (6* - 6 11 3 “11)3“11 ' 23 11 11) (a-a 5*) 21) “24 21 3 “2113“21 ' 6 6 3Y13 3 Y11 11) 26 ' Y23 3 Y21 21 1. — * 3“10 “1o 3 “20 3 Y10 Y12 “ 1 5* 0) ll 27 “20 ' “20 3 Y20 3 Y22 11 11 28 ' 257 = — * = “33 “11(“11 “ ) “55 1 3 Y13 3 Y11 “11 = *_ = — * “34 “21 (5 “21) “57 Y1o Y12 “ “37 3 “20 3 “1o 3 n10 “66 3 1 3 Y23 3 Y21 “21 6* “67 3 Y20 3 Y22 And in matrix notation it becomes: 258 u an u 2% MN uN H-u~u m Alum» IUN“ HIuN Hindu Alu~ ... E ~-u- ”.6“ I» A we d~> -> so. “a. un- “N ha AH» L u~ “an u~ an u~ 'D‘ uu 'D‘ can «A an nu ma -u «an- “H.oa- - s s (a + a > + a a (s - 3Tifl' 11( “22 333 “23 21 11 21 23 33 13 11 21 1 (a B a ) 3T6T 24 213 “14 22 “22)] 260 _ 1 Q36 3 TET [“11(“22“34 3 “24“21) 3 “11“21 (“34 3 “24) 3 “14“11 (“21 3 “22” Then the reduced form is: 261 Huuflu Huu~o L H He «H». mm mm mm nH anumu uu as as u HHMZ u 1% EH o o o 0 mac can. man. «an man «No man can A o o o o ~na ”no. Nno. sac -o Hue «Ho Ado 1 o «no «no- «sou. o and Add o o HNo Add AN NM NM NN NH 566 “no 1296- 51:63-26- HnO+H HnO+H Mme an «A AH an AM AN AM a aux z. u~ as ”N uH Aw.OAIaq onu .owuuuaaho mu Aomoo.umv conga .uuowon ad I Aomuc.uflv 271 where ID- 2 2 2 “11 3 q11Q11 3 q22Q21 3 (“33 3 q44>Q31 IO: 2 2 2 “22 3 q11Q12 3 q22sz 3 (“33 3 q44)Q32 g 2 “333 (“33 3 “44)n 11 d “213 q11Q11Q12 3 q22Q21Q22 3 (“33 3 q44>Q31Q32 £1 b31 3 (“33 3 q44)“11Q31 9. “32 3 (“33 3 q44)“11Q32 Substituting the b 's and b1 (i = 1,2,3; 3 = 1,2,3) by their I 11 j 8 corresponding values, we get: 2 2 2 g (“11“22 3 “21) “33 3 “11“32 3 b22“31 3 2“21“31“32 3 q11q22 (“33 3 “44) 2 2 n11 (Q12Q21 3 Q11Q22) 2 g 2 2 2 2 “22“33 3 “32 3 q11 (“33 3 “44) n11 Q12 3 q22(q33 3 q44)”11Q22 on! 0" 0" I d 2 2 2 2 11 33 3 31 3 q11(‘133 3 q44)"11Q11 3 q22(‘133 3 “44) n11 Q21 _ _ g _ 2 2 11 22 21 q11‘122 (Q12Q21 Q11Q22) 3 q11(q33 3 q44)(Q12Q31 3 Q11Q32) + 2 q22(‘133 3 q44)(Q22Q31 3 Q21Q32) g _ 2 “31“32 3 “21“33 3 (“33 3 “44) n11 (“11Q11Q12 3 “22Q21Q22) IID- b22“31 3 “21“32 q11(‘133 3 “44) 011012(012031 3 Q11Q32) q22 (“33 3 “44) n11Q22 (Q22Q31 3 Q21Q32) 22712 NANNoNNouNNoNNaVchqu NaNNaNNonNNaNNavNMCNNv NMcaeov+nnuo NNNNaNHaINNaNNooNNcNNcNNe NNNNaNNo 4 NNoNNodecNNeNNu g .NnaNNo+NNoNNcV + NANnoaNa+NnoNNav + A ANnaNNc+NnaNN8NNaNNe .NnoNNo+NnoNNo.NNaNNou