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'\'.' . «,0. -~' ." u". “ - I "' ,‘5‘ l J, C. ‘ 1: 29.2.3" - h . .f. ‘ '35: ,5? c‘. . . ‘. fly...) ‘ --.,', V ,, ‘35:“ it t This is to certify that the dissertation entitled AN APPLICATION OF CATASTROPHE THEORY TO THE CLOSED AND OPEN MACROECONOMY presented by HAS SAN KHADEMIAN has been accepted towards fulfillment of the requirements for DOCTORAL degreein ECONOMICS M Major professor I (LAWRENCE H . OFFICER) Date MAY 15, 1984 MSU is an Affirmative Action/Equal Opportunity Institution 0-12771 MSU LIBRARIES .——_ \— RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. AN APPLICATION OF CATASTROPHE THEORY TO THE CLOSED AND OPEN MACROECONOMY BY Hassan Khademian A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Economics 1984 Copyright by HASSAN KHADEMIAN 1984 ABSTRACT AN APPLICATION OF CATASTROPHE THEORY TO THE CLOSED AND OPEN MACROECONOMY By Hassan Khademian The existing literature in economic theory explains the con- tinuous dynamic equilibrium values of economic variables. However, economic variables in general and the foreign exchange rate in particular may display discontinuous changes in their dynamic equilib- rium values. The existence of such a phenomenon warranted this author and his dissertation committee members to search for the roots of this phenomenon for almost two years. In my dissertation, I model both closed and open economies in the spirit of catastrophe theory. This theory argues that in a system (model), catastrophes occur when the equilibrium values of endogenous variables experience sudden discontinuous changes (jumps) as a result of continuous change in exogenous variables. Catastrophes are not sudden discontinuous changes of great proportion. But simply, they are discontinuous changes in the equi— librium values, regardless of their magnitudes. Real economic vari— ables, such as real income, may show discontinuous changes (catas- trophes) of small magnitude. But financial variables, such as foreign exchange rate may show these characteristics in a large value. V6} 1e: bre Hassan Khademian This dissertation is a new approach to an old problem (i.e., exchange rate fluctuations). It will stir both anxiety and contro— versy. And this is the purpose of this dissertation: to produce at least a viable intellectual explanation of the problem and at most a breakthrough. To My People With Love it Of my ev th tY ACKNOWLEDGMENT I remember I read somewhere that 'school is not a preparation for life,‘ it is 'life.' I came to Michigan State young, inexperienced, and seeking adventures. I am leaving this beautiful campus young, experienced, and seeking adventures. I am happy that I came and sad to go. I would like, first of all, to thank my parents, my sisters (Golnar and Sousan), and my brothers. It would be fair to say that it would have been much harder to finish my graduate work without Golnar's support. I would also like to thank my thesis advisor, Lawrence H. Officer and other members of the dissertation committee: Stephen Martin and James Johannes. Professor Officer's intellectual and maverick approach allowed my mental agility to thrive while holding a firm grip on the course of events. Professor Martin's astute perception of economic theory was a key instrument in the production of this work. However, I am personally responsible for any flaw in this thesis. I thank Ms. Jo McKenzie for assuming the painstaking task of tYPing my dissertation. iv LIST Chapt TABLE OF CONTENTS LIST OF TABLES. . . . . . . . . . . . . . . . . . Chapter 1 INTRODUCTION. . . . . . . . . . . . . . . Footnote . . . . . . . . . . . . 2 INVESTMENT FUNCTION . . . . . . . . . Introduction . . . . . . . I. General Literature on Conventional Theories of Investment Behavior. II. Investment and Theories of Income Determination. IIa. Real Theories of Invest— ment Behavior. . IIb. Monetary Factor in Short Run and Long Run Models of Investment Behavior . III. The Modified Investment Function and its Mathematical Derivation. Footnotes. . . . . . . . . . . . . . 3 PARTIAL EQUILIBRIUM ANALYSIS OF INCOME DETERMINATION . . . . . . . . . . . Introduction . . . . . . . . . . . . . . Real Theories of Income Determination. Footnotes. . . . . . . . . . . . 4 CATASTROPHE THEORY. Footnotes. . . . . . . . 5 MACROECONOMICS AND CATASTROPHE THEORY Introduction . I. IS—LM Again . . . . . . . Ia. Short Run. Ib. Long Run . V o Page viii 14 15 20 22 26 28 28 30 41 42 49 50 50 54 59 61 Chapter II. Aggregate Demand— —Aggregate Supply . . Footnotes. . . . . . . . . 6 THEORIES OF EXCHANGE—RATE DETERMINATION . . . Introduction . . . . . . . . . I. Flow Market Models of Exchange- Rate Determination . . . . . . . Ia. Elasticity Approach. . . . . Ib. Absorption Approach. II. Internal and External Balance Approach. III. Asset Market Models of Exchange— Rate Determination . . . . . IIIa. Purchasing Power Parity Theory (PPPT) . . . . . . . IIIb. Monetary Model . . . . . IIIc. Portfolio—Balance Model of Exchange—Rate Determination . Footnotes. . . . . . . . . . . . . . . 7 THEORIES OF EXCHANGE—RATE AND REAL INCOME DETERMINATION . . . . . . . . Ia. Goods Market . . . . . . Ib. A Model of Real Income and Exchange—Rate Determination . II. Exchange—Rate Behavior and Perfect Substitution . . . . . . . . . III. Equilibrium Solution. . . . . . . . . IIIa. Comparative Statics. IIIb. Equilibrium Behavior of Exchange Rate . IV. Exchange Rate and Rational Expectation. V. Exchange-Rate Behavior and Inflation. Va. Exchange Rate and Inflationary Shocks. . . Vb. Price Level as an Endogenous Variable. . . . . . . . . VI. Exchange— —Rate Movement in a Dynamic Model. . . . . . . . . . . . . . VIa. Stability. . . . . . . . VIb. Steady State and Cour parative Statics. . . . . Footnotes . . . . . . . . . . . . . . . . . 8 EXTENDED MODEL: A THEORY OF EXCHANGE RATE, REAL INCOME, AND INTEREST RATE DETERMINATION. . . . . Ia. Imperfect Substitution Among Financial Assets. . . . . . Vi 64 71 72 72 74 74 77 78 79 80 83 87 88 89 94 95 101 106 112 115 118 123 130 133 135 140 141 141 Chapter Ib. Asset Market . . . . . . . . . Ic. Goods Market . . . . . . . . . II. Equilibrium Solution. . . . . . . . . . III. Stability Analysis. IIIa. Dynamics of the Model in an E—Y Plane. . . . . IIIb. Dynamice of the Model in an r— —Y Plane. . . . . . . . IV. Comparative Statics . . . . . . . . . . IVa. Goods Market . . . . . . . . . IVb. Foreign Bonds Market . . . . . IVc. Home Bonds Market. . . . V.‘ Comparative Statics: Effects of Finan— cial Policies on Exchange Rate, Real Income, and Interest Rate; Combined Effects. . . . . . . . . . . . . . Va. Monetary Policy: Discount Rate and Reserve Require— ment Fluctuations . . . . . Vb. Monetary Policy: Open Market Operations in Home Bonds Market. . . . . Vc. Intervention in Foreign Bond Market. . . . . . . . . . . Vd. Fiscal Policy. . . . . . Footnotes . . . . . . . . . . . . . . . . 9 CONCLUSION. . . . . . . . . . . . . . . . . APPENDIX A . . . . . . . . . . . . . B . . . . . . . . . . . . . . . . . . . . . . . . BIBLIOGRAPHY. . . . . . . . . . . . . . . . . . . . . . . . 150 153 156 158 159 160 161 162 166 170 173 178 179 185 189 192 LIST OF TABLES Table Page 3.1 39 7.1 Effects of Monetary, Open Market Operations, and Budgetary Policies on Real Income and Exchange Rate in a Short-Run Model. . . . . . . 106 7.2 Effects of Financial and Inflationary Shocks on Real Income and Exchange Rate in an Inflationary Model. . . . . . . . . . . . . . . 122 7.3 Effects of Open Market Operations and Budgetary Policies on Real Income, Exchange Rate, and Overall Price Level. . . . . . . 129 7.4 Effect of Financial Policies on Real Income, Exchange Rate, and Wealth in a Dynamic Model in the Long Run . . . . . . . . . . . . . . . 139 8.1 Shifts in Real and Financial Markets as a Result of Financial Policies (Markets are Examined Independently from Each Other) . . . . . . 161 8.2 Effects of Open Market Purchase of Bonds on Real Income and Financial Markets . . . . . . . . 166 8.3 Effects of Open Market Purchase of Foreign Bonds on Real Income and Financial Markets. . . . 171 8.4 Effects of Mbney Financed Fiscal Policy . . . . . . 173 8.5 Effects of Bond Financed Fiscal Policy. . . . . . . . 175 viii ’i—r'“ .. .1- .. ...- 1-1... -11....“ , CHAPTER 1 Introduction The foreign exchange rate, or simply the exchange rate, is the price or value associated with a currency being acquired or bought in terms of another currency. Small fluctuations in exchange rates are common phenomena. However, sudden large fluctuations, distinctively different from the rest, have warranted scholars' time and effort in the last decade or so. "News" or "new information," by changing expectations, is con- sidered a major contributing factor to these fluctuations.l However, this approach relies on the theoretical manipulations of ad hoc assump— tions about expectation. An alternative approach is the application of catastrophe theory to the theory of exchange-rate determination. De Grauwe (1983) is the only application of catastrophe theory (CT) to the theory of exchange—rate determination. He applies CT to a monetary model of exchange-rate determination. His work is interesting and enlightening. However, it suffers from some shortcomings. First of all, the oscillatory fluctuations and the occurrence of catastrophes were explained in the framework of some ad hoc expec— tations formation assumptions. In this model, expectations formation 3 comprised of two parts: (a) non—linear regressive (b) extrapolative xpectations. De Grauwe postulates the existence of an extrapolative 1 2 expectations around the long run equilibrium values of exchange rate (E). As a result, exchange rate moves away from E (due to extrapolative expectations) until regressive expectations become dominant. But, Salin (1983) raises some doubt about the occurrence of extrapolative expecta- tions around E. Secondly, De Grauwe (1983) applies two restrictive assumptions of interest parity and purchasing power parity conditions. Chapter six deals with these conditions and their shortcomings in models of exchange—rate determination. Third, De Grauwe takes prices and out— put as "slow" variables in a dynamic model of CT. I believe that this implies a wrong perception and understanding of the theory. "Slow" vari- ables are parameters of a system in CT. Catastrophes (or discontinuous changes) in the equilibrium values of some endogenous variables occur as a result of continuous changes in the parameters of the system. Real income and price level are not slow variables. They can be easily considered endogenously determined variables in De Grauwe (1983) by relaxing some assumptions. But financial and open market operational policies are the parameters of the economic system. Finally, De Grauwe explains the oscillatory fluctuations of exchange rate within the framework of a catastrophic model. But, as was correctly understood by Salin (1983), it is the existence of large swings with short—run oscillations in the values of exchange rate which warrant our search and effort. The purpose of this thesis is to deal with the subject of ex- change rate, real income, and interest rate fluctuations within the mathematical framework of CT. I concentrate on the real sector of the economy rather than regressive and extrapolative expectations formulations as the major contributing factor in these cyclical —_—'* ..- “flaw”--- --., . . 3 fluctuations of exchange rates. Furthermore, exchange rate and interest rate behavior are explained according to portfolio-selection criteria. As explained in Chapter six, this model has a better capability than monetary models of exchange-rate determination in explaining the behavflmrof our variables. In addition, the model explains large swings in the values of the exchange rate and other economic variables as a result of continuous changing financial policies. This thesis is comprised of two parts. Chapter two through five comprise part one. In this part, the occurrence of catastrophes in a closed economy will be examined. The investment function plays a key role and provides an essential building block in this dissertation. Chapter two deals exclusively with the shape and importance of the investment function. In Chapter three an examination of macroeconomic model is conducted. Based on the shape of the investment function derived in Chapter two, goods market equilibrium conditions are speci- fied. Chapter four examines the essentials of CT. Its examination is in a non-technical, non-mathematical term.2 In an economic system, catastrophes occur when the equilibrium.values of endogenous variables experience sudden discontinuous changes (jumps) as a result of con— tinuous change in exogenous variables. In the jargon of CT (and in my odel), the exchange rate, real income, and interest rate all are "fast" variables and financial policies are the "slow" variables. The haracteristics of a Keynesian macroeconomic model are examined in hapter five. This chapter shows that financial policies can cause he equilibrium value of real income to move in a discontinuous manner. Chapters six through eight comprise the second part of the hesis. In this section, real and financial models of exchange rate, 4 real income, and interest rate determination are examined. This is an extension of the closed economy to an open (i.e., international) economy. Chapter six presents a brief review of the exchange-rate deter- ination literature.3 This chapter lays a foundation for the next ones. he theoretical building blocks of exchange-rate behavior, to be eveloped in the next two chapters, are based on the concept that the rious theories are supplementary rather than mutually exclusive in plaining exchange-rate behavior. Chapter seven constructs a real and financial model of income d exchange-rate determination. This model is based on equilibrium nditions in both goods and financial markets. The goods market .earing conditions in conjunction with the portfolio selection equi— bria conditions (deve10ped in Chapter six) will be used to determine uilibrium values of real income and exchange rate. In this chapter, ne and foreign bonds are considered perfect substitutes. The expec— :ions, inflation, and the dynamic behavior of this model are explained :pectively. In Chapter eight, I relax the assumption of perfect substitu- n (interest parity) between home and foreign bonds. Private home Lth holders face a portfolio selection problem of choosing between :real balances, home bonds, and foreign bonds. Thus this chapter ents an extended version of real and financial model examined in ter seven. A simultaneous solution of goods, home bonds, and .gn bonds market clearing conditions determines equilibrium values a1 income, home interest rate, and exchange rate, respectively. Chapter nine provides my conclusions and suggestions for _ -~ .y, future research. The possibility of changing signs in the slope of the LM curve can contribute to further investigation of the occurrence of large swings in the equilibrium values of an economic system. This is ex— amined in Appendix A. Appendix B presents a critical examination of De Grauwe (1983) in explaining cyclical fluctuations of the exchange rate. CHAPTER l--Footnotes 1See Frenkel (1981) and Mussa (1979). 2But the chapter provides references to the mathematical treatment of the theory. 3See Whitman (1975), Dornbusch (1980a), Murphy and Duyne (1980) and Pearce (1983). CHAPTER 2 Investment Function Introduction The investment function plays a key role in this dissertation. Being of such vital importance, a thorough analysis is imperative. This chapter will endeavor to serve two purposes. First, in section I, I will specify the investment function. A survey of empirical studies n conventional theories of investment behavior indicates that capacity itilization or output and the interest rate are the two major explan- atory variables in the function. Second, sections II and III present :he structure and characteristics of the investment function which I e1y upon to explain drastic changes in economic variables (both in losed and open economies). Linearized theories of investment behavior present researchers Lth a much simplified paradigm of the economic world. However, these eories fail to explain turning points in investment and employment. remedy to this situation is to base my investigation on a non—linear iel. Section II will deal with this. This will involve a discussion the impact of real economic variables on investment behavior. Also :influence of monetary factors on investment and other economic iables will be examined in section IT. AS a final note, it is important to stress that investment is anded to be the unique paramount actor on stage in Chapter two. ————M‘MWL .1- ._._..,._._.--. 8 Any reference to other issues—-such as economic fluctuations or the trade cycle—~are used only to clarify the nature and role of the investment function. In section III, I attempt to produce an inte- grated synthesis of both real and monetary factors in the theory of investment behavior. The rationale for this modification will be made clear throughout the rest of this work. I. General Literature on Conventional Theories of Investment Behavior A starting point for the investigation of alternative theories of investment behavior is found in Jorgenson and Siebert (1968a) and Jorgenson (1971). It is a valuable starting point to become familiar ith these theories, even though their empirical conclusions have been challenged and rejected by Elliot (1973) and Eisner (1974). The following review of the literature is based on Jorgenson and Siebert (1968a), and Jorgenson (1971). The flexible accelerator model of investment was originated by henery (1952) and Koyck (1954). According to this model, the actual evel of capital (Kt) is adjusted toward its desired level (Kt) by a ertain proportion of the discrepancy between desired and actual capital 1 each period. * Kt-Kt_l = (l—A)(Kt —Kt_l) ich implies that actual capital is a weighted average of all past rels of desired capital: (D K = _ t (l A) céo Kt-g ‘ " .. ‘_A1'__" _‘-.-‘.A .'..