NNnaNNo+NnaNNavNNoNNu+NNnoNNo+NnoNNoVNNoNNu NNNNaNNauNNoNchaacNNvNNo NfiNNoNNouaNaNNaVNNuNNc . NaNNaNNouNNaNNaVNNaNNu ANnoNNo+NnaNNavNNaNNuuNNNoNNc+NnaNNoVNNoNNcu NwaNNu + NMaNNe ANNoNNaNNENNoNNozuT - Nuaunuo.unv NNNNoNNaINNaNNo.NNcNNuNNu NaNNoNNauNNaNNaoNNvNNc NaNNoNNauNNaNNaVNNuNNu ANnoNNo+NnoNN8NNaNNviNnazamnoNNSSade “NNoNNoNNc+NNoNNaNNuV- NMaNNu + NMoNNu L 1:. «nonwo + unouuav Macaw: Ac}. .9 nnvv «Nv ... Annodo + HndNHOvHHozc Ace—v + nnsuuv m unnHNA I annum.“- 273 mm mm ma o o «m o o «N o o «N Aequ+mmvaNc mmw MM¢ o 0 NM Mme o 0 mm mac 0 0 NH OHH C ANNONN0+NNONNovu .mw. NNou NHO Hmm NNm u < o m A m o No AN.NNI 31 3 n11 313 “11 Q22Q313Qzl“32 21 Q12“313“11Q32 :‘ Q egl—A+A-n < > + 33—1- 30 r1110 [ 11 Q22‘1313‘121Q32 21 Q12‘2313‘111‘132] 011 A g—1 < >- < 5' + 13331 31 nllQ [Q11 Q22Q313‘121Q32 Q21 Q12Q313Q11Q32‘ n11 - Q q d 1 31 33 A =—Q (QQ'1'QQ )‘Q (QQ+QQ ) +_+ 41 11110 [11 22 31 21 32 21 12 31 11 32_ n11 (q33+q44) Q q d 1 32 44 A=_‘Q (QQ+QQ)+Q(QQ+QQ)+—+ 32 “11Q[ 12 22 31 21 32 22 12 31 11 32] "11 ((133.4144) d 1 1“32 “33 3 0110 [“12(Q22Q313Q21Q32)3“22(“12“313Q11Q32)] 3 011 d 1 “33 “34 3 "11“ [“13(“22Q313Q21Q3213D23(Q12Q313Q11Q32)] 3 011 A d 1 36 A =‘-“—A(QQ+QQ)"A (QQ+QQ)+— 35 an I 16 22 31 21 32 26 12 31 11 32] 011 0 d 1 34 “36 3 r1110 [“14(Q22Q313Q21Q32)3“24(“12“313Q11Q3211 3 011 d 1 “35 “37 3 nllQ E315(022031+021032)-025(012031m11032'fl 3 011 Q q d 1 32 44 A =——'Q (QQ+QQ )‘Q (QQ+QQ ) -_" 38 11110 [12 22 31 21 32 22 12 31 11 32] 011 (q334-q44) Q q d 1 31 33 A =——-Q(QQ+QQ)+Q(QQ+QQfl-—- 39 "11“ [ 11 22 31 21 32 21 12 31 11 32 "11 (q33+q44) Then the optimal control is: 227717 u H I“ u N Io u H O :63qu“ n!" I 9 F. i I u N u u-I II» [IV 6N: HHH HHH “Nap “N .N NLNh Neow+nnucNNc ON on _Nuumu NIumc In N H6 Nuums N-umu Naumn In N “a Nuum» N-u«m A2029:0281 «nOHNYNMONNo oNNNa aNNNo. 278 — _ ~ Since the targets 3 YZt’ Blt’ B2t are defined as follows 11’ “11 3 “10 3 “12 L11 ““21 3 “20 3 “22 L21 “11 = 0 “21 = 0 after substitution and rearrangement of terms, equation A-ll.4 becomes: 2Z7!) N . Haw: u NNmu uNuu uNa UN: NIuNu UN» uNa “Nan NN u I c Nn Nn NN Nn NN meI. NNH: fiwan NNNI NNNI NnaI onaI nan N ean nnNI NNNI Nqu «I a o+ a a NNNI 0N NNNN o u NN .IIIIII NuN NNNI quI NN