-..wr-.. /~ _"' ‘. here the weights decline geometrically. The latter form of the flex- ‘ble accelerator is referred to as a ”distributed lag function," and he average lag of adjustment is A/(l—A). In order to turn this model into a complete theory of invest— ent behavior, additional specifications for replacement investment and esired level of capital are required: KEKt—l = It-(SKt-l ere 6 is a fixed rate of replacement of capital.3 Therefore: at It = (1-A)(11ders' equity by reducing its stock of fixed assets.9 Clark (1979) >nsiders the Q theory as a "supplement" rather than a direct competi— r to output based models. The security value model does not show a 0d performance compared to output—based models in Clark's paper. deed Clark argues: " ... output is clearly the primary determinants of non- residential fixed investment" Clark (1979), p. 103. Investment and Theories of Income Determination It is at this point that I have to consider the impact of real nomic and monetary factors on investment and, hence output fluctu- ans. Specific references to theories of trade cycles or income :tuations are meant to clarify the role of the investment function I respect to them. As shall be demonstrated shortly, investment is idered the most significant factor in explaining employment fluctu— ns. This led to an explosion of research on theories of investment —l 7 15 behavior is a non—linear form. Hicks', Goodwin's, and Kaldor's non— linear sigmoid—shape investment function revolutionarized this par- ticular field of economics. In this section and the next, it will be evident that my investment function is a modified version of the Kaldorian investment function. IIa. Real Theories of Investment Behavior Besides Haberler (1941), Hansen (1951) is one of the most authoritative work on cycle theory. As explained by Hansen, Tugan— Saranowsky (1901), at the turn of the century, recognized (correctly) :he fact that fluctuations in the rate of investment are an essential :haracteristic of the cycle. This made a drastic departure from revious theories of turning points. It is the inherent character- stics of the modern economy which make economic fluctuations inevi- able. The great influence of Tugan—Baranowsky on the literature of )cles after the turn of the century can be clearly recognized in the brk of Spiethoff (1902) and Cassel. The early roots of the acceler— lion principle can be found in Cassel (1932), where he makes a dis— hction between those durable means of production which work for the Lsumer and those employed in the production of further means of pro— :tion. The rate of interest occupies a central role in explaining :les by influencing investment expenditures. But to Cassel, the :1e is a recurrent phenomenon caused by growth and progress. Wicksell (1934) ascribes fluctuations in output to real causes, hnological and growth factors. In explaining investment determina— 1, Wicksell (1936) comes very close to what was later developed by w r' ‘r ‘ "'---- . ..~_.,__, .-. 7, 1,, l6 Keynes, i.e., the marginal efficiency of capital. Fisher (1907, 1930) seems to agree with Wicksell that techno— logical progress, discoveries, and inventions have something to do with fluctuations in output and the increase of the rate of return aver cost--Wicksell's natural rate, Keynes's marginal efficiency of :apital. However, in his subsequent work he ignores the importance of 10 >utput fluctuations. In A Treatise on MOney (1930) and General Theory of Employment, gnterest and Money (1934), Keynes argues that cycles were believed to te essentially caused by a fluctuation in the rate of investment. He ought the causes of output changes, as well as cyclical changes in he marginal efficiency of capital, in changes in the state of liquidity references (through expectations). He attributes recessions not to a ise in the rate of interest, but to a sudden collapse in the marginal fficiency of capital.11 Aftalion attributes fluctuations in the rate of investment to e dynamics of consumer wants.12 It is the capitalistic, time con— ming process of production that transforms the small oscillations of sumer demand into large swings of output fluctuations (the accelera— n principle). He perceives a violent fluctuation of aggregate demand und a relatively stable growth in consumer demand. Pigou (1927) insists that variations in profit expectations [tribute to the cycle. Autonomous monetary changes, real factors, psychological factors cause the variations. In his theory of les, expectations, though influenced by the pattern of the modern unique of production, still predominate. Monetary factors play a ge conditioning factor. 17 The essential elements of the modern theories of business cycles have been developed through an evolutionary process. Tugan— Baranowsky, Spiethoff, Cassel all stressed the role of fluctuations in the rate of investment as an important factor in explaining cycles. Wicksell and Keynes contributed greatly to the analysis of the deter- minants of investment. Spiethoff and Harrod enhanced the process by discussing the rate of dynamic factors, such as technology, resources, and population growth as determinants of investment. Aftalion, Pigou, and Clark developed and stressed the role of technique or production (time lags involved)anni the principle of acceleration; Keynes and Kahn developed the concepts of the investment multiplier and investment function. Therefore, modern theory of cycles evolved around three main building blocks (forces). These self—limiting forces are: the falling marginal efficiency of capital, the acceleration principle, and the slepe of the consumption function. The theory reveals that, so long as the economy remains dynamic, and so long as the requirements of growth and progress require investment expenditure, these strong forces make the occurrence of investment and output fluctuations inevitable. As mentioned, Aftalion's theory of business cycle was the orerunner of the accelerator principle. It was fully developed by lark (1917). These theories of the determinants of the optimal stock 3f capital were criticised by Goodwin and Chenery because of the msence of an adjustment process. The deficiencies in the dynamic djustment process of the original acceleration principle formulation ave since been largely cleared up. This was done partly due to the istributed lag accelerator of Hicks (1950), but especially through 1e work of Goodwin (1951a) and Chenery (1954). They developed an 3,- 1m. l8 adjustment process designed to eliminate the disequilibrium between the desired and actual capital stock according to a distributed lag pattern rather than instantaneously as spelled out in the original acceleration theory. At this stage, I examine the characteristics of the investment function. I rely upon this investment function to explain sudden :hanges (i.e., catastrophes) in economic variables (in closed and open economies). Kaldor (1940, 1954) presented a non-linear theory of invest- ent and output fluctuations. Goodwin (1948, 1951b, 1955) developed non-linear investment function broadly resembling Kaldor's investment inction. Kalecki's (1939, Chapter 6), Tinbergen and Polak (1950), :himura (1954, 1955), Black (1956), Rose (1967), Bober (1968), Evans .969), Klein and Preston (1969), Chang and Smyth (1971), and Varian 979) have elaborated on the non—linear sigmoid—shape investment nction and output fluctuations of Kaldor's model. The Hicks and odwin models have striking similarities.13 First, I introduce Kaldor's model. In this model, it is :umed that gross investment (It) depends positively on the level of 1PUt (Yt) and negatively on the amount of existing capital stock (Kt): It = f(Yt) - mKt, m > 0 (2-1) (BI/BY) > o, (BI/3K) < 0 :e f(Yt) is a non—linear function of Yt' The investment function . shift down as capital stock increases, other things being con- it: and is shown in Figure 2.1. IIIIIIIIIIIIIIIIIIIIIIll-ll-::::———————————s.~~«&uv~nmw~se~ 19 In Evan's terminol— , ogy, the investment func- tion is likely to be income inelastic at low levels of income because of the exis- tence of excess capacity. Due to high costs of con- . t struction and the increased Figure 2.1 (Ichimura's Figure 11.1) difficulty of borrowing, it is likely that the investment function is also inelastic at very high levels of income. Kaldor has defended this type of function: " . it is probable that dI/dx (the marginal propensity to invest), will be small, both for low and for high level of X (the level of economic activity), relatively to its "normal" level" Kaldor, 1940, p. 81. The main differences between Hicks' and Goodwin's models is ound in the functional forms of their investment behavior. Consider— ‘ng the acceleration principle: I=F(Y)+B, where F(Y) is the induced nvestment and B is its autonomous component. Goodwin's (Hicks') in- estment function can be explained through Figure 2-2, in blue (red), here y(t)=Y(t)-Y, and Y is the equilibrium level of income (Y). denotes the rate of change of the deviation of income (Y) from the quilibrium level (Y), y=dy/dt and t denotes time. Ichimura (1954, p. 216-218) compares and contrasts these eories of output fluctuation which are put forward by Hicks, Goodwin, d Kaldor. According to Ichimura, the Hicks—Goodwin theory is an celerator model of investment in a context of "overinvestment —-—I——'“ ‘ “‘“’" w ‘ '”"‘ ”‘" ‘ " 20 / .. . r NW 57 red blue Figure 2.2 theories" framework; while the Kaldorian non-linear investment func- tion is flavored with the color of underconsumption theories, in [chimura's analysis. IIb. Monetary Factor in Short Run and Long Run Models of Investment Behavior In the previous section, I examined the investment function in .real sense. The Goodwin—Kaldorian investment function provided a edium—run explanation of employment fluctuations. Therefore, the im- ct of monetary factors on the investment function was ignored. In is section my objective is to depict the role of interest rate in e functional behavior of investment. Short—term fluctuations in employment and output can be ex— ained by distinguishing between investment in inventories (working pital) and investment in fixed plant and equipment. Abramovitz (1950) cognized the importance of inventory investment in business cycle avior. Metzler (1941, 1946, 1947) developed an analysis of entory behavior within the framework of the multiplier and _- - . , .__ - - Mflg , all ———“‘T' ”IRA... 4-H“ .. - _—_ Mn— 21 accelerator principles. Metzler's theory has been developed and elabor— ated by Klein (1950), Nurske (1952, 1954), Darling (1959), Fromm (1961), and Lovell (1961). Bryant (1978) supports the hypothesized effect of inventory stock upon output. Blinder (l980:4) develops a macro model in the line with Metzler's model of inventory fluctuations. Blinder (1981) reaffirms the importance of inventory fluctuation in business cycles. But Gordon (1952) criticized the Metzler theory of short term cycles. He argues that: " ... the trouble with such a theory is its artificially precise character and its attempt to explain the minor cycle in purely mechanical terms. So far as minor cycle is concerned, Keynes was certainly much closer to the truth, if less precise, in his emphasis on "minor mis— calculations" and changes in short run expectations" Gordon (1952), p. 315-316. In spite of the above argument, it would be a serious theoreti- cal error to ignore the crucial role of the interest rate in explain— . . . 15 . 1ng 1nvestment behav1or. The monetary theories of investment and employment changes emphasize the impact of inventories rather than investment of a fixed nature (machinery and equipment). The latter as been largely stressed in real theories of employment fluctuations. uthors of the monetary school correctly recognize the short run henomenon of the interest rate and its impact on inventory and other on—fixed investment.l6 To be more specific, at lower levels of nterest rate, sellers augment their orders for manufactured goods. his leads to higher demand for money and, therefore, higher prices or goods. Sellers continue to increase their orders, which generates re employment, output, money income, higher demand for money, and 'gher prices for goods. This process continues in a cumulative manner til the capacity of banking system to supply loan decreases. This 22 creates an upward pressure on the interest rate in the face of ever— ‘ncreasing demand for loans by sellers and merchants and a declining apacity of the banking system to provide these loans. At this point ure monetary theories explain the downswing of the economy. Money ncome, demand for money and goods, prices for goods, all decline. The rocess continues this pattern in a cumulative manner until the iquidity positions of bank improve with their capacity and desire to nd to the merchants. Besides an ever—increasing optimism among mer- ants and in the industry will have convinced them that the worse has ssed. This (i.e., the new lower level of interest rate) would set in. tion forces that will turn around the economy toward higher employ- nt. In this theory, the interest rate is considered as the true use of fluctuations. Once the interest rate activates, it is the onomic system itself (the demand for money and the velocity circula— on of money) that generates the cumulative process of upward or inward motion. . The Modified Investment Function and its Mathematical Derivation I have set the stage for my main theme. In section IIa, I .ained the influence of real factors on investment and employment vior; in section IIb, I presented the monetary side of the story. of these alternative theories explains turning points in invest— and output. Individually, they are significant contributions to dvancement of the theory. But they fail to provide a complete sis of the phenomenon. It is my aim to produce an integrated synthesis of real and ry theories by applying monetary market disturbances in terms ——'— ..1..u_-.....,. I.._._ 23 of interest rate to the real theories of investment and output fluc- tuations. I choose this model because, first, this model is based on a tractable function which makes mathematical work on it possible; second, Koyck's empirical investigations (the only empirical work) on the time—shape of the reaction of capacity to changes in output led him to believe that: ” ... the results for the industries investigated are not favorable for the acceleration principle but are much more in accordance with the theory of Kaldor" Koyck, 1954, p. 109. Thus, I modify the sigmoid-shape investment function in order to accommodate with these matters:17 It = f(Yt) — mKt — nrt m,n > O (2.2) he first two terms of the right hand side of the above equation explain ovements in fixed part of gross (or net) investment (I), while the 11rd element (nrt) explains mainly inventory fluctuations in 1.18 lerefore, whereas the Kaldor's sigmoid—shape investment was able to ift upward or downward over time (in the long run, due to the process capital decumulation or accumulation reSpectively), my investment lction can also shift upward or downward in the short run, Figure 2.3. se short run shifts are the result of monetary disturbances. The mathematical exposi— 1 of the investment function ased on Allen (1967). He ies the following mathemat- structure (Equation 2.3) to 19 It=f(Yt)-mKt—nr —linear accelerator model. t Figure 2.3 24 f(Y) = b((a+b)/(ae‘VY+b)) — 1 (2—3) here a denotes the net capacity of the capital goods industries (in llen's terminology), 1: denotes the scraping rate of capital equipment, nd v denotes the capital output ratio. Both equation (2-3) and igure 2.4 define the investment unction for both positive and gative values of Y.20 Mathe— f(Y) tically, f(Y)+b is the same f(Y)]bl units up if b > 0; uation (2-4) and Figure 2.5. other words, I redefine the “““““““ b igin with respect to which 2 function f(Y) is defined: Figure 2’4 (Allen, 1967: p. 379) . "VY ‘VY f(Y)+b=b((a+b)/(ae +b)—b)+b=b(a+b)/(ae +b), , (2’4) 4v’J'Y' // f(Y)+b Again, mathematically / I +£)+b is the same as a/ T ,’ f(Y) +b units lilto the right /’ J>0, equation (2—5) and ‘,/”’ re 2.6. Figure 2.5 25 f(Y—£)+b=b —V?Y?Z) = b 0 Thus, investment as a function of income, the stock of capital, and interest rate is rewritten in the form: I=q/(l+de-VY)—mk—nr (2‘6) q,d,v,m,n > 0 Based on the shape of investment function, derived in this chapter, goods market equilibrium conditions will be examined in the next chapter. IIIIIIIIIIIIIEIII:_________________________________—_T""W ‘ 26 CHAPTER 2--Footnotes See, for example, Blinder and Solow (1973, 1974), Tobin and 3uiter (1970), and Tobin (1979). See Gordon and Klein (1965) Readings in Business Cycles, p. 3 3 See Jorgenson (1963), p. 254, Jorgenson and Stephenson (1967), M 192—212, and for individual firms see Meyer and Kuh (1957), p. 91—94 4Jorgenson and Siebert (1968a), p. 683. 5 For full detail on the neo-classical theory of investment ehavior, see Jorgenson (1963, 1967), Jorgenson and Siebert (1968a, 968b), Jorgenson (1971), and Clark (1979). For more detail, see Johansen (1959), and Bliss (1968). 7 Examples of theories based on putty-putty technology are to e found in Jorgenson and Siebert (1968a, 1968b) which are already dis— ussed. 8 Modigliani cities Ando, Modigliani, Rasche and Turnovsky Ll974). loSee Fisher (1925). For more detail, see Tobin (1969), Brinard and Tobin (1976), .ucas and Prescot (1971), and Ciccolo (1978). 12 See General Theory of Employment, Interest and Money, Ch. 22, notes on Trade Cycle," p. 313—332. Cited in Hansen (1951), p 13 . 348. Ichimura (1954), p. 200. 1['Evans, M. (1969). SHajela, P. D. (1952), Chapters VIII, IX, and K provide a .mple version of the impact of interest rate on investment function. l6Mass summarized views of the researchers who empirically esti— ted the important role of interest rate: "Abramovitz recognized that inventory changes accounted for 23 percent of the total change in output during business-cycle expansions and for 47 percent of the change in output during contractions before World War II. ...T. M. Stanback, Jr., found that changes in manufacturers' inventories accounted and 1957/58. for 79, 56, and 25 percent, respectively, of the change in gross national product during recessions of 1948/49, 1953/54, Finally, L. R. Klein and J. Popkin argue that 27 the business cycle could be virtually eliminated if inven— tory fluctuations were reduced by 75 percent, ... Mass, N. J. (1975), p. 19—20. 17F orsake Of simplicity, I pursue my investigation of the vestment function, in terms of explanatory variables f(Y), K, and in a linear fashion. 18 Besides monetary theory (in support of the important role the interest rate in explaining investment behavior), prominent .onomists of the past such as Cassel and Fisher believed that interest rte plays a major role in explaining the behavior of investment func— .on. "Fisher comes to very much the same conclusion as Wicksell: that the initiating impulses for expansion or contraction have in the past come in large measure from technical progress, discoveries, and inventions, which raise the "rate of return over cost"-—Wickse11's natural rate or Keynes' marginal efficiency of capital” Hansen (1951), p. 333. 19 R. G. Allen (1967) Macroeconomic Theory, p. 379—380. 20Recall that Allen (1967) defines investment outlays in terms of the rate of change in income (Y) in an accelerator model, rather than in terms of the level of output or income (Y). Therefore, even though, his diagram holds true for an accelerator model, I need to introduce some modifications. These specifications do not change the mathematical structure of the investment function, but make the investment function both up and to the right so that real level of income could assume positive values only (in the north-east quadrant). This task will be done without disturbing the mathematical structure of the function. CHAPTER 3 Partial Equilibrium Analysis of Income Determination uction This chapter derives the IS and LM curves. I present them Ltely, in section one and two, respectively. A simultaneous sis of real and monetary equilibria will be postponed until er five; and examination of foreign repercusions on domestic Lcial macro-economic policies will be dealt with in the second of this dissertation. Generally speaking, the IS curve gives the equilibrium pairs he interest rate (r) and income (Y) in the product market, where iemand for and supply of goods are equal; Figure 3—1: Y = C(Yd) + I(Y,r,K) + G (3.1a) Y - C = I + G and S (3.1b) Y = real income (real gross national product, GNP) Yd = real disposable income = Y - T + R R = transfer payments I = interest rate on home bonds K = nominal values of equities issued by firms to finance their investment C = a consumption function, real consumer expenditure I = real investment demand (equation (2.6)) 28 29 G = real government purchases of goods and services = real tax revenue as a function of real GNP — tY C(Yd) in equation (3.1a) explains consumption as a function of Josable income. The investment function I, as explained in the previous chapter, a function of income (Y), interest rate (r), and stock of real capital ds (K). Each element in the above equation is in a planned or ex ante el. Thus, I is the level of planned, fixed and inventory investment, :re al/aY >0, BI/Br <0, and BI/BK <0. Government expenditure is con- iered as an exogenous variable. Consumption is related positively to sposable income. Figure 3.1 The LM curve gives the equilibrium pairs of Y and r in the .nancial market——where the supply of money is equal to the demand for ney, Figure 3.2. The demand function for real balances is given by: Figure 3.2 ound = was») (3.2) 3O 1 am/BY >0, Bm/Br < 0, am/Bw > 0. Where w denotes real wealth. : the stOCk of real balances. P is home price level.1 Assuming an exogenously fixed supply of money, equilibrium in financial market is realized by: M/P = m(Y,r,w) (3.3) A simultaneous solution of equations (3.2) and 3.3) gives Librium in both goods and financial markets and thus in the economy. The above apparatus is a popular model of income determination. 'der to move from a static analysis of economic activity to a dynamic tigation of output fluctuations, we introduce a specific non—linear of investment function, was explained and analyzed in the previous er. This investment function (sometimes called the "sigmoid—shape" tment function) is closely associated with N. Kaldor; other vari— s > of it have been used by Goodwin and Hicks, as discussed in 2r tWO . 'heories of Income Determination In this section, I depict the role of the sigmoid—shape invest— unction in explaining income fluctuations in a simplified economy. aim the relationship between the real theories of income fluctu— and non—linear investment functions. A starting point is Kaldor's model of the trade cycle (Kaldor His model is a very simplified commodity market which is not try consistent with a Keynesian or IS model. According to Kaldor, :e of change of output is proportional to the difference between tent and saving: 31 Yt+l — Yt = y(It-St) YI>0 (3.4) re Y is constant and is called the "response coefficient" of supply "effective demand." The long run equilibrium is defined where = I = ' St’ and t (Ir)t where Kt and Yt do not change and (Ir)t is the Lacement function. Kaldor defines the replacement investment func— 1 (Ir) as a linear function of Y, as shown in Figure 3.3. I = u-Yt + z, (3.5) e z is a constant depreciation per unit period. Y * * * Y Ya Yb C figme33 At point A and C in Figure 3.3, marginal propensity to invest is less than marginal propensity to save (mpS), the opposite is It B. Therefore A and C are stable equilibria, while B is an un— : equilibrium point of this simplified model of income determina- This is true because at A, the saving function cuts investment on from below--mpS > mp1. At point B the investment function he saving function from below—-mpI > mpS. If the economy is 11y at A (or C), any disturbance which moves the economy beyond 32 * r Yc) is self—correcting. This is true because levels of saving reater than investment. Aggregate demand is less than aggregate y. Therefore, output levels shrink. The mechanism works in . . * * se, for disturbances which shift the economy below Ya (or YC). : point B, if the initial equilibrium condition is disturbed, is no self—correcting mechanism to restore the initial equili- * Yb. This is true because, beyond Yb the saving level (and ore, aggregate supply) is less than investment (aggregate demand). * * he economy continues the course of expansion beyond Yb' Below Yb, * onomy continues the course of contraction until it reaches Ya’ 3.3. * Point B gives a long run equilibrium value of Yt’ Yb. The model :able at this point. Thus, the economy is unlikely to remain ’or a prolonged period of time. But at C (or A) gross investment (is less than) replacement investment. Therefore, over time, estment curve shifts down and capital accumulates (up and capital ates). Thus, output declines (increases) toward its long run * * 3 Yb. But the long run equilibrium level of output, Y , is not and therefore, output tends to move away from it. This produces . 2 :al pattern in output levels. At this stage I would like to present my own version of income tation in a Keynesian framework. In a good market, disposable income is specified as: C+S=Yd=Y—T+R (3.6) ium is given as: Y=C+I+G (3'7) 33 s is: S = sYd — f f > 0 (3.8) nsumption is: C = Yd — S = Yd - sYd + f = f + ch (3.9) s - marginal propensity to save c — marginal propensity to consume f = autonomous consumption. vestment function, I, is the modified version of sigmoid—shape nent developed in Chapter two. Replacing Yd in equation (3.9) 1uation (3.6) we get a conventional consumption function: C = f + c - (Y + R - T) (3.10) :uting C from equation (3.10) and I as the non-conventional func- ,erived in the previous chapter) into equation (3.7) I get the IS n implicit form. Y=f+c.(Y-T+R)+G+I(——9:—Y,K,r) (3.11) l+dev Following the non—linearity of the investment function, the IS 5 also non-linear. I can examine and analyze its characteristics graphically or mathematically. Graphically, equilibrium in the goods market is achieved when G. This is so because: Y = Yd + T—R = C + I + G, and Yd = C + S. Therefore, 34 + T - R = C +'I + G, or S + T — R = I + C, Figure 3.4. S+T-R ;+T-R b Figure 3.4 A, B, and C, where leakages are equal to injections (aggregate equals aggregate supply) are equilibria. Points A and C are goods market equilibria while point B is an unstable equilibrium. . In order to derive the IS curve, I should trace out equilibrium pf Y against different rate of interest. If the real interest uctuates (goes up or down), the investment function and thus the we will shift (down or up respectively). To derive the IS curve ally, suppose that the economy is represented, initially, by 1d Go+I(Yt’ Ko’ r0) in Figure 3.5. Equilibria a0, b0, and co 'epresented in Figure 3.6 for the same rate of interest, r0. rhe interest rate rises to r1, the G+I curve would shift down 'equilibria, bl’ and c1, in Figure 3.5. They are redrawn 1’ and c1 in Figure 3.6 for interest rate r1. Suppose r2 is rest rate at which G+I curve is tangent to the S+T-R curve 35 S+T—R C2/ 1 S+T‘R ’ ‘- t 0 I I C G+I(Y,K,r) / o r‘Gr+-I(Y,K,r) t o o t o o / ’r’ ’pt‘". I, I’c ” i, If I . I I I I / b0, ’ l I [I / §2/ 1,, I ’ I [I I ) ”v I I ’4 " / 30 I I! a" I L”’ I a ’l a "r 'd”' Y r t Figure 3-5 ‘A at 42 9 6" ‘II— ‘-—- 'o ST‘s E 9—*—-9-—§—~ a —2 Y Figure 3-6 rows indicate dynamic behavior of real income. (at P Figur furth outpu For e tange curve the p for a can b secti 310pi Figur equil inves lead A dec the I will Both in pr , Slope 36 nt c2). Then a2 and c2 are the only equilibria at this rate, 3.5 and 3.6. Any increase in interest rate beyond r2 induces a downward shift in G+I curve with only one equilibrium level of (which declines as r rises). A reduction of interest rate causes an upward shift in G+I curve. lple, r_2 is an interest rate, lower than r0, which makes G+I to S+T-R curve at point a_2. By connecting such points in Figure 3.6 I develop a non—linear 'S(Ko)' This curve gives the equilibrium pairs of Y and r in not market, where the demand for and supply of goods are equal, ven level of capital stock, KO. An interesting characteristic of the IS(KO) curve, is that it ivided into three sections, two (conventional) downward sloping and one (unconventional) upward sloping part. The downward arms are loci of stable equilibria (such as AB and CD, in 5). But the upward sloping arm is the locus of unstable a (such as BC, in Figure 3.5). enerally speaking, in a conventional downward sloping IS curve, t is a function of interest rate alone, I=I(r). This would he kind of IS curve which appears in most of the literature. in the rate of interest would cause investment and therefore urve to shift up. The increase in the level of investment ce an increase in equilibrium income through the multiplier. g (S) and taxes (T) increase. There is again an equilibrium market at a lower interest rate and higher income level. curve depicting equilibrium pairs of Y and r must be negatively in Figure 3.1 or downward sloping arms of IS(KO) in Figure 3.6. ——— r mas 37 But as seen above, if investment is a function of both the ‘est rate and the output level, then the whole picture changes. lope of the IS curve is greatly changed by making investment d on both Y and r. A decline in the interest rate would lead to crease in investment, and that in turn leads to higher levels of e which leads to higher investment (i.e. I=I(r,Y)). The upward ng section of IS(KO) is quite plausible if investment, I(r,Y), fficiently responsive to increases in income. A rapid expansion some could lead to a surge in S+T level much higher than the 11 increase in I+G. To restore equilibrium in the goods market atween G+I and S+T, an increase in r, to cushion the surge of a is needed. Arrows in Figure 3.6 indicate the dynamic behavior 11 income. The mathematical investigation of the IS curve, equation (3.11), rms the nature of IS curve derived graphically, in Figure 3.5 and Rewrite the IS curve, equation (3.11), in a linear form in terms nd K, if T=ty. Y=f+c(Y-tY+R)+G+—-—-g~_—_——-mK-nr (3.12) l+de v (1_C(1-t))y = q/(l + de‘VY) — mK - nr + G + f + cR (3.13) me rearrangement, we derive an equation for the IS curve: r ._.. (l/n) {q/(l+de-VY) _ (1-C(1—t))Y—mK + G + f + CR} (3.14) In order to verify the nature of the extrema of the above equa- e take a partial differentiation of (3.14) with respect to Y: WhI 38 am = (NH) r(qdve‘VY)/<1+c1e"’Y>2 - <1—c(1—t)>} set l-c(l-t) = fl, then 3r/3Y = (l/n) {(qdvé'Vh/t(c»a"Y+c1)/(e""">12 - 2} = (1/n) {(qdvé’Y)Mm”)2 - 2} (3.15) 3r _ ‘57 - 0 - {- qdve"Y + £(d+eVY)2} = o - {2 (d+eVY)2 - qdveVY} = O - {2 d2+2£deVY+£-eZVY-qdveVY} = 0 — {2 eZVY+(2£d-qdv)eVY¥£d2} = O . . vY lic1ty we define e =X (3.16) e, 2 2 {z X +(22d-qdv)x+2d } = 0 (3.17) he roots: {qdv—szi[(2d-qdv)2—422d2]%} /(22) >4 II 1, 1 {qdv-ZfidtIqV(qv-4£)]6} /(2£) (3.18) , the necessary condition for the existence of extrema is that: V > 4% (3.19) of the partial derivative equation (Br/BY) is the same as of the equation -(X-X1)(X-X2). Where X1 and X2 are roots of 39 n (3.17) and suppose Xl < X2. To determine if X1, X2 are ve) maximum or minimum of the above equation I construct table TABLE 3.1 0 X1 X2 00 ---- 0 +++ +++ ++++ -...-- _...__ _____ 0 ++H. -X2) ' -H-H- 0 —————— 0 'H-H- —X2)=(3r/8Y) -——— O ++++++ 0 ———— : row shows that (X—Xl) is zero at X=Xl’ positive (negative) at X greater than (less than) X1. The second row shows values of ll to, greater than or less than zero as X is equal to, greater less than X2. The third row shows different values of (X—Xl)- different levels of X1 and X2. These values are derived by ation of the first and second rows. As is indicated in the w, the partial derivative Br/BY values at different level of n opposite to (X—Xl)(X-X2). ntitutively, the fourth row indicates that the partial deriva- he IS curve is zero at X1 and X2 (extrema points), negative ) at values of X less than X1 and greater than X2 (between ). In other words the IS curve is sloping downward (upward) at X less than X1 and greater than X2 (between X1 and X2). . X1(X2) is a relative minimum (maximum) point of the IS curve. w' ' f2, Acc 0116 shc Th1 40 'he discussion can be extended from values of X to income, Y. le vY2 ; to equation (3.16), e =X1 and e =X . There is a one—to- 2 .ionship between X and Y. Therefore, the IS curve (3.14) has minimum at Y1 and relative maximum at Y2, while Y1 < Y2. 'herefore, the IS curve, r(Y), behaves something like the graph Figure 3.7. r 1'. IS t Y1 Y2 t Figure 3.7 . above mathematical investigation confirms the graphical of the IS curve. ad‘ 41 CHAPTER 3—-Footnotes lReal wealth is defined as: P {1:1 P EF + +— P 2} n le ml where w - real wealth % = real money balances held by home private citizens BP §_ = real home bonds held by home private citizens EFP —5— = real foreign bonds held by home private citizens and denominated in home currency E = Exchange rate, home currency price of foreign currency *6 II Home price level. 7 'Readers interested in following the Kaldorian model on an level are advised to consult Ichimura (1954). whi: tual hig} 0011: mm met‘ fer. whe1 ECO] We] CHAPTER 4 Catastro he Theor My objective, in this thesis, is to build economic modeb are capable of explaining large swings and oscillatory fluc- ns of economic variables. Catastrophe theory (CT) provides a complex mathematical framework in achieving this objective. r, I stress the economic rationale, rather than mathematical irations per se, to explain the equilibrium path of real , exchange rate, and interest rate.1 Therefore, this chapter ended to provide only the essentials of catastrophe theory. :erials explained in this chapter will not be dealt with Ltly in the rest of the dissertation. However, I believe that :k of mastery in CT may damage this dissertation‘s effort in ring equilibrium behavior of economic variables. Varian (1979) describes catastrophe theoretic models as para— ed dynamical system, described explicitly by a system of dif— al equations: X = f(x,a) is an n—vestor of state variables (endogenous variables in : models), x is the n-vector of their time derivatives, and a rector of parameterS. In these models, parameters may change 1e, but at a much slower rate than the state (endogenous) 42 variab econou label] demand Here x rate a variah and is face. Stewaz “3mm 43 les. The exposition of the theory is pursued through a macro— ic model, similar to the one developed in Chapter three. Suppose there are n commodities, real and financial, 2d 1, ..., n. The equilibrium conditions, E, make excess , 2 for each commodity zero: (E): el(al, ..., am, x1, ..., xn) = 0 e2(al, ..., am, x1, ..., xn) = 0 e (al, ..., am, x1, ..., xn) = 0 (x1, ..., xn) is a n—vector of endogenous variables (interest 1 income in Chapter three) "a" is an m—vector of exogenous as (i.e., financial policies). S denotes the set of solution a1, ..., am; x1, ..., xn of (E) :alled the equilibrium surface (manifold) or stationary sur— 1 point (ao,xo) in the surface S is singular if: det(3ei/3xj) = 0 ), given eis are continuously differentiable. a0 is defined 0 astrophe if there are xo such that (ao,xo) is a singular Sussman (1975), Lu (1976), Golubitsky (1978), Poston and (1978), and Sussman and Zahler (1978) all use the following :o depict the simplest type of catastrophe: the fold catas— SuPpose the equilibrium equation is: Then, Figure left c tions, the rj qualit of sol of a g zero. trophe catasl scien< which funct: Where ables (20113]:I §, Wh 44 e(a,x) = a-x2 = 0 Then, the equilibrium surface is the parabola a=x2 in the a,x plane, Figure 4.1. As shown in Figure 4.1, there exists no solution to the left of a=0, but two solu- x tions, given a value of a, to the right of a=0. Therefore, qualitative characteristics catastrophe of solution change for values )f a greater and less than Figure 4.1 zero. Then, a=0 is a catas- :rophe point. The second type of catastrophe is the cu p. This type of :atastrophe has widespread applications in both natural and social 3 :ciences. The cusp equilibrium surface, S, is equivalent to: ll 0 3 x -a x-a hich is the first derivative, at extrema, of a function called energy unction, 6.4 _ 3 _ (BC/3x) — x -alx—a2 — O lere one dependent variable, x, is explained by two independent vari- lles (parameters), a and a Figure 4.2. l The equilibrium surface (stationary surface), S, in Figure 4.2, 29 nsists of two qualitatively different parts. The intermediate part which lies between the fold curves, F1 and F2, is called repellor Pro j e \ Slll Whl b0! 45 I I 51 / origin S2 F 1d u Equilibrium sur- : g c rve, face or Equilibrium Old curve, 2 Manifold or Station— ? F1 . ary Surface, S >jection , T2 U]. I- - a -- - —- -" - — - / -" Catastrophe set ya P, cusp point Parameter set P = origin projected Figure 4.2 urface where (BZG/Bx2)=3x2—a <0 so that G is at maximum.5 The l emaining S—S of equilibrium surface, is called attractor surface, Jere (32G/3x2)=3x2-a1>0 so that the energy G is at a minimum.6 The )undary between the attractor and repellor surfaces is the fold 2 1rve depicted by F and F2, where (32G/3x2)=3x -a1=0. Projections 1 3 fold curves from the equilibrium surface onto the catastrophe set .ve a cusp-shape bifurcation set depicted by BIPBZ‘ They are loci— " catastrophes. The significance of a cusp in explaining the behavior of 3 del (system) can be studied further. Suppose 81’ t1 (or 52, t2) om the parameter set correspond to Sl’Tl (or 82, T2) of the the 3110 The one sta 00C 0C( IE] eq1 46 >r set (equilibrium set), in Figure 4.2. By changing parameters . you could change the value of the endogenous variable from S1 The transition occurs in a continuous and smooth manner. The .lds true if by changing parameters, the system moves from S1 to .1 to T2, on the equilibrium surface. The picture is quite :nt if by changing parameters you wish to move the system from 2. The system moves smoothly from S to U1 and back, T2 to V1 2 .k. If at Ul you increase the value of parameter (beyond U1), tem has no choice but to jump from a stable state at Ul to stable state at U2 (point u on B curve, in catastrophe set). 2 to U is fast. The transition from ement of the system from U1 2 ble equilibrium to another stable equilibrium, through an un- equilibrium, is defined as catastrophe. The same phenomenon when the system reaches the lip of the pleat at V1 from T2 to An important characteristic of the catastrophe theory is the ace of the so called "delay rule." In the jargon of CT, the Lay rule means that the system stays in the original stable rium surface as long as it can. Suppose the system was at 2 in the equilibrium surface S in Figure 4.2. By changing para- 11, 32 you can change the equilibrium value of the endogenous : from S to T . The catastrOphe occurs at point U1. However, 2 2 :conomy was at point T in the equilibrium surface S of Figure 2 a picture is different as far as the occurrence of catastrophe 2 to $2 and the phe occurs at point V1 (v) in the equilibrium surface S rned. In this case the system moves from T Dphe set P) rather than at U1 (u). alas (:th trol refe 47 The fold and the cusp are two of the seven catastrophes :lassified by Rene Thom. He proved that these seven (elementary) :atastrophes depict all possible discontinuities in a paradigm con— :rolled by no more than four parameters. Interested readers are referred to Lu (1976), Golubitsky (1978), and Poston and Stewart [1978) for a complete exposition of the seven elementary catastrophes, 7.1. Arnol'd has classified catastrophes up to at least 25 dimen- sions.7 However, natural and social phenomena can best be explained >y the seven elementary catastrophes. The mathematical structure of catastrophe theory has been explained by Lu (1976), Golubitsky (1978), and Poston and Steward (1978). To avoid confusion and for the sake of simplicity, the theory :ould be defined in non-technical terminology. It asserts that while :here are an infinite number of ways for a system to change continuously {i.e., moving from S to T Figure 4.2); there are only seven struc- 1 l’ :urally stable ways for it to change discontinuously (passing through Ln unstable state).8 The nature of the catastrophe depends on the Lumber of exogenous variables in the model. Debreu (1976) argues that an economy that has an equilibrium such that in any neighborhood of e, there are infinitely many equilib- ia, is unstable and "... the explanation of the equilibrium is essenti- "9 He continues that "... the economic system 11y indeterminate ... s unstable in the sense that arbitrary small perturbations from e to neighboring equilibrium induce no tendency for the state of economy 3 return to e."10 Therefore, through an elaborate mathematical expo- Ltion he shows that unique equilibria is not necessarily considered : a good assumption. Furthermore, equilibria are not continuous in the par is a us rim va 48 >arameter of the paradigm at all time. Thus, catastrophe theory useful technique to explain discontinuous changes in the equilib- values of a system. Golubits (equilib vant (er constrai 49 CHAPTER 4——Footnotes 1 . For a mathematical treatment of the theory, see Lu (1976), :sky (1978), and Poston and Stewart (1978). 2Sussman and Zahler (1978), p. 141. 3See Sussman and Zahler (1978). 4In economics, behavioral functions such as demand functions ,brium in a portfolio—balance model) are derived from the rele- :nergy) objective function of utility maximization subject to tints (risk minimization subject to wealth constraint). 5A locus of unstable equilibria is defined as repellor surface. 6A locus of stable equilibria is defined as attractor surface. 7Arnold (1972, 1974, 1976) cited in Golubitsky (1978), p. 353. 8Woodcock and Davis (1978), p. 42. 9Debreu (1976), p. 281. loIbid. m cal l chap! clo s will open 38ng the‘ work the econ in t (198 11011. deve neit come CHAPTER 5 Macroeconomics and Catastrophe Theory ac_t1_or1 The economic analysis of this chapter is based on the theoreti- Lndations that were derived in the previous chapters. In this ' I end the theoretical investigation of discontinuity in a economy. The building blocks developed for a closed economy : used as a spring board into theoretical investigation of an onomy's interactions and foreign exchange market. This introduction presents a formal discussion of conventional te demand and supply models. In section one, I develop further LM framework of Chapter three. It will incorporate that frame— to a catastrophe-theoretical model. In section two, I derive ernative (unconventional) aggregate demand curve, and discuss 2 equilibrium within an aggregate demand-supply framework. The macroeconomics of aggregate demand and supply presented section is based on Sargent (1979) and Dornbusch and Fischer They are developed here in order to set the stage ready for rentional analysis of aggregate demand and supply to be ad in section two. As far as aggregate demand model building is concerned, classical nor Keynesian economists have any disagreement mg the structural equations. The introduction to Chapter 50 three 1 tions shown to der gate d the ec equili A pric P1 imp real b right 0f rez tatior Preset leave: With‘ unknm KeYne Probl. a 10w deter respo meat rtimed 51 three reviewed aggregate demand in terms of IS and LM curves; equa— tions (3-1) and (3-3). For convenience, these IS and LM curves are shown in Figure 5-1. I now use Figure 5—1 r to derive the standard aggre— gate demand curve. Suppose o the economy is in an initial r1.... equilibrium, depicted by E0. A price reduction from P0 to P1 implies an increase in Figure 5-1 real balances, holding other things constant. LM curve shifts to the right to LM(P1). Thus, a decrease in price level leads to higher level of real income, Figure 5-2. The Hicksian interpre- P tation of income determination, presented in IS-LM framework, leaves the economic system with two equations and three unknowns, Y, r, and P. The Y Keynesian approach to this Figure 5-2 problem is to assume P to be exogenous. Therefore, for an economy at a lower level of employment than full employment, aggregate demand determines the equilibrium level of real income. Monetarist might respond to the same problem by assuming Y to be fixed at full employ— ment level (so that P is determined endogenously). Generally speaking, the limitation of an exogenous price is remedied by amending this model with an aggregate supply function that relates (W/P Equatior tion of run. Ec generate physical work co] economi. 1: P, a‘ Point i is trea model i equatio Keynesi aggrega aggrega the fle EQUilit 18b0r I] 52 relates real income and price level: Y = F(K,N) (5-1) (W?) = (aF K1) investment outlays increase and the investment curve shifts up, for each level of interest rate, Figure 5-6. S+T—R S+T-R .. G0+I(Yt,Kl,ro) G+I . ( ) GO+I(Yt,KO,rO) Yt Figure 5-6 Assume that there are two economies which are exactly alike except in their level of capital stock (K1KO>K1 Y t Figure 5—7 The converse is true for capital accumulation (Ko —-—> K2). I can add further insight to our understanding of this economic model by investigating fiscal policy. An increase in government spending, G, (revenue, T), implies an upward (upward) shift of G+I (S+T—R) curve. This adjustment is reflected in an upward (downward) shift of IS curve, respectively. S+T-R S+T-R -—- Gl+I (Yt ,Ko ,ro) G'I'I(.) GO+I(Yt,KO,ro) ' Y A1 : Yl 2 Y t Figure 5—8 The IS(K0,GO) curve, in Figure 5—9 is derived by tracing out equilib— rium levels of output (in Figure 5-8) as the interest rate changes. As the interest rate rises, the investment curve and thus, the whole GO+I(Yt,KO,rt) curve shifts down, holding Go and Ko constant. The critical point Y1 (Y2) of the IS(KO,GO) curve in Figure 5—9 corresponds to the tangency of S+T—R and Go+I(Yt’Ko’rt) curves as interest rate dec] that siox to ( the cri exe cur tha whe ThJ' tic 57 r \\ ,I’I-‘\ \ b I , \\ \ a. I I C r \‘l }’/\ \ l O a ~- b : C ‘\ o ' o . o : ‘LS(KO,Gl) E IS(KO,GO) Yt Figure 5-9 declines (increases) and Go+I(Yt’Ko’rt) shifts up (down). Suppose that the initial conditions of this economy are changed by an expan— sionary fiscal policy. The GO+I(Yt,KO,rO) curve would shift upward to Gl+I(Yt’Ko’ro)' In order to derive the new IS curve I trace out the equilibrium level of output as interest rate is changing, Figure 5—8. Points ai, bi’ C1 in Figure 5-9 correspond to points Ai’ Bi’ and C1 in Figure 5—8 respectively, which in turn corresponds to G1 fiscal policy for a given r.. Figure 5-8 and 5—9 clearly show that a smaller (greater) reduction of interest rate would bring our system to the critical point Y1 (Y2) after the expansionary fiscal policy is executed than before the expansionary fiscal policy. Thus, our new IS curve, 15 (Ko’Gl) lies above the initial one IS(KO,GO), Figure 5-9. Ignoring the unstable arm of the IS curve temporarily, I see that the stable arm shifts rightward (leftward) as capital decumulates (accumulates) and investment increases (decreases). This is true when government spending, G, increases, or its revenue, T, decreases. This phenomenon is in complete harmony with the conventional applica— tions of IS curve. Conventional IS curves shift to the right as in- vestment and/or government outlays rises or government revenue HBt of tr using CEIDG 38831 Grap' upwa fixe both solv equa equ: Sta] 0n- 58 of transfer (T—R) declines. Now I turn to LM curve. I assume a normal-shape (conventional) LM curve. Appendix A presents an additional source of catastrophes, using a non-conventional LM curve. However, this thesis is not con— cerned with this second source of catastrophes. The LM curve gives the equilibrium pairs of r and Y in the asset market——where a portfolio equilibrium is achieved; equation (5—5). fi/P = m(r.Y) (5—5) Sraphically, portfolio equilibrium in the asset market is shown by an Jpward sloping LM curve in Figure 5-1. A simultaneous solution of these two (equations) curves—~18 and .M-—gives equilibrium in both markets and thus in the economy, with fixed prices. We can find the pair (r,Y), that gives equilibrium in >oth markets by these two curves (IS&LM) in the same quadrant, that is ;olving equations (5—4) and (5-5) simultaneously. This is shown in :quations (5—6) and Figure 5—10. r = l (———‘1— — Y+Co(Y-tY+R)+G—m.k+f)) (5—6a) t n —vY l+de l - _ = _ __ x o Ownward shift of the IS curve and a rapid jump (catastrophe) of the :onomy from point F on the LM curve to the point g on the same :rve. Further continuation of fiscal policy reduction makes the onomy contract smoothly and continuously after it passes the point an the LM curve. Ib. by t ore econ the in I cor: lib] At 1 men‘ Fig shi out dec IS dow thr 3—6 the on] C0] tht Wit De: 61 lb. Long Run The long—run equilibrium path of the economy is influenced by the change in the stock of capital and is depicted by Figure 5-12 or equation (5-6). Suppose the economy is initially at the equilibrium point P, in Figure 5—12, which corresponds to the equi— librium point C in Figure - f h o 3 3 o C apter three Figure 5_12 At point C gross invest- ment exceeds replacement investment (i.e., positive net investment), Figure 5-13. Therefore, due to capital accumulation, investment curve shifts downward, and output and employment : decline. Thus, the IS curve is pressured downward (see Chapter Yt three, FiguresB-S and Figure 5‘13 3-6). For the moment I assume a stable (fixed) monetary policy. As the curve shifts downward, the economy moves along the stable LM curve until the economy reaches the point q in Figure 5—12. Point q corresponds to the point c2, in Figure 3-5 of Chapter three (where :he equilibrium points B and C of Figure 3—4 in Chapter three coincide rith each other). Since q is not a stable equilibrium point, a small terturbation would shift the economy to the point U in Figure 5-12. The pas: (as the sup cor poi the Ch; poi ju de 62 The converse is true when the economy is in a recession. Another factor must be considered. In the above diagram (5-12), the equilibrium interest rate declines along the Pq line, as the IS curve shifts down. This produces a resistance against the falling investment curve and IS curve. Thus, whether the economy passes through the critical point of q and produces a catastrophe (as indicated by Kaldor, Ichimura, and Varian) or not, all depends on the investment sensitivities to interest rate and capital. Besides, supportive (expansionary financial) policies if diagnosed and pursued correctly could postpone or even eliminate the occurrence of the turning points and sudden jumps. Ill—planned financial policies could quicken the occurrence of downswings or prolong it. In terms of catastrophe theoretical concepts, the model can be discussed as follows. Two endogenous (state) economic variables Y and r are explained by financial policies in the short run and long run. A continuous change in these policies leads to a smooth and continuous change in the economy. Nevertheless, at certain levels of these policies (i.e., LM3) the state of equilibrium undergoes a discontinuous jump (from S to q in Figure 5-10 or q to U in Figure 5-12). Figure 5-10 depicts different LM curves associated with different level of nominal (or real with fixed prices) balances, given a fixed fiscal policy (IS(KO,GO)). Figure 5-14 shows the equilibrium path for real income, Ye, as the supply of money is changing, M4>M3>M>M2>M1. Arrows above the ACT curve depict the equilibrium path of real income due to an expansionary monetary policy, given GO. Arrows underneath the curve iepict the equilibrium path of real income due to a contractionary lonetary policy, given Go' Points H, g, S, F, q, and E of Figure 5-14 corres points far a: conce: omy i conti monet to a the I the 6 At S conti afte1 as 1111 cont isti cont furt g fr cont The 1ng, cat; phEI in 63 3 corresponds to the same points in Figure 5-10, as 3’32 warr far as real income is {Z |—' concerned. If the econ— omy is initially at H, a continuous expansionary monetary policy, leads e to a rightward shift of Figure 5_14 the LM curve. Therefore, the economy traces out an equilibrium path until it reaches point S. At S a further pursuit of expansionary monetary policy causes a dis- continuous jump of the system to point q, Figure 5-10 and 5-14. Then, after this sudden jump, the system behaves smoothly and continuously as money supply expands. However, as the system contracts due to contractionary monetary policy, a qualitatively different character— istic develops. The economy contracts, starting from E, smoothly and continuously until it reaches point F as money supply shrinks. A further reduction in the supply of money makes the economy to jump to g from F, Figure 5-10 and 5-14. After this sudden jump, the system contracts again smoothly and continuously as money supply declines. The economy is not contracting along the path it followed while expand- ing. While contracting, the economy postpones the occurrence of catastrophe until it reaches point F rather than q, Figure 5—14. This phenomenon, which is called the "delay rule," is explained and pre- dicted by catastrophe theory (see Chapter four). The dynamic analysis of equilibrium in the economic system is, in a sense, limited by the assumption of a fixed fiscal policy, G0. Ic tro EVE fun tic the det II. on: de 311 to h 64 could embark a more complicated analysis of equilibrium and catas- ;r0phe by taking both fiscal and monetary policies into account. How— ever, the main purpose of this chapter is the construction of solid fundamental building blocks for a theory of exchange rate determina- :ion. This chapter and the previous chapter provide a framework for the application of catastrophe theory to models of exchange rate determination. II. Aggregate Demand-Aggregate Supply In this section I take the analysis of income determination one step further. First a non-linear (non—conventional) aggregate demand curve that follows from the non-linear investment is derived. Then, I will use this demand curve, in conjunction with an aggregate supply curve, to build a macroeconomic model. The purpose, thus, is to shed some more light on the theory of catastrophe in a closed conomy. But I always believe in the law of diminishing marginal eturns. As a result, this section is short and brief. It may answer ome theoretical curosity of the application of catastrophe theory to n aggregate demand concept. The rest is left to readers' imagination. Equations (5—6) and Figure 5-15 present the structural equa- ions of the aggregate demand curve. Interest rate and real income re assumed to be endogenous variables. Price level was assumed con- tant. NOW suppose the economy is initially at the equilibrium point 3f T in Figure 5-15. Given a nominal supply of money, an increase in :he price level reduces real balances. In order for the public to told a lesser quantity of real balances, income would have to decline tr interest rate to rise. Therefore, LM curve shifts to the left. If we aggre Figu: upwa1 shif poli comd Pos: Prie 65 we trace out the equilibria, given a fiscal policy, we derive an gregate demand curve of the type depicted in Figure 5—16. In gure 5—15 I assume initially that the LM curve is flatter than the vard sloping part of IS curve. r Figure 5-15 Expansionary Fiscal and monetary policies imply an upward Lift of the aggregate demand curve. A contractionary financial >licy causes a downward shift of the curve. The full equilibrium condition of the system is achieved by bining aggregate demand and supply schedules, Figure 5-16. Here (YE) depicts a short-run (long-run) aggregate supply curve. Its sitive (zero) slope reflects the sluggish (rapid) adjustment of ices and wages to output. P Figure 5—16 Arr riu CUI car ma] re 11 Th sl 66 Arrows in Figure 5—15 indicate the stability (unstability) of equilib- rium for a conventional (unconventional) downward (upward) Sloping IS curve. A conventional aggregate demand curve is sloping downward. This can be explained in economic terms. Suppose that the goods and money markets are initially at equilibrium. Thus, the economy is at a point on the aggregate denand curve. A lower price level leads to higher real balances and therefore creates an excess supply of real balances at the given level of income. The excess supply of real balances reduces interest rates in return. The decline in interest rates cause a surge in real spending by encouraging investment. Equi— librium level of income rises as a result of this surge in investment. Therefore, I can conclude that aggregate demand curve is negatively sloped. But the aggregate demand curve derived in Figure 5—16 is non- linear. It comprises two conventional downward sloping and one un— conventional upward sloping sections. I know that the aggregate demand curve is a combination of price level and real income where both goods and money markets are at equilibrium. The money market clearing conditions are met along the upward sloping section of aggre- gate demand. The goods market equilibrium conditions are also met, but are unstable. This is true because along the upward section of aggregate demand curve households' marginal propensity to save is less than their marginal propensity to spend (i.e., point B in Figure 3-4 of Chapter three). Therefore, points El and E3 (E2) are stable (unstable) equilibria. Suppose the economy was initially at equilibrium point E3 sio shi rit C01 sir 67 in Figure 5-17. Expan- P sionary financial policies shift YD upward. Equilib- rium point E3 moves upward as expansionary financial policies continues. If the course of financial expan- Figure 5-17 sionary policies are continued beyond the ones associated with aggre- gate demand YD'and equilibrium point E4, the economy suddenly jumps from equilibrium point E4 to E5, The catastrophe nature of the model is better explained through Figure 5-18. Equilibrium path of P (or Y) E / 5 e ’ ‘ / E4 M and/or G Figure 5—18 Fiscal (monetary) policies In the long run the picture changes. The level of real income is fixed in the long run. As a result of expansionary financial >olicies YD shifts upward. Real income does not change. However >rice level rises, Figure 5—19. In Figure 5-15 the LM curve was flatter than the upward :loping part of IS curve. In this part, I reverse that assumption. assu were the tio gat sam As in Figure 5—20, I assume that the LM curve is steeper than the up— ward sloping section of the IS curve. The deriva— tion mechanism of aggre— gate demand curve is the same as explained before. 68 Figure 5—19 As prices rise, the supply of real balances declines. rise and hence discourages the real spending on investment. come declines. A rise in price level is indicated by an upward (leftward) shift of LM curve, Figure 5—20. To de— rive the aggregate demand curve, I trace out the equilibrium level of in- come correspondent to changing price levels, Figure 5—21. Aggregate demand curve, YD, derived in Figure 5-21 looks like a conventional downward sloping demand curve. curve in Figure 5—21 which corr Interest rates Real in— / IM r / / I l / IS Y Figure 5—20 \ S P \. "””’,de \. SK A \‘N ‘\‘~YD' B \YD Y Figure 5-21 However, the AB portion of aggregate demand esponds to upward sloping part of IS (n 69 curve is unstable. Along AB section of YD marginal propensity to save is less than marginal propensity to spend (i.e., point B in Figure 3—4 of Chapter three).2 The equilibrium path of real income and price level are clear as long as the economy lays outside of AB portion of aggregate demand curve, Figure 5—21. Expansionary financial policies shift YD curve up. Both price level and real income rise. However, equilibria are unstable if aggregate demand and supply curve on the AB portion of YD curve, Figure 5-21. Figures (5-22 and 23) specify the dynamic behavior of economy in such a case. Suppose r aggregate demand and supply curves intersect at point U on the unstable part of aggregate demand curve in Figure 5—23. Point W, in Figure 5—22 is the intersection P of IS and LM curves which corresponds to point U in Figure 5—23. By disturbing the system, the economy moves to region I, II, III, or IV in Figure 5-22. The Figure 5_23 economy could not remain in regions I and III. In these regions both goods and money markets clearing conditions are disturbed. Therefore, economy moves from regi move the libr leve and pat (or 70 region I (III) into II (IV) or lays on LM curve. If the economy moved to region II (IV) excess supply (demand) in goods market causes the real income to decline (rise). To keep the money market in equi— librium as income declines in region II (increases in region IV) price level should rise (fall). LM curve shifts up (down) in region II (IV) and the economy moves away from point U in Figure 5-23 toward point C (D). The equilibrium path of real income or price level is explained through Figure 5-24. Equilibrium path of Y III/J',/’ (or P) D1 I’l’,/;"/’C Figure 5-24 C (and/or M) Fiscal (monetary) policies 71 CHAPTER 5——Footnotes lPatinkin, Don. 1956. As we know, an economic model which is comprised of both a (conventional) downward sloping demand curve and an upward sloping supply curve is stable in both Marshallian and Walrasian sense, Figure 5F-l. Point E in Figure SF—l is a stable equilibrium point. Therefore, the economy remains there as long as structural changes in supply and/or demand curve is assumed. P But as readers notice, the economy is unstable in the Figure 5F-l AB portion of the aggregate demand curve, YD, even though this demand curve is downward sloping, Figure 5-21. This apparent paradox may be explained in a static- dynamic dichotomy. The stability of equilibrium of Figure SF-l is explained within a static framework. However, this is not true for my model, Figure 5F—2. In this model, the dynamic equilibrium be- havior of real income and prices are studied. Beside, the equilibrium instability is an inherent character— istic of my model. Both aggregate demand curves derived in Figure 5-16 and 5-21 are reproduced in Figures 5F-2 and 5F—3. These aggre- gate demand curves show identical stability characteristics. The economy would be stable at levels of real income below Y (Ya ) or above Yd (Y ), in Figure 5F—2 (SF-3). The economy is unstable in the YC- da(Y -Y ) range of real income in aFRgure SF-Z (SF-3). Y Y YD is sloping upward in C d Figure 5F-3 since LM curve is assumed flatter than the Figure 5F'2 upward sloping (unstable) stable section of IS curve in P Figure 5-15. While, YD is sloping downward, because LM A \ curve is assumed steeper than ‘~B the upward sloping (unstable) I‘Nfifiizie section of IS curve in Figure Y W D 5-20. In either case, a b marginal propensity to save is less than marginal pro— Figure 5F’3 pensity to spend, CD (AB) in Figure 5F—2 (SF-3). CHAPTER 6 Theories of Exchange-Rate Determination Introduction Chapter two through five developed a non—conventional closed economy model. In this second part of the dissertation the non—con— ventional macroeconomic model is extended to allow for international economic repercussions. In Chapter seven a (non-conventional) macro- economic model of an open economy will be developed. Once this model has been developed, it will then be applied to explore the short run and long run effects of macropolicies on both home and international (i.e., exchange rate) economic variables. It will be argued that no single theory completely explains the foreign exchange rate, the price or value associated with a currency being acquired or bought in terms of another currency.1 The main objective of this chapter is to review the various theories of exchange-rate determination. This task will be accom- plished in a concise and brief manner, since there are numerous authoritative and critical surveys in the literature.2 This chapter lays a foundation for the next ones. The theoretical building blocks of exchange—rate behavior, to be developed in the next two chapters are based on the concept that these theories are supplementary rather than mutually exclusive. In the nineteenth century there were two views on the theory 72 73 of the flexible exchange rate. One view was the monetary approach to exchange rates. This View asserted that foreign exchange rates are determined only by the money supply (Ricardo) or mainly by the money supply (Christiernin, Thornton). The other view was the balance of payments (BOP) theory. According to this view, exchange rate fluc- tuations are explained in terms of the demands for and supplies of currencies in the foreign exchange market. International trade flows lead to demands for and supplies of foreign exchange in these models. This dichotomy (stock vs flow) of theories of exchange-rate behavior has been an enduring characteristic of this area of inter- national economics. The hypothesized relationship between capital and current accounts (stock vs flow) has changed considerably over time. After World War II, due to the obstruction of the free flow of funds and other international financial barriers, the emphasis of exchange-rate determination models was on the current account. As a result the flow BOP theories of exchange-rate behavior were developed and applied. However, in contemporary and modern international finan— cial system, the core of prevailing exchange-rate determination models originates from the portfolio equilibria selection and monetary con— siderations. This led to the development of a vast theoretical and empirical pool of literature on (stock) asset models of exchange-rate movements. One branch of asset models, called the monetary approach, describes exchange-rate behavior mainly in terms of changes in demands for and supplies of money stocks. The other asset model, the port- folio-balance model, is a more generalized version of the monetary approach. As wealth holders realign their wealth among different financial assets, exchange and interest rates are determined 74 imultaneously. In section one of this chapter I explore the flow market (BOP) Ddel of exchange-rate determination. In section two, I will briefly .iscuss the external and internal balance theory of BOP adjustment 1nd exchange-rate determination. This model is based on an IS-LM nodel extended to an open economy. In section three I examine the LSSGt market models of exchange—rate behavior. 2. Flow Market Models of Exchange—Rate Determination The flow market model of exchange-rate (or traditional text- »ook analysis of foreign exchange market) uses a Marshallian demand ind supply analysis. The foreign exchange rate like any other price, Ls determined by the demand for and supply of foreign exchanges (currencies). Several different approaches have been posed to explain :he effects of exchange rate movements upon the BOP and international economic phenomena. Perhaps the most widely discussed approaches are elasticities and absorption approaches. Ia. Elasticigy Approach The essence of this approach is the relation between the slope ef the demand and supply curves of foreign exchange and various lemand and supply elasticities of tradable commodities. This model lttempts to link the effects of exchange rate changes in the foreign exchange market to i) the relative price effect, ii) the internal >rice effect, iii) the income effect and iv) the distribution effect. The analysis is limited to the current account portion of the BOP. The main question that this approach seeks to answer is: "what are 75 he necessary and sufficient conditions for devaluation to achieve an mprovement in the current account balance?" The elasticities condition to improve the BOP has been speci- ied in many writings and may be repeated here as: nx—l nm+nm/em d(TB)/dB = W + arm-7;— (6‘1) x x m here d(TB) = the change in the home country's trade balances, denominated in home currency nx(nm) = foreign (home) demand elasticity for its export (import) ex(em) = foreign (home) elasticity of export supply. The famous Marshall—Lerner condition is derived if ex=em=w. fhe size of elasticities is very important in the stability analysis rf this approach.3 In response to the critiques, the elasticity approach has been extended to include secondary income effects. These effects are »rought on as the multiplier works through the economy because of the xpansion of income generated by an increase in net export (i.e., X = xports minus imports). These secondary effects are offsetting in the ense that an increase in income implies an expansion in imports. Thus, he Keynesian revolution served as an extension of the domestic income ultiplier (developed basically by J. Robinson and R. Harrod) to the rade section. b. Absorption Approach The restrictive nature of traditional elasticities models led o the further modification and consideration of income/price 76 interactions by Sohmen (1957), Harberger (1950), Vanek (1962), and Clement, Pfister, and Rothwell (1967).4 While the proposed modifica— tions made the above approach more complex, it failed to provide remedies for it§§limitations. Thus, Alexander's absorption approach was a response in eliminating the complexity of proposed modified elasticities approach by focusing on the macroeconomic View of the problem. This approach begins with national income accounting which can be expressed as: Y = C+I+G+X (6-2) where the left (right) hand-side of the above equation implies aggre- gate supply (demand). the value of real aggregate output, the value of real aggregate consumption outlays, the value of real investment expenditure, the value of real government expenditure, the value of real net export expenditure. NOHOM Illlll Equation (6—2) can be rewritten as: Y = A + x (6-3) where A:C+I+G is defined as the national expenditure on domestic goods and services, or absorption. The current account balance, B, is de- fined as export of goods and services minus their imports, X, which is also equal to Y—A. X = Y-A (6-4) It can be easily deduced from equation (6-4) that if domestic absorp— tion, A, exceeds output then more goods are being consumed than pro— duced and thus B luperscripts "+" ("-") above an explanatory variable indicate how that ’ariable is related to stock demand functions. These demands are 'estrained by the wealth budget constraints, equations (6-17) and 6-21). m(.)+b(.)+b*(.) = 1 (6—21) In a financial portfolio-balance model, equilibrium is chieved when the desired stock demand for these assets equal their pplies. The price level and real income are assumed given. The uilibrium values of exchange and interest rates are determined multaneously: — * A; = m(r,r ,Y)w (6‘22) - * ‘BP = b(r,r ,Y)w (6‘23) P 85 fi' * * '3: = b (r,r ,Y)w (6‘24) The left-hand side of equations (6-22), (6—23) and (6—24) denotes stock supplies of real money balances, home bonds and foreign bonds respectively. By implication of the wealth definition (equation 6-17)) equilibrium.condition for home cash balances is dropped, equation (6-22). Thus, equations (6—23, 24) determine r and E simultaneously. Yield on foreign bonds, r*, is given by invoking a small country assumption. As noted by Pearce (1983), the model is concerned with the allocation of the net wealth of all private home wealth holders. Thus, the monetary base rather than the supply of money enters into this model. This is true because demand deposits are the liabilities of commercial banks. The outstanding stock supplies of financial assets are held constant in a time period. Thus, in the short run, exchange and “nterest rates move to equalize desired demand and stock supplies of hese assets. However, through a positive net investment, government eficit spending, and current account imbalances, stocks of these ssets change over time. A money-financed deficit would increase the domestic monetary ase and home wealth. In realigning their wealth, wealth holders ould like to redistribute their wealth increase toward home and oreign bonds. The exchange rate goes up (home currency depreciates), hile home interest rate drops. The foreign interest rate is assumed onstant. A current account surplus would increase wealth and disturb bortfolio selection. To readjust their portfolio, home wealth holders 86 would like to realign their wealth toward home assets. Exchange rate declines (home currency appreciates) and home interest rate falls. A bond financed deficit would increase the outstanding supply of home bonds and thus private citizens wealth. This government policy has two offsetting impact on exchange rate. On the one hand, wealth holders realign their wealth in favor of foreign assets. This would lead to an increase in the exchange rate (depreciation of home currency). At the same time, a bond financed deficit increases home interest rate and thus makes foreign assets less appealing. Thus, the outcome of a bond financed deficit on exchange rate is ambiguous and depends on whether the wealth effect or the substitution effect is dominant. This finishes the survey of different models of exchange-rate determination. Traditional flow models gave their way to asset models. This was mainly due to a stronger forecasting capability of the latter models. Both Murphy and Duyne (1980) and Pearce (1983) indicate the superiority of portfolio—balance model in explaining exchange-rate behavior. Monetary models can be considered a special case of port— folio-balance models where financial assets are perfect substitutes and the wealth effect is ruled out. At the same time, portfolio- balance models includes elements of flow models by allowing current account imbalances to affect wealth and hence exchange and interest rates . 87 CHAPTER 6--Footnotes lPearce (1983), p. 16. For a comprehensive and authoritative survey on theories of exchange-rate determination see Whitman (1975), Dornbusch (1980), Murphy and Duyne (1980), and Pearce (1983). 3For more detail see Grubel (1981) Chapter 14; Vanek (1962), p. 56-65; Kreinin (1983); G. Orcut (1968); and Yeager (1976), Chapters 8, 10-11. For critic of the elasticities approach, see Yeager (1976), p. 172-178. SFor further detail see Meade (1951); Johnson (1958); Mundell (1968); and Willett and Forte (1969). There has been a major empirical concern about the validity of PPPT in the short run. For further detail on this matter see Kohlhagen (1978). 7Lindbeck, A. Scand. J. Ec., 1976, Vol. 78, No. 2 Private citizens are not holding other countries' currencies. CHAPTER 7 Theories of Exchange-Rate and Real Income Determination My objective in this chapter is to construct a real and finan— cial model of income and exchange-rate determination. This model is based on equilibrium conditions in both goods and financial markets. In the previous chapter both the traditional flow and asset market models of exchange—rate determination were explored. It was shown that real economic variables (i.e., real income) were the major factors in explaining exchange—rate behavior in flow models. In the asset market models, monetary and portfolio-selection considerations mainly explained exchange—rate movements. The portfolio-balance model of asset market models emerged as a theory capable of explaining exchange—rate changes. This model includes some elements of the traditional flow market models, by allowing current account imbalances to change wealth and hence interest rates and exchange rates. In this chapter, in contrast to a restricted financial port— folio-balance model of the exchange rate, real income is assumed to be an endogenous variable.l Its equilibrium value is determined when the demand for and supply of goods and services are equal (i.e., IS curve). In Chapter three I discussed and developed equilibrium conditions for the goods market in a closed economy. In this introduction, the equ111b— rium condition for the goods market is extended to an open economy. 88 89 Then, the goods market clearing condition, in conjunction with the portfolio-selection equilibria condition (developed in Chapter six), will be used to construct an income and exchange—rate determination model. To derive conclusive results about exchange-rate behavior I explore in section two a simplified version of this model. The equi- librium solution, stability conditions, and comparative statics of the (simplified) model are presented in section three. Extended versions of this model will be discussed in the next chapter. Links between exchange rate and real income on the one hand and expectations, infla- tion, and growth (dynamic), on the other hand will be discussed in section four, five, and six respectively. Ia. Goods Market This section presents a general examination of the goods market. In Chapter three I presented my version of income determination in a Keynesian framework. Here, I extend this model to the open economy. Real disposable income in the goods market is given as: C+S=Yd=Y-T+R+r (BP/P) +Er* (Fp/P) (74) where C = real consumption expenditure S = real saving Y = real disposable income Y = real income I = tY = government revenue as a function of real income R = government transfer payments I = interest rate on home bonds f(Bp/P) = real interest payment received on home bonds held by home private wealth holders 9O * Er (Fp/P) = interest payments (reaI)received on foreign bonds held by home wealth holders in home currency. r* = interest rate on foreign bonds E = exchange rate, home currency price of foreign currency Furthermore, the equilibrium condition for the goods market is: Y=C+I+G+X (7-2) where = real investment expenditure G = real government expenditure X = real net export expenditure We know that saving is: s = sYd-f f > o (7-3) and consumption is: - - = - = + Y (7-4) c — Yd s Yd sYd + f f c d where marginal propensity to save 0) ll marginal propensity to consume 0 ll autonomous consumption The investment function, I, is the modified version of the sigmoid- shape investment function developed in Chapter two: I = (q/(1+de'VY)) — mK - nr (7—5) n, q, m, d, and v are parameters. Replacing Yd in equation (7-4) from equation (7-1) I get a con— sumption function: G = f + cs[Y + R - T + r (Bp/P) + Er*'(Fp/P)] (7-6) 91 Replacing C from equation (7-6) and I from equation (7-5) in equation (7-2) I get the IS curve for an open economy: Y = f + c-[Y + R - T + r (Bp/P) + Er*-(Fp/P)] + (q/(l + de'VY)) - mk - nr + G + x (7—7) Net exports, X, are a function of both real income and the exchange rate: X = x(Y,E) xY < 0, xE > 0 (7—8) The equilibrium condition for the goods market in an open economy may be rewritten as: (n - c(Bp/P))r = (q/(l + de’VY)) - (1 — c (1 - t)) Y - mK + c + x(Y,E) + f + cR + cEr* (Fp/P) (7-9) where tax revenue, T, is a simple proportional function of real income, T = tY. The equilibrium condition for the goods market in the open econ— omy version, equation (7-9), can be graphically represented in a 3-dimen- sional diagram. Real income, Y, real interest rate, r, (with given prices), and exchange rate, E, all are depicted by this equilibrium surface. However, for the sake of simplicity, I take a cross section of this equilibrium surface, once with respect to r and Y, and once with respect to E and Y. This leads us to the goods market equilibrium condition (i.e., IS curve) in r-Y and E-Y planes which will be examined. First, I totally differentiate equation (7—9): - Y -vY 2 (n - c(Bp/P))dr - (rc/P)dbp = ((qdve v ) / (l + de ) - (1 - c (1 - t)) + xY)dy + deE + (cr*Fp/P)-dE + (cEr*/P)de + dG <7-10> 92 The slope of the IS curve in a r-Y plane is: (3r/8Y)|IS = ((qdve-VY)/(l + deIVY)2 - (l — c(1 - t)) + xY) / (n - c) <7-11) The slope of the same IS curve in an E-Y plane is: vY (as/2m]IS = -((qdve‘VY)/(1 + de- )2 — (1 - c (1 - tY)) + xY) / (XE + er* (Fp/P)) (7-12) The term, (n - c(Bp/P)), of the denominator in equation (7-11), is con- stant and positive. Therefore, the sign of the slope (Br/BY)IS depends 2 on the sign of nominator and is denoted by Z: 2 = (qdve'VY) / (1 + de'VY)2 - (1 - c(l — t) + xY (7-13) Chapter three examined both mathematically and graphically, the charac— teristics of Z in equation (7-13). In a r-Y plane, Figure r 7-1, the goods market equilib- rium curve, IS, shows combina— tion of interest rates (on home bonds) and levels of in- come for which the goods market Figure 7-1 clear, holding everything else particularly exchange rate constant. The (conventional) downward sloping sections of this IS curve represent locus of stable equilibria. But the upward (unconventional) sloping section of the IS curve (Figure 7-1) is a locus of unstable equilibria. The upward sloping section of the IS curve is quite plausible if investment, I(r,Y,K), is sufficiently 93 sponsible to increases in income. A rapid expansion of income could ad to a surge in S+T-R level much higher than the initial increase in G+X. To restore equilibrium in the goods market and between G+I+X i S+T-R, we need an increase in home interest rate, r, to cushion the rge of income. In an E—Y plane, Figure 7-2, the goods market equilibrium medule, IS curve, shows the locus of the exchange rate and levels of :ome for which the goods market clears, holding everything else 1rticu1arly the interest rate) constant. However, the relationship :ween the exchange rate, E, E 1 real income, Y, needs some Lboration. For a conven- >nal IS curve, as E in- eases (i.e., home currency breciates) so does the land for goods (through net Figure 7-2 torts). To restore equilib— 1m in the goods market, a rise in real income, and thus imports, is uired. This is clearly shown by the upward sloping sections of the curve in Figure 7-2. The downward (unconventional) sloping sections of the IS curve, ure 7-2, is a locus of unstable equilibria. The downward sloping tion of the IS curve is quite plausible if investment, I(r,Y,K), is ficiently responsive to increases in real income. A rapid expansion income could lead to a surge in S+T—R level much higher than the tial increase in I+G+X. To restore equilibrium in the goods market between G+I+X and S+T-R, there must be a decrease in exchange rate 94 ., home currency appreciates) to cushion the surge of income. Mathematically the slope of IS curve in E-Y plane is a nega- multiplication of the same IS curve in r-Y plane (equations 7-11 7-12). Thus, for the same level of real income, while the IS - is an increasing function of real income in r-Y plane, it also decreasing function of Y in E-Y plane, Figures 7—1 and 7—2. A Model of Real Income and Exchange-Rate Determination In this section, I summon up the theoretical building blocks -oped in section IIIc of Chapter six and section one of this chapter. A portfolio—selection equilibrium requires that home stock ldS for home real balances, home and foreign bonds be equal to f stock supplies, equations (6-22, 23, and 24). At the same time .ibrium in the goods market requires that the supply of goods be . to their demand, equation (7—9). Thus, I have a total of four ets in this model. They are rewritten as: (M/P)= m(r,r*,Y)w (7—14) (Bp/P) = b(r,r*,Y)w ASSETS MARKETS (7-15) (EFp/P) = b*(r,r*,Y)w (7-16) (n-c(Bp/P))r = (q/(1+de‘VY)) - (l-c(l-t))Y - goons MARKET mK + G + x(Y,E) + cR + f + cEr*(Fp/P) (7—9) >erscript "p” above B (F) denotes stock supply of home (foreign) 1 denominated in home (foreign) currency and held by home private .ens. Equilibrium in any two of the asset markets (stocks) insures .ibrium in the third asset market. However, equilibrium in the 95 three asset (stock) markets does not guarantee equilibrium in the flow, goods market.2 By implication of the wealth definition, equation (6-17), I drop the equilibrium condition for home real balances market, equation (7—14). Thus, I rewrite the model as: (Bp/P) = b(r,r*,Y)w (7—15) (EFp/P) = b*(r,r*,Y)w (7-16) (n-c(BP/P>> = - <1—c<1—t>>Y - * mk + G + x(Y,E) + cR + f + cEr (Fp/P) (7-9) The above set of equations determined the equilibrium values of the rates of return on home bonds, r, exchange rate, E, and real income, Y, respectively and simultaneously. The policy determined variables are the stock supplies of real money balances, M/P; home and foreign bonds held by central bank, Bc/P and FfilP. Where superscript "c" denote central bank's holding of a variable. II. Exchange-Rate Behavior and Perfect Substitution To derive conclusive results about exchange-rate behavior, this section explores a simplified version of the former model (i.e., equa— tions 7—15, 7-16, and 7-9). The simplifying assumption will be relaxed in the next chapter. In this chapter, I assume that home and foreign bonds are per— fect substitutes. Therefore, in the financial markets, home citizens face the choice between two assets: money and bonds. The demands depend on rates of return, income, and wealth: (7-17) (1 * (M/P) h(r,r ,Y)w (7—18) (139/P)d g(FP/P)dY + (1/(l-g)(Fp/P))((g/P)dM — (1-g)(1/P)dsp - E(1-g)(l/P)de) (724b) ~ slope of BB curve is given as: (as/air) = (ng)/(1—g>(FP/P> > 0 0-25) 98 Slope=eng>/(1-g>»0 The goods market equilib- Figure 7-3 rium curve, equation (7-23), which was examined in section one of this chapter, is illustrated in Since home Y , , Figure 7-4 interest rate is assumed constant, examination of the model will be continued in an EY plane in the rest of this chapter. To examine stability of equilibrium, a dis- equilibrium behavior model for exchange rate and real income is given as: ['1'] ll a[g(r.r*.Y>w — (Bp/P> - (EFF/P)1, a>0 (7-26a) K! II B[c(Y-tY + R + r(Bp/P) + Er*(Fp/P)) + I(Y,r,K) + G + x(Y,E) — Y], B>0 (7—26b) where a and 8 are adjustment coefficients. Equation (7-26a) indicates that the rate of change in exchange rate is proportionally related to excess demand for bonds. There are two sources for the supply of bonds 99 (i.e., home and foreign bonds). One is a government bond—financed deficit which leads to an increase in stock supply of outstanding home bonds. The second source is a current account surplus which increases home supply of foreign bonds. If the excess demand for bonds (in equa— tion (7—26a)) is not met by either sources, then private wealth holders will realign their portfolio by supplying their real money balances for bonds in foreign exchange market. Home currency depreciates and E rises. Equation (7-26b) is an excess demand equation for goods. The basic underlying premise for this equation is the producers response to unexpected inventory changes. As the firms find their inventories are decreasing, because demand exceeds their production, they increase their output to meet the demand for goods. A linearized version of(7-26),in the neighborhood of equilibrium is given as: R —a(Fp/P)(l-g) gYw E- E (. ) = * -VY ) /(H _) (7- 27) 8(xE + cr XFp/P) (c(l-t) + (qdve (1+de""")2 - 1) + xY where E and Y are equilibrium values for E and Y. Change in real wealth is substituted from equations (7-20). The necessary and sufficient conditions for the stability of the system requires that trace of the coefficient matrix J, on the right hand—side of (7-27) be negative and its determinant positive. a 0) -(FP/P)(1-g) gYw (xE+cr*/(1—de‘ ) - (l-C(l-t)) + xY 100 The trace and determinant of J are: TraCe(J) = - e(Fp/P) + 8[(qdve‘VY>/(1+de‘VY)2— (l-c(l—t) + XY] (7-28a) det(J) = eel-(Fp/P)((qdve‘VY)/<1+de‘VY>2— (l-c(l-t)) + x _ (XE +(cr*Fp)/P)gYw] (7—28b) Y The (2,2) element of matrix J, (qdve-VY)/(l+de-VY)2—(l-c(l—t)+xY=Z determines the slope sign of the IS curve. This was examined in sec— tion one of this chapter, equations (7-11, 7-12, and 7-13). For the IS curve to slope upward, (3E/3Y)lISI>O, Z must be negative. Otherwise, IS curve slopes downward, (3E/8Y)IIS <0. Therefore, along the upward sloping sections of the IS curve, where (BE/3Y)|IS is positive and Z=(qdve-VY)/(l+de-VY)2-(1-C(l-t))+XY is negative, trace(J) is negative. To meet the sufficient condition for stability, it is required that det(J) should be positive. In other words: det(J) >0 if: det(J) = -(Fp/P)Z—gYw(xE+cr*(Fp/P)) >0 (7-29a) where z = (qdve‘VY)/<1+de"VY>2 — <1—c<1—t>> + XY or -(Z)/(xE+cr*(Fp/P) >(ng)/(Fp/P) (7-29b) The left-hand side of inequality (7-29b) is the slope of IS curve, (BE/3Y)|IS, equation (7-12). The right-hand side of inequaity (7—29b) 101 is the slope of bond market equilibrium condition, equation (7—25). Thus, I conclude that both necessary and sufficient conditions are met when our IS curve intersects BB curve (i.e., bond market) from below, Figure 7—5. By considering the downward sloping section of the IS curve, I realize that the trace(J) :0, while the det(J)<0, so the Y system is unstable. Figure 7_5 Comparative Statics To examine comparative statics, total differentiation of equa- tions (7-22) and (7-23) provides: ngdY + gdw = (dBP + Ede + deE)/P (7-30a) p -VY -VT 2 _ l- )) + )dY + ((-crdB )/P) = ((qdve )/(l+de ) - (l C( t KY * dG + xEdE + cr (Fp/P)dE + (7-30b) cr*E(de/P) where the change in wealth is substituted from equations (7-20). In equations (7—30), the change in the home country's rate of interest is zero, by invoking the small country assumption. In the set of equations (7—30) the term dB—dBC (dF—dFC) can be substituted for its equivalent values dBp(de). The set of equations (7—30) could be rewritten in a matrix format: 102 dM dB [3..] l ) = dBc 13 CH [bij] 32C (731a) where -(FP/P)(1-g) gYw A = [31.] = J (xE+cr*(Fp/P) Z -(g/P) -((l-g)/P) -E(l-g)/P O B " [bi‘] = * J O cr/P cEr /P —l The determinant of matrix A is written as: * det(A) = Z(Fp/P)(l—g) - gYw(xE+cr (Fp/P)) (7—31b) Along the downward sloping section of the IS curve (i.e., when Z>>O ==:> (3E/8Y)|IS (BE/3Y)Iisi>0), we have the following circum— stances for the determinant of matrix A. p * p det(A)>>O if |z(F /P)(1-g)|>|gYw(xE+cr (F /P)l p * p det(A)‘<0 if IZ(F /P)(l-g)l —(1/P> (BY/3B ) = (l/det(A)) * = x +cr (Fp/P) cr/P E (l/det(A))(l/P)(-cr(Fp/P) + g(Fp/P)(cr/P) + xE+cr*(Fp/P)) = (l/det(A))(l/P)(xE+gcr(Fp/P) = (+)/(det(A)) * * :r(Fp/P) and cr (Fp/P) are cancelled out because r and r are equal (and fixed) in this model (i.e., due to perfect substitution between nome and foreign bonds). -(Fp/P)(1-g) -E/P (BY/arc) = (l/det(A)) = 7': p 9: xE+cr (F /P) cEr /P (l/det(A))(E/P)(—cr*(Fp/P) + gcr*(E/P)(Fp/P + xE+cr*(Fp/P) = (l/det(A))(xEE/P+gcr*(E/P)(Fp/P))= (+)/(det(A)) -<1—g> o (3Y/3G)=(l/det(A)) = (+)/(det(A)) x —cr*(Fp/P) -l E -(g/P) gYw (BE/8M)=(1/det(A)) = (l/det(A))(-Zg/P) 0 Z -(1/P)(l-g) gYw (BE/BBC)=(l/det(A)) cr/P Z (l/det(A)){-Z/P)(l-g)-CrW8YP = (?)/(det(A)) 104 To find the sign of nominator for the above partial derivative, c BE/BB , two cases should be examined (a) determinant of A is positive, (b) determinant of A is negative. If det(A) >0, I know from (7—29b) that: p * p —Z:>(l/(F /P)(l-g))(gYW(xE+cr (F /P))) or * 7': p p Z < -gchr -((gYw)(xE+cr g(F /P))/(F /P)) or * ‘ * Z+gchr < —(1/> (gYwO then I can write: *‘O partial derivative, 3E/3BC, has a positive sign for its nominator: aE/aBC = (+)/det(A) (7-323) However, if det(A)‘-gchr*-(l/(Fp/P))(gYw(XE+cr*g(Fp/P))) or 105 z+gYwer* > - (1/ (FP/P)) (gYwexE+cr*g>> -E/P ng (BE/BFC)=(1/der(A)) * = cEr /P Z (l/det(A))(—E/P)(Z+chr*w) = (?)/det(A) The only difference between the Sign of BE/BFC and {BE/BBC is that the former is multiplied by a positive term (i.e., E/P). Thus, as far as the sign of BE/BFC is concerned, the argument made for the partial derivative BE/BBC is applicable to partial derivative BE/aFC. Thus, when det(A)=>0 and the system is stable, I have: (as/arc) (+)/det(A) (7-32b) But when det(A)<<0 and the system is unstable, I have: (BE/BFC) (?)/det(A) For the partial derivative aE/BG, I have: I 0 ng (3E/8G)=(l/det(A)) = (+)/det(A) -1 Z c The result of monetary and open market purchase of bonds (+dB =dM or EdFC=dM) and expenditure generating policies are recorded in Table one. This table depicts all possible influences (sign patterns) on equilib— rium values of real income and exchange rate as a result of govern- ment financial policies when the system is stable. Rows two and three describe the system in unstable positions. 106 TABLE 1 EFFECTS OF MONETARY, OPEN MARKET OPERATIONS, AND BUDGETARY POLICIES ON REAL INCOME AND EXCHANGE RATE IN A SHORT-RUN MODEL det(A) > 0 1 :) z<0 ==>(3E/3Y)IIS >0 Y - _ _ _ det(A) <0 2 E - r r — Y - _ _ _ LI) z>0 ==>(8E/8Y)IIS/P>dF°- w0, O0, Ofazfl, Ofaz+a3il To relate home produced goods price level, Ph’ to overall home price leve, P, Turnovsky (1977) assumes the following equation: * P = ‘1’(Ph,EP ) (7-48) This equation asserts that overall home price level,P, is a function (W) of home produced goods price level, Ph, and price of foreign pro- * duced goods expressed in home currency, EP . The Complete Model and Its Solution In this subsection, I, first, put the whole model together: g(r-n,r*-n*,Y)w = (Bp+EFp)(l/P) (7—49a) VY) _ (n-c(Bp/P))(r-h) = q/(l+de— (l-c(l-t))Y-mK+G+f+cR+x(Y,(EP*/Ph)) + cr*E(Fp/P) (7—49b) 118 _ * Ph = a0+al(Y—Y)+a2EP +a3fl (7-47) * P = W(Ph,EP ) (7-48) Equation (7-49a) is market clearing condition for bonds (i.e., equa- tion 7-42). Equation (7-49b) is the goods market equilibrium condition. Net exports, X, is value of exports (EX) minus value of imports (IM): imY > 0, im(EP*/Ph) < 0, XY < O, x(EP*/Ph) > 0 (7—50) At this point I continue the investigation of this model on two fronts. First, I take overall and home price levels (P,Ph) as exogenous variables. This would allow me to measure and explore real income and exchange rate responses to changes in inflation rates and price levels. Then, in the second approach, price levels are taken as endogenous variables as specified by equations (7-47) and (7-48). Va. Exchange Rate and Inflationary Shocks In this subsection, price levels are considered exogenous. I totally differentiate equations (7-42 and 7-43) respectively: wg dn+ngdY+gdw+(Bp+EFp)(l/P2)dP = (l/P)(dBp+deE+Ede) (7-51a) n [C(Bp/P2)dP-(c/P)dBp](rJfi) — (n-c(Bp/P))dn = [qdve—VY/(l+de—VY)2 - (l-c(l—t)) + xY]dY+dG+xE°(P*/Ph)dE * it +xP odP +cr*(Fp/P)dE-(cEr'/P2)deP+(cEr /P)dFp h h (7-51b) 119 or ngdY-(Fp/P)(l-g)dE = [(-BP+EFp+g(M+Bp+EFp)/P2-g]dP - (g/P)dM+(1/P)(l-g)dBp + (E/P(l-g)de (7-52a) where the change in wealth is substituted from equation (7—20). [(qdve‘VY)/(1+de“’Y)2 - (l-c(l-t)+xY]dY+[xE(P*/Ph) + cr*(Fp/P)]dE = (((CBp/P2)(rrfi) - (n-(CBp/P)) - (Er*e/P2))dP-xP dPh - (c(r—n)/P)dBp - (cEr*/P)de—dG h (7-52b) The expected overall home rate of inflation,1r, is considered as a func- tion of overall home price level, P: do = 9(P) (7-53) 9 could be a function which measures the deviation of the overall home price level, P, from its steady state value, an equation like (7-34). Thus: = 7-54 dn GPdP 9P/P * cEr /P The element a22 determines the sign pattern of the slope of IS curve, derived in section Ia of this chapter (i.e., Z in equation 7-13). The sign pattern of changes in exchange rate and real income with respect to financial and inflationary shocks are given by the following partial derivatives and table two. - + (aY/aP)=(1/det(A)) =(f)/det(A) + + - 0 (eY/aPh)=(1/det(A)) I=(-)/det(A) + + (aY/aM)=(1/det(A)) ' =(+)/det(A) + o (BY/BBC)=(l/det(A)) '=(+)/det(A) 121 '(BY/BFC)=(l/det(A)) =(+)/det(A) + _ - 0 (BY/BG)=(1/det(A)) =(+)/det(A) + _ f + (BE/3P)=(l/det(A)) =(i)/det(A) i i 0 + (BE/BPh)=(l/det(A)) =(-)/det(A) + e - + _ _ + for the stable economy (BE/3M)-(l/det(A)) O + —(f)/det(A) - for the unstable economy - + c _ _ + for the stable economy (BE/3B )_(1/dEt(A)) _ + —(i)/det(A) f for the unstable economy — + c _ _ + for the stable economy (BE/8F )-(l/det(A)) _ + —(t)/det(A) i for the unstable economy 0 + (BE/BG)=(l/det(A)) =(+)/det(A) — + Partial derivatives of real income and exchange rate with respect to financial policies in this section are exactly identical to those de— rived in static models (section three). Thus, from now on, I continue my discussion by focusing on inflationary impacts on real income and exchange rate. (3Y/8P)=(:)/det(A), (BY/aPh)=(~)/det(A) 0 E i — + + + + I) a <0==>(eE/3Y)| >0 22 IS Y i + _ _ _ _ det(A) < O 2 E i + ‘ i i ‘ Y -f + - - - - 3 11) 322>0==>(aE/3Y)IIS<0 & det(A) <0 E i + + i i ' The sign patterns of partial derivatives of real income and exchange rate with respect to inflationary shocks are ambiguous (table two, column one). However, this makes my analysis more interesting. An inflationary shock (P+) would reduce the real supply of bonds and creates an excess demand for bonds. At the same time the demand for bonds declines through rational expectation assumption, dn=9PdP. Therefore, the direction of change in E depends on the relative strength of these two factors. If private wealth holders have a very elastic price expectations (i.e., as price, P, rises so does n) then demand for bonds falls faster than their supply. Wealth holders re— align their portfolio toward home real balances and therefore, price of home currency in terms of foreign currency increases (i.e., home currency appreciates) and E declines. E rises if home wealth holders have a very inelastic price expectations. 123 Whether real income rises or declines as P changes all depends on price elasticities of demand for investment and net exports. Prices increases lead to an increase in investment outlays (i.e., through price expectations w) and a decrease in net exports. If price elasticity of demand for investment is greater than price elasticity of demand for net exports, then income should rise. As far as home produced goods inflation rate, Ph, is concerned, we should bear the following fact in mind. Changes in Ph have two sig- nificant fallouts: one is the impact on inflation rate (P) and the other is the impact on price expectations. The former leads to a reduction of real supply of bonds while the latter contributes to a surge in demand for bonds, equation (7—49a). Thus, in row one of table two, price expectations are very inelastic (weak). Demand for bonds declines drastically and hence private citizens realign their portfolio toward home currency. Home currency appreciates and E declines. But row two has a different story. Here, price expectations are very elastic (strong) and that leads to a great surge in demand for bonds. Private wealth holders realign their portfolio toward bonds away from real balances. Home currency depreciates and E rises. Vb. Price Level as an Endogenous Variable In this subsection, I assume that home overall price level, P, and home produced goods price level, Ph, are endogenous variables. They are given by equations (7—47) and 7—48). I can rewrite my model, in this context, as: s * g(r-n,r —w ,Y)w = (Bp+EFp)/P (7-49a) 124 (n—c(Bp/P))(r-TT) =[q/(1+de‘VY) - (l—c(l—t)lY—mK+G+cR+x(Y,EP*/Ph) + * p cEr (F /P)+f (7-49b) 7’: P = T(Ph,EP ) (7-48) P — ‘ * w 47 h — aO+al(Y—Y)+a2P +a3 (7- 7 I totally differentiate the above set of equations to get: ngdY-(Fp/P)(l—g)dE+(9Pg"1‘T‘+(Bp+EFp)/P2 - g(M+Bp+EFp)/P2)dP = ((1-g>/P)(dB-dB°) + (E/P)(1-g)(dF-ch) - (g/P)dM <7-56a) ((qdve-VY)/(l+de—VY)2 - (l—c(1—t)) + xY+aleh)dY + (XE(P*/Ph) + (cr*/P)deE — ((ch/P)(r-fl) — (n-c(Bp/P))gP _ * 2 p _ c (cEr /P )F -a3xP )dP — - (c(r-r)/PXdB — dB ) - h * c (cEr /P)(dF-dF ) (7-56b) dP = WP .dPh + TEdE (7-56C) h * dP = a dY+a dn+a P dE (7-56d) h" 1 3 2 I substitute (7-56d) into (7—56c) to get: at P +11 )dE+(l-9 ‘1’ .a3)dP = 0 (7-56e) h h E P Ph 2 where dPh from (7—56d) is substituted into equation (7—56b) and (7-56c). The solution for changes in exchange rate, real income, and overall home price level is given as: 125 G dE dM A(g;)=B dBC (7-57) dFC where {'(l‘gMFp/P) ng QPgflw+(Bp+EFp)/P2 ' '. + +[ * A: xEu) /Ph)+cr*(Fp/P) Z+aleh '(( on/P)(r-h)-(n-(ch/P)9fi-== + t I * (cEr /P2)FP—33xP ) h is “1’"an -a‘¥ (l—aO‘P) __+ 1 E 2 Ph 1Ph 3PPh .. b _ The jacobian matrix A, in this section differs from its counterpart in the previous sections only by the introduction of price level as the third endogenous variable. If a =Z+a xP is negative 22 l h then the IS curve is sloping upward (stable) in an EY plane. This is * true because the slope of IS curve in an EY plane is —(Z+alxP )/(xE(P /Ph) h n + cr (Fp/P)). The denominator is positive. Thus, the sign of the slope depends on the sign of a22=Z+ale . If Z is negative, then the slope h * n "' — p ' o o of IS curve, (BE/3Y)|IS— (Z+aleh)/(xE(P /Ph)+cr (F /P)), 15 p081t1ve and IS curve is sloping upward in an EY plane, Figures 7-5 and 7-6. I proved, in section three, that the system is stable when the IS curve is sloping upward and cuts BB curve (bonds market equilibrium) from below in an EY plane. “b -(g/P) -(l-g)/P -(E/P)(1‘8fl (-0 ' _ '1 * —l O c(r-n)/P cEr /P = — O + + LO 0 O 0 ‘1 LO 0 0 0.1 L_ 126 In the jacobian matrix A, the element a is Z+a . I know that 22 -f%h xP < 0, as home produced goods price level increases net exports h declines. Thus: a22 = Z+aleh< 0 if Z < O In that case, the IS curve is sloping downward (upward) in an r-Y (E—Y) plane, Figure 7—1 (7—5). a22 = (3r/BY)IIS+aleh§ O lf(Br/8YTIS > 0 In other words, the element a22 or the sign pattern of the slope of IS curve may be either positive or negative and the system is unstable. Thus, I continue the discussion by focusing on the stability conditions of the system (i.e., Z+alxP < 0). h The sign pattern of determinant of jacobian matrix A depends on the sign pattern of element 323. If a '>0, I can write matrix A determinant in an expanded form 23 as: p _ . P _ _ -(1—g)(F /P)[(Z+aleh)(1 a39PTPh) + alTPH (C(B /P)(r W) 9% (n-c(Bp/P)9P-cEr*(Fp/P2) - a3xP )1 - (xE(P /Ph)+er*(FP/P) h 2 [ng(l-a3GPwPh)+alTPh) + aleg(9Pgflw + (Bp+EFp)/P )] + (-YE-a2P*YP )[WgY(-C(Bp/P)(r—n)- (n—c(Bp/P)9P - h 2 cEr*(Fp/P2) - a3xPh) - (Gpgflw+(Bp+EFp)/P )(-z+aleh) (7—58) But if a < 0, then det(A) is rewritten as: 23 127 -<1—g)ep - h * p 2 * * p cEr (F /P ) — a3xP )] - (xE(P /Ph)+cr (F /P))[ng(l—a39P‘i’P ) + h h a T .(9 g w+(Bp+EFp)/P2)] + (—Y —a P*TD )[W. (—c(Bp/P)(r—n) - 1Ph PW E2.th (n—ch/P)0P-cEr*(Fp/P) - a3xPh) - (Z+aleh)(9Pg£w+(Bp+EFp)/P2) (7-59) By comparing equations (7-58) and 7—59) I realize that in the latter equation all expanded elements have identical signs except the second and fourth ones which are positive. Thus, in order to have the system in a stable position (i.e., det(A) >0) we need to have a23 to be negative. This is true because the determinant of the jacobian matrix A is greater when a23 <0, equation (7-59) than when a23>0, equation (7—58). The partial derivatives of changes in real income, exchange rate, and overall home price level can be given as: - 0 + (BY/3M)=(l/det(A)) ‘+ - — ‘ = (t)/(det(A)) — 0 + — — + (aY/eBC)=(1/det(A)) + - — = (i)/det(A) - 0 + \- — + (aY/aFC)=(l/det(A)) + + — = (i)/det(A) 1-.. (3Y/3G)=(1/det(A)) ' + (3E/3M)=(1/det(A)) O (BE/BBC)=(l/det(A) + (BE/BFC)=(l/det(A)) + (8E/3G)=(l/det(A)) - (3P/8M)=(1/det(A)) + (BP/BBC)=(1/det(A)) + (BP/BFC)=(l/det(A)) + (3P/3G)=(l/det(A)) + =(i)/det(A) =(+)/det(A) =(f)/det(A) =(f)/det(A) =(+)/det(A) =(+)/det(A) =(f)/det(A) =(f)/det(A) =(+)/det(A) 129 TABLE 3 EFFECTS OF OPEN MARKET OPERATIONS AND BUDGETARY POLICIES ON REAL INCOME, EXCHANGE RATE, AND OVERALL PRICE LEVEL M BC FC G Y i i’ i i a22 <0 ==> det(A) > 0 E + i I + P + f f + P C C- Stable Economy —dB =dB =dM, -EdF=dF —dM The results obtained from table three do not predict, in a determinate manner, the direction of change in real income, exchange rate, and overall price level as a result of financial policy changes. Part of this ambiguity could be explained by complications in estimating sign patterns of coefficient matrix in partial derivative equations. It could be also explained by certain assumptions regarding speed of adjustment in real and financial markets. For example, as far as real income is concerned, expansionary fiscal policy leads to a higher income. But having assumed price level a function of real income (i.e., equations 7—47 and 48), overall price level increases in the home economy. This increase in price level leads to an excess demand for bonds, equation (7-49a). To restore asset in market equilibrium condition, Y should decline or price level expectation, should rise. The outcome depends on the speed of adjustment as far as price expectations are concerned. If price expec- tations are elastic, then there is no need for Y to decline. And 130 therefore, Y would rise as G rises. Otherwise, real income, Y, declines as G rises. An Open market purchase of home bonds (i.e., Bp+, Bet) leads to an excess supply for real balances. To restore equilibrium in money market, either interest rate (r) should decline or real income, Y, should rise. But in this model, r is fixed. Therefore, Y would rise. Meanwhile, this open market purchase of bonds reduces supply of bonds and creates an excess demand for bonds. To restore equilibrium in the asset market either P or Y should decline. Again the speed of adjust— ment is very critical. The same could be argued for the effect of open market opera- tion of foreign bonds on real income. VI. Exchange-Rate Movement in a Dynamic Model To close in the model in a dynamic sense, I need to specify the nature of wealth over time and the role of central bank and government over monetary and fiscal policies. In the equilibrium analysis of section III, the change in wealth was taken into account. Usually the stock of real wealth is considered constant in a very short run and long run model. The change in real wealth, n, which is equal to saving is given as: u=s=Yd-c (7‘60) where *4 2:an II = real disposable income real consumption expenditure = real saving expenditure = real wealth 131 real consumption is: C = ch+f where C f marginal propensity to consumer autonomous consumption I substitute (7-61) in equation (7—60) to get: w = Yd-ch-f with some rearrangement I get: w = (l—c)Yd-f Real disposable income is: Y—tY+R+(r-n)(Bp/P)+E(R*-n)(Pp/P) Yd = where Y = real income T = tY = simple proportional tax R = transfer payments (r-fi)(BP/P) = interest earnings on home bonds held by home private citizens * E(r -w)(Fp/P) = interest earnings on foreign bonds held by home private citizens and denominated in home currency (7—61) (7—62) (7—63) (7—64) E = exchange rate, home currency price of foreign currency To get an equation for the change in real wealth, I substitute equation (7—64) into (7-63). I get: n = (l-c)[(1—t)Y + (r—n)(BP/R) + e(r*-n)(Fp/P)1 — e A balance on official settlement is defined as: BOP = x(E,Y)—EF (7—65) (7-66) 132 under a flexible exchange rate regime, BOP=O. Therefore, current account imbalances, x(E,Y), are equal to foreign asset flow, F. . up 0C x(E,Y) = EF=E(F +F ) (7-67) (EFp/P) = x(E,Y)-(EFC/P) (7-68) The central bank's asset include home bonds, BC, and foreign reserves, FC. Central banks usually do not hold currencies of other countries. Their liabilities are the banking system reserve deposits with the central bank, <(B°+EFC)/P> = M/P (7-69) where M denotes the monetary base. Changes in central bank stock of bonds depend both on its monetary and foreign exchange objectives. The government deficit, D, is defined as: D = G—(T-R—r(Bp/P)) (7—70) where D = government deficit G = real government expenditure T = real government revenue as a function of real income R = government transfer payments to private citizens Deficits are defined either by printing money and/or issuing new bonds: D = G+R—T+r(Bp/P) = M+Bp (7-71) where (B/P) denotes real stock supply of both home and foreign bonds held by home private citizens. If [m (Kb) denotes the fraction of 133 deficit financed by printing money (issuing new bonds) by government, then I have: u = 10, then trace(J)::0 Thus, the first condition of stability should be met if IS curve were sloping upward in an E—Y plane. * k * det(J) = g1+c(1-c><1-t>-z0 OI' 135 -2/[xE(P*/Ph)+c} > — <1—c) <1—t) / (FF/P) The above inequality holds for an upward SIOping IS curve in an E-Y plane. Therefore, both necessary and sufficient conditions of dynamic stability are met if IS curve slopes up in an E-Y plane. Steady State and Comparative Statics The rates of growth of real income, the exchange rate, and wealth declines to zero in a long run stationary state (i.e., equilib— rium). Thus, the equilibrium solution of this dynamic model is given by the following equations. g(r-h,r*-W,Y)w = (B—BC+E(F-FC))/P (7-78) (n-e((B—B°)/P)(r—n) = q/(1+de‘VY) — (1—c-flrt))Y-mK+G+f+cR+x(Y,EP*/Ph) + cEr*((F-FC)/P) (7-79) (l-c)[(l-t)Y+c(r-n)(Bp/P)+E(r*-h)(Fp/P)] — f (7-80) where equations (7-78 and 79) are market clearing conditions for bonds and goods respectively. Equation (7-80) states saving (i.e., change in wealth) equals zero. These equations determine long run equilibrium values of E, Y, and w simultaneously. I totally differentiate equations (7-78, 79 and 80) to get: ngdY-(Fp/P)dE+gdw = (—dBC-EdFC)/P (7—81) ZdY+(xE(P*/Ph)+c(r*—n)(Fp/P))dE = (cr/P)dBC+(cEr*/PHFC—dG (7—82) * c * C (l-c)(l-t)dY+[(r —n)(Fp/P)]dE = (r/P)dB +(E(r -n)/P)dF (7-83) The solution for changes in exchange rate, real income, and 136 real wealth is given as: C dE dB A (dY) =B (dFe) (7-84) dw dG D-(Fp/P) ng g' '— + +' A- xE(P*/Ph)+e(r*_n)(FP/P) z —1 = + i o * p .(r -w)(F /P) (l-c)(l-t) 0‘. .+ - 01 '-(l/P) -(E/P) 0‘ B== c(r-n)/P cE(r*-r)/P -l _(r-n)/P E(r*—n)/P 0 d The jacobian matrix, A, in this section, differs from its counterpart in previous sections only by the introduction of wealth as the third endogenous variable. It has a positive determinant (proved in the stability section), if the IS curve slopes upward, in an E-Y plane. The partial derivatives may be written as: —(l/P) ng g (BE/BBC)=(l/det(A)) e(r-n)/P z 0 (r—h)/P (l—c)(l-t) 0 O g[c«r-n)/P)(l-c)(l-t) - Z(r-W)/P]/det(A) > —E/P ng g (BE/BFC)=(l/det(A)) e(r*-n)E/P z 0 E(r*—n)/R (1—c)(l-t) o )1g/P1 137 The partial derivative (BE/BFC) is equal to E(BE/BBC). Therefore, it has the sign pattern of (BE/BBC): (BE/BFC) = (+)/det(A) O ng g (BE/BG)=(1/det(A)) —1 Z 0 = 0 (l-c)(l—t) O (l/det(A))g[-(l-C)(l-t)] = (-)/det(A) -(FP/P) —1/P g (BY/BBC)=(l/det(A)) xE(P*/Ph)+c(£:r)(Fp/P) e(r-n)/P 0 = (r*-n>(Fp/P> 0 (8-8) and consumption is: = : — = — = (8‘9) C Yd S Yd sY+f f+ch where marginal propensity to save marginal propensity to consume 8 C The investment function, I, is the modified version of sigmoid—shape investment function developed in Chapter two: 144 I = q/(l+de—VY) - mK - n(r—h) (8—10) where, q, d, v, m, and n are parameters. Replacing Yd in equation (8—9) from equation (8—6) we get a consumption function: * C=f+c[Y+R-tY+(r-h)(Bp/P)+E(r -h)(Fp/P)] (8-11) substituting C from equation (8—11) and I from equation (8-10) into equation (8-7) I get IS curve for an open economy: p * p -vY Y=f+c[Y+R—tY+(r—h)(B /P)+E(r —n)(F /P)]+q/(1+de ) -mk—n(r-w)+G+X (8-12) Net exports, X, are a function of real income, Y, and the ratio of is overall foreign and home price levels, EP /Ph: * = * _ X x(Y,EP /Ph) fl <0, AXEP /Ph>0 (8 13) where * P = overall foreign price leval Ph = home goods price level The equilibrium condition for the goods market in an open economy can be written as: (n-e(Bp/P))(r-n)=q/(1+de‘VY) - (l-c(l—t))Y-mK+x(Y,EP*/Ph) + G+f+cR+cE(r*—h)(Fp/P) (8-14) The relationships between E, Y, and r need some elaboration. In a r-Y plane, the downward sloping sections of IS curve shows com— binations of interest rate (r) and levels of real income (Y) for which the goods market clears, holding E and everything else constant, Figure 8-1. 145 However, the (uncon- r ventional) upward sloping part of the IS curve, in the same r—Y plane, is a locus of unstable equilibria, Figure 8-1 as shown in Chapter seven, section one. Figure 8-1 In an E-Y plane, the curve slopes in an opposite direction as compared to the direction of IS curve in a r-Y plane, Figure 8—2 as shown in Chapter seven, section one. The upward sloping sections of the IS curve Y in an E—Y plane can be ex— Figure 8-2 plained, first, mathemati- cally. A total differentiation of equation (8—14)is given as: * * p p (xE(P /Ph)+c(r -fl)(F /P))dE+(Z)dY-(n—c(B /P))dr = p * p -c(r-n)(1/P)dB -cE(r —n)(l/P)dF -dG (8-15) where Z = (qdve-VY)/(1+de-VY)2—(l-c(l-t)+xY In a r-Y plane, the slope of IS curve is given as: (or/aY) 1s = Z/(n—c(Bp/P)) (8-16) 146 The denominator, n—c(Bp/P), is constant. Thus, the sign pattern of the slopes depends on the sign of Z term. In an E-Y plane, the slope of the IS curve is given as: . * * p (BE/BY) IS 13 -Z/KxE(P /Ph)+c(r —h)(F /P)] The denominator, in equation (8-17), is constant and positive. Thus, the sign pattern of the slope, depends on the sign pattern of —Z term. From the economic point of View, the (conventional) IS curve slopes upward in an E-Y plane. This is true because as E rises, so do net exports. An increase in real income would raiSe imports and hence offset the surge in net exports and equilibrium.would be restored in the goods market. This requires that the term -Z/KxE(P*/Ph+ c(r*-n)(Fp/P)] (or the slope of IS curve) be positive for a conven— tional upward sloping IS curve. The denominator is positive. Thus, the term Z should be negative to guarantee a (conventional) upward sloping IS curve. Equation (8—16) is the slope of IS curve in a r-Y plane. It must be negative for a conventional downward sloping IS curve (in r—Y plane). From the previous discussion, the term Z is negative. This is true in order to guarantee a conventional upward sloping IS curve in an E-Y plane. For equation (8—16) to be negative, having the term 2 already negative, requires that the term (n-c(Bp/P)) be positive. Thus, it can be concluded that (conventional) IS curve slopes downward (upward) in r-Y plane, Figure 8-1 (8—2). Unconventional IS curve sloped upward (downward) in r—Y (E-Y) plane, Figure 8—1 (8-2). 147 II. Equilibrium Solution With a given home price level, P, equilibrium in asset mar- ket requires that the existing stock of financial asset be equal to their stock demands, equation (8—1, 2, and 3). Equilibrium in any two of the asset markets (stock) insures equilibrium in the third asset market. However, equilibrium in the three asset (stock) mar- kets does not guarantee equilibrium in the flow, goods market.1 By applying the asset constraint to the asset markets, I drop the equilib- rium condition for the home real balances, equation (8-1). Thus, I rewrite the model as: b(r—n,r*—N,Y)w = Bp/P (8-2) b*(r-n,r*-n,Y)w = EFp/P (8—3) (n—c(Bp/P))(r-h) = q/(1+de-VY) - (1—c(1-t))Y—mK+x(Y,EP*/Ph)+G+cR+f+ cE(r*-n)(Fp/P) (8-14) The above set of equations determine the equilibrium values of the rates of return on home bonds, r; exchange rate, E; and real income, Y; respectively and simultaneously. By invoking a small * * country assumption, r and P are considered constant. The policy , 'c c c c determined variables are M, B , F , and G. B and F denote central bank's holdings of home and foreign bonds. III. Stability Analysis A disequilibrium behavior for exchange rate, real income, and interest rate is given as: 148 ' n n E = a[b (r-W,r— h,Y)w-(EFp/P)] u3>0 (8—18) Y = B[c(Y-tY+R+(r-n)(Bp/P)+E(r*-h)(Fp/P))+I(Y,r-n,K)+G+x(Y,EP*/Ph)+ f—Y] B>>o (8—19) r = AT(Bp/P)—b(r—h,r*-W,Y)w] A.>O (8-20) where d, B, and A are adjustment coefficients. Equation (8-18) indicates that the rate of change in the ex- change rate is proportional to excess demand for foreign bonds. This causes home private wealth holders to realign their portfolio by supplying their real cash balances and/or home bonds for foreign bonds in foreign exchange market. Home currency depreciates and E rises. Equation (8-19) is a goods market adjustment equation. The basic underlying hypothesis for this equation is producers' response to unexpected inventory changes. As firms find their inventories are decreasing unexpectedly, they increase their output to meet the demand for goods. Equation (8—20) states that the rate of change in home interest rate, r, is proportional to excess supply of home bonds. If the stock of home bonds held by home private wealth holders, Bp/P, rises rela- tive to its demand, b(r—h,r*-w,Y)w, the treasury must offer higher interest rate (r) to borrow from private wealth holders. A linearized version of (8—18, 19, and 20) in the neighborhood of equilibrium is: E a O O -(Fp/P)(l-b*) wb: wb: ‘ E-E Y = o e o xE(P*/Ph)+c(r*-1r)(Fp/P) z -(n-c(Bp/P)) Y—Y r 0 0 A —b(Fp/P) ewa dwbr j r—f (8-21) 149 where E, Y, and f are the long-run equilibrium values for E, Y, and r respectively. To derive the coefficient matrix, the change in wealth, dw, is replaced by differentiating equation (8-4). The necessary and sufficient conditions for the stability of the system are that the trace of coefficient matrix J be negative and its determinant positive: P 7': * it a 0 O -(F /P)(l-b ) wa wbr J=ljij1 0 s 0 xE(p*/ph)+e(r*—n)(FP/P) z -(n-c(Bp/P)) o o A —b(Fp/P) -wa -wbr The trace and determinant of J are: trace(J) = —d(Fp/P)(1—b*)+BZ—war, a, B, A>O If Z<0, then trace(J) <0 If Z>O, then trace(J) §0 Therefore, the necessary condition for stability would be met if the IS curve was sloping downward in a r—Y plane, Figure 8—1. The above statement is equivalent to the following one. The necessary condi— tion of stability is met if the IS curve was sloping upward in an E—Y plane, Figure 8-2. det(J) = aBAK—Fp/P)(-Zwbr)dw(n—c(Bp/P))wa)—(xE(P*/Ph)-+ * 7k c(r*-n)(Fp/P)(-wb;bf+w2b:bY)-b(Fp/P)(dw(n-c(Bp/P)bY-Zwbr] (8—22) Under the necessary conditions (i.e., Z0 (8-24) 151 On the FF curve, in Figure 8—3, the rate of change in ex— change rate is zero, E=O. A11 points below the FF curve depict ex- cess demand for foreign bonds. Points above the FF curve depict excess supply of foreign bonds. Therefore, exchange rate tends to rise (decline) for points below (above) the FF curve. The dynamic behavior of real income was specified by equation (8-19) and Figure (8-2). Figure (8-2) is represented here as Figure 8—4. E The IS curve, in Figure Igfy=0 8—4, depicts combinations of E .___;___) ._, ... ...);— ——>(_. and Y, where the rate of change ._, ¢_.. —4» 4— e in real income is zero, (i.e., (—__ Y=0) and demand for goods equal ' Y Figure 8-4 their supply. The sign pattern of the slope of the IS curve were explained in Chapter three. A11 points above the IS curve depict excess demand for goods. Thus, the real income tends to rise (decline) for points above (below) the IS curve, Figure 8—4. The dynamic behavior of home interest rate was explained by equation (8-20). This equation is an excess supply of home bonds. The BB curve, in Figure 8-5, depicts combinations of exchange rate and real income where demand for E bonds equal their supply for given r*. The BB curve has a negative slope. An increase in real income causes an ex- BB cess demand for home bonds. Figure 8—5 152 Exchange rate should decline to restore equilibrium in this market. A decline in exchange rate would reduce the home currency valuation of foreign bonds and thus real wealth. The demand for home bonds decline as wealth declines. The slope of the BB curve can be derived by taking a total differentiation of equation (8—2): -(Fp/P)de-wadY-wbrdr = (b/P)dM-((1-b)/P)(dB—dBC) + (Eb/P)(dF-dFC) (8—25) The slope of the BB curve is given as: (dB/cm = c-wa)/< ((wa)/(F /P(l-b )) (8-27) as long as Z is negative. The left (right) hand side of inequality (8-27) denotes the slope of the IS (FF) curve. The stability condition of (8—27) denotes the fact that home foreign bonds market is more responsive to a given change in exchange rate than home goods market. In other words, the real income must rise more in home foreign bonds market than in the home goods market for a given change in the exchange rate in order to restore the equilib- rium in both markets. IIIb. Dynamics of the Model in an r-Y Plane The dynamic behavior of home interest rate was explained by the equation (8-20) and Figure (8-8). The BB curve, in Figure 8—8, r Excess Demand for Home Bonds 1 above the BB curve are character- Y depicts combinations of the home interest rate and real income where the rate of change in interest rate is zero (i.e., r=O). All point on W Ill fl 0 II C ized by an excess demand for home Figure 8-8 154 bonds. As a result, interest rate is rising. The slope of the BB curve can be derived from equation (8-25): /(br) < 0 (8—28) The dynamics of real income can be represented by equation (8—19) in Figure 8-9. At point r again I assume that the first (necessary) condition of sta- bility is met and the IS curve is sloping downward. Points below the IS curve indicate Y an excess demand for goods. Figure 8—9 Points above the IS curve represents an excess supply of goods. Thus, real income tends to rise (fall) for points below (above) the IS curve in Figure 8-9. The dynamic behavior of exchange rate was explained by equa- tion (8-18). This equation was an excess demand for foreign bonds. The FF curve in Figure 8— r 10. depicts combinations of FF interest rates and real in- comes where demand for home bonds equal their supply. The FF curve has a posi- tive slope. An increase in real Figure 8-10 income causes the excess demand for foreign bonds. Interest rates should rise to restore equilibrium in the market, holding everything else constant. A rise in the home rate of interest reduce demand 155 for foreign bonds. The slope of the FF curve is derived from equation (8-23): * * (dr/dY) — -(bY/br) > 0 To determine the stability conditions, I superimpose Figures (8—8, 9, and 10).3 In Figure 8-11 (8-12) the IS curve slopes down steeper (flatter) than the BB curve. r Figure 8—11 Figure 8—12 Arrows in Figures (8-11 and 12) indicate the direction of change in real income, Y, and interest rate, r. Figure 8—12, where the BB curve is sloping downward and steeper than the IS curve, depicts a clear picture of unstable dynamics for interest rate and real income. But when the IS curve slopes down steeper than the BB curve, our economy behaves in a stable manner, Figure 8-11. The stability condition of Figure 8-11 can be written in terms of respective slopes: Z/(n—c(bp/P)) < -(bY/br) (8-30) as long as the necessary condition of stability (i.e., Z <0) is met. The left (right) hand side of inequality (8—30) denotes the slope of 156 the IS (BB) curve. The stability condition of (8-30) denotes the fact that the home bonds market is more responsive than home goods market to a given change in interest rate. In other words to restore equilibrium in both markets, real income rises more in the home bonds market than in home goods market for a given rise in interest rate. IV. Comparative Statics To examine the comparative statics, I take the total differen— tiation of equations (8-2, 3, and 14), with the change for real wealth can be derived equation (8-4): wbrdr+wadY+(b/P)(dM+dBp+Ede+deE) (1/P)dsp (8—31a) wb:dr+wb:dY+(b/P)(dM+dBp+Ede+deE) (E/P)dFP+(Fp/P)dE (8-31b) (xE(P*/Ph)+c(r*—n)(Fp/P))dE+(Z)dY-(n-c(Bp/P))dr = -o(r-n)(1/P)dBp - cE(r*—n).(1/P)de-dc (8-31c) The set of equations (8-31) can be rearranged as: -(Fp/P)(1-b*)dE+wb:dY+wb:dr= —(b*/P)dM—(b*/P)dB—dBC) + (s/P(1—bTXdF—dsc) (8-32a) (xE(P*/Ph)+o(r*—n)(Pp/P))dE+(Z)dY-(n-o(BP/P))dr-o(r-nx1/P(dB-db°) + (Eo(r*-n)/P)(dF—dF°)—dG (8-32b) —(Fp/P)de-wadY—wbrdr = (b/P)dM—((l—b)/P)(dB—dBc) + (Eb/P)(dF—dFC) (8—32c) 157 The solution for changes in exchange rate, real income, and interest rate is given as: dM dE c —A (dY) = dB dr dFC dG where F’(FP/P)(l-b*) ‘wb: wb* - r * A = xE(P /Ph)+c(r*-n)(Fp/P) z -(n—c(Bp/P)) --(b/P)(Fp/P) .wa -wbr J and F—(b’VP) b*/P -(E/P)<1-b*) o- B = 0 c(r—h)/P cE(r*—o)/P -l _b/P (1-b)/P o . The jacobian matrix A is identical to coefficient matrix J (derived for stability analysis) except for the positive parameters a, B, and A. However, since it was not possible to prove the sign of coefficient matrix A, the comparative analysis of partial deriva- tives is pursued geometrically rather than mathematically. The graphical investigation of comparative analysis will be based on three building blocks. These building blocks are the equi- librium conditions of the goods, home bonds, and foreign bonds mar- kets. The comparative statics of each market will be examined inde- pendently from the other markets, in the following subsections. In other words, the impact of financial policies on each market is examined independently from the others. However, the theoretical 158 implications of these analysis will be combined in order to give us an overall view of the economy's response to changing policies. Goods Market IS Equilibrium in the E goods market is depicted by Figure 8-13 and 14. * Monetary policies Slope=-(Z)/(xE(P /Ph) + * P leave both IS curves un- c(r '")(F /P)) disturbed. Y But open market Figure 8-13 operations and budgetary policies have reverse effects on the IS curves in Figures 8-13 and 14. An open market pur- r chase of home and/or foreign bonds (i.e., Bci, Fc+) re- duces home citizens interest earnings and hence their Slope=(Z)/(n—c(Bp/P)) IS disposable income and aggre— Y Figure 8-14 gate demand holding every- thing else constant (including GNP). Exchange rate should rise to restore equilibrium in goods market through a surge in net exports. Therefore, a purchase of bonds would shift the IS curve in Figure 8-13. But as we will see this effect is not very strong. Holding everything else (including real income and exchange rate) constant, the contractionary impact of bond purchase could be offset through a reduction in home interest rate. Therefore, the IS v—‘a—-—o—~ 159 curve would shift downward in Figure 8-14 as a result of bond pur- chase (i.e., Bc+, FC+). This effect is not strong either as we will see in the open market operation subsection (Vb) of section V. An expansionary fiscal policy disturbs equilibrium in goods market. Either the exchange rate should decline or home interest rate should rise to restore the equilibrium in that market. There— fore, the IS curve shifts down (up) in Figure 8-13 (8-14) as a result of expansionary fiscal policy. Foreign Bonds Market Equilibrium in the foreign bonds market is depicted by Figures 8—15 and 8—16. B r FF FF * Slope=(wb:)/(Fp/P)(l-b*) Slope=-(b;/br) Figure 8—15 Figure 8-16 An expansionary monetary policy (i.e., printing money) dis- turbs equilibrium in this market by increasing wealth and hence demand for all financial assets including foreign bonds. In the foreign bonds market either E and/or r should rise to restore equilib- rium. An increase in exchange rate would increase the home currency value of foreign bonds held by home citizens. An increase in home interest rate would reduce the home demand for foreign bonds. Thus, the IS curve shifts upward in Figures 8—15 and 8-16. «usury-hm -HHun—H-ug—«HL -__._'-..__._..-.-.._-— «wank-w 160 An open foreign bond purchase (i.e., FC+) disturbs the market by reducing the outstanding supply of foreign bonds. As the case of monetary expansion, either E and/or r should rise to restore equilib— rium in the market. An open home bond purchase (i.e., Bc+) disturbs the market by decreasing wealth and hence the demand for foreign bonds. To restore equilibrium E and/or r should decline. Therefore, the FF curve shifts down in Figures 8—15 and 8—16. A fiscal policy by itself has no effect on the location of the FF curve. Home Bonds Market Equilibrium in the home bonds market is depicted in Figures 8-17 and 8—18. E r Slope=—(wa)/b(Fp/P) BB Slope=-(bY/br) BB Figure 8—17 Figure 8-18 An expansionary monetary policy (i.e., M+)disturbs this mar— ket by increasing real wealth and hence demand for home bonds. Either E and/or r should rise to restore equilibrium in this market. A re— duction in r reduces demand for home bonds directly. A reduction in E reduces home currency values of foreign bonds and therefore the total wealth. Thus, a reduction in E reduces demand for bonds *4 161 indirectly through a change in wealth. In both cases the BB curve shifts down in Figures 8-17 and 8—18. An open foreign bond purchases (i.e., Fc+) affects the market indirectly through wealth. Real wealth declines so the demand for home bonds. Either E and/or r should rise. Thus, the IS curve would shift up in Figures 8—17 and 8—18. An open purchase of home bonds (i.e., BC+) reduces the out- standing supply of home bonds. Either E and/or r should decline. The BB curve shifts down in Figures 8-17 and 8-18. Table one sums up the conclusion of our discussion. TABLE 1 SHIFTS IN REAL AND FINANCIAL MARKETS AS A RESULT OF FINANCIAL POLICIES (MARKETS ARE EXAMINED INDEPENDENTLY FROM EACH OTHER) M BC FC G EY Plane - up up down IS curve rY plane — down down up EY Plane up* down* up - FF curve rY Plane up* down* up - EY Plane down* down up* — BB curve * 4‘ *4~ rY Plane down* down up* — V. Comparative Sgatic§;_ Effects of Financial POlicieS on Exchange Rate, Real Income, and Interest Ratei_Combined Effects The effects of financial policies on real and financial markets were examined independently in each market. This section 162 examines such effects (interdependently) on all markets. I make the following assumptions about real and financial markets' speed of adjustment. I assume that financial markets respond faster than the real sector to government policies. This is a standard assumption made in the literature. Va. Monetary Policy: Discount Rate and Reserve Requirement Fluctuations An application of an expansionary monetary policy to the eco- nomic system can be explained by a decrease in discount rate, reserve requirement ratios and/or government purchase of home bonds. In this section, I examine the impact of an expansionary monetary policy, through discount rate and reserve requirement ratio manipulations, on the economy. Any change in the supply of home bonds held by private citizens is sterilized. As explained in Table one, the IS curve does not respond to an expansionary monetary policy. Expansionary monetary policy shift the equilibrium locus (FF curve) up. Home bonds market equilibrium conditions (BB curve) shifts down, Figure 8—l9. Suppose the economy was initially at point E in Figure 0 8~l9. With an expansionary monetary policy, the financial markets would be at equilibrium point E But at E the goods 1' 1 market (i.e., IS curve) is out Figure 8-19 of equilibrium condition. There is an excess demand for goods. This % 163 isbased on the assumption that financial markets respond faster than real markets to financial policies. However, because of an excess demand for goods at E , real income starts to rise. El starts to 1 move to the right. At equilibrium point E1, demand for foreign bonds equals their supply. However, this demand for foreign bonds rises as real income rises. The exchange rate rises and home currency depreciates. The supply of foreign bonds declines as net exports (as a function of real income) declines.A Exchange rate surge keeps rising. But this rise is partially dampened by the very effect of rise in E on supply of foreign bonds denominated in home currency. As far as home bonds market is concerned, a rise in real income requires a decline in exchange rate in order to remain in equilibrium. However, exchange rate equilibrium value is determined in the foreign bonds market. Home bonds respond to this circumstances by shifting to the right. This happens as supply of foreign bonds decline through a surge in imports. Thus, the economy starts to move away from El along the FF curve toward E2. But the economy can stop short of E2 as the IS curve starts to shift upward. An upward shift in the IS curve can be explained by a reduction in interest earning of private citizens on foreign bonds. However, this shift is not likely to be a major one as this interest earnings makes a small fraction of real income. The large swings and cyclical behavior of exchange rate and real income might become reality if an expansionary monetary policy passes certain levels. Suppose the economy is initially at poine E3 in Figure 8-20. A further expansionary mone- tary policy can cause a sudden jump (large swing or catastrophe) in equilibrium values of real income and exchange rate. As BB' BB 4’ Y explained before, at E real income will rise and the Figure 8-20 BB' curve shifts upward along the FF' curve. However, the economy will move from E4 to another equilibrium point, E5, on the other stable arm of the IS curve in Figure 8—20. The cyclical behavior of exchange rate (real income) might become more clear by the use of picture (8-21). E (or Y) curve, in Figure 8-21, shows the equilibrium path of exchange rate (or real income) as a result of continuous change in monetary policy. Arrows in Figure 8-21 depict the direction of equilibria as a function of monetary policy. The interest rate impact E (or Y) ., n 0" ,1 of monetary policy can be illus- J:;"”="’f’ trated by the use of Figure 8— i L 1 22. As a result of expan- sionary monetary policy, the economy will move to equilib- Figure 8’21 rium point E1 (for financial markets only) if it was initially at E0. At E1 financial markets are in equilibrium but goods market is not. At E1, because of excess demand for goods, real income rises. Demand for home bonds rises and therefore, interest rate declines. Net eXports decline as real income goes up. Supply of foreign bonds decline. This leads to a downward shift in the FF curve. Therefore, the economy Y moves away from.El, along BB' Figure 8—22 curve toward E2. However, the economy can stop short of equilibrium point E as the IS curve shifts down. The shift in the IS curve is 2 caused by the reduction of interest earnings of private wealth holders on home bonds. The cyclical behavior of interest rate and real income can become clear by the use of Figure 8-23 similar to Figure 8—20. Suppose the economy r was initially at equilibrium point E An expansionary 3. monetary policy insures equilib- rium for financial markets first. But at E4 the real Y income starts to rise and Figure 8-23 economy moves to an equilibrium point such as E5 in Figures 8—23 and 24. At M1 level, the economy r experiences a sudden jump (i.e., catastrophe), as money supply rises continuously. But the economy does a — .00000000000 0 O. .‘M.... not experience large swings . - Y Figure 8-24 1% (i.e., catastrophes) at the same level of money supply (i.e., Ml) as money supply shrinks. This has been explained by the "delay rule" in theory of catastrophe. 166 Figure 8-25 Vb. Monetary Policy: Open Market Operations in Home Bonds Market Government open market purchase of home bonds leads to a reduc— tion of outstanding supply of home bonds held by home private wealth holders and an expansion in the supply of cash balances. The effects of open market operations (in home bond) on the economy are summarized in Table two. Information, given in Table two, is obtained from Table one. This table provides information about the economy in both E—Y and r-Y planes. TABLE 2 EFFECTS OF OPEN MARKET PURCHASE OF BONDS ON REAL AND FINANCIAL MARKETS E-Y Plane r—Y Plane Bc+=========> M'f BC+==========> M+ IS curve shifts up no change shifts down No change FF curve shifts down* shifts up* shifts down* shifts up* shifts down* shifts down shifts down* BB curve shifts down* * 167 The comparative statics of exchange rate and real income can be illustrated by Figure 8—26. IS' 0 E '\ I IS FF 0 Bb \ 33a 3 Ba 3 Figure 8-26 An open market purchase of home bonds does not cause any shift in the foreign bond equilibrium condition. The FF curve shifts up and down by (b*/P)/((Fp/P)(l—b*)) amount (equation 8-23). BB curve shifts down by (l/pr) (equation 8-25). The IS curve shifts up by (c(r-fi)/P)/(xE(p*/Ph)+c(r*-w)(Fp/P)) amount (equation 8—32b). Suppose the economy was initially at equilibrium point A0 (B0), in Figure 8—26. An open market purchase of home bonds causes the IS curve to shift up and the BB curve down. Since the absolute amount of change in location of curves can not be determined, I drew l 2 3 l 2 three different shifts for the BB curve, Ba’ Ba’ and B8 (or Bb’ Bb’ and B3). Equilibrium conditions in financial markets would be 1 2 3 . achieved at points Al, A2, and A3 (or B , B , and B ) respectively. At equilibrium point A1 (or B1) the goods market is in disequilibrium. . 1 The supply of goods exceeds their demand at equilibrium point A (or 1 . Bl). Real income declines. The B: (or Bb) curve shifts down along the FF curve until it reaches point A2 (or B2). If the initial open market purchase of home bonds implied a financial equilibrium 168 condition of point A3 (or B3), demand for goods would exceed their supply. Real income would rise and the B: (or B2) curve would shift upward. The economy would settle at equilibrium point B2 (or A2). The catastrophic nature (large swings) of real income and exchange rate can be illustrated by the use of Figure 8—27. Suppose the economy E 151 was initially at point E0. An open purchase of home bonds leads to an upward (downward shift of the IS (BB) curve from 13° (313°) to Is1 (BB1) in Figure 8-27. Financial markets Figure 8—27 will be in equilibrium at point E1. However, at El’ excess supply of goods is positive. Real income would decline. The BB curve would shift down along the FF curve until it reaches point E3. At E3, all markets are in equilibrium. Equilibrium path of exchange rate (or real income) and the occurrence of catastrophe is depicted in Figure 8-28. The effects of an E (or Y) open market purchase of home ‘\\“~‘~E bonds on the interest rate 5 J t can be illustrated by Figure E 8—29. Suppose that the economy 3\\\“~.‘~ was initially at point A0 (or c Bo). An open market purchase Figure 8—28 of home bonds does not cause any shift in foreign bond markets. The FF curve shifts up as shown Figure 8-29 by (b*/P)/(wb:) amount (equation 8—23). The BB curve shifts down by l/war amount (equation 8-25). The IS curve shifts down by (c(r-n))/ (n-c(Bp/P)) amount (equation 8—32b). An open market purchase of home bonds causes the IS and BB curves to shift down. The analysis is similar to the one discussed for Figure 8—25. The economy will move from equilibrium point Ab (B0) to A2 (or B2). Interest rate and real income decline. At this stage I skip the dynamics of the economy in moving from the initial equilibrium point A.O (or B0) to A2 (or B2). The argument is similar to the one given for E-Y model (i.e., Figure 8—26). The catastrophic nature of interest rate is illustrated by Figure 8—30. r Figure 8-30 170 The economy was initially at 30' An open purchase of home bonds leads to downward shifts of both the IS and BB curves. Point B1 would be a financial equilibrium if the IS curve shifts down further than the BB curve. The economy can not sustain itself at point B1. It moves further westward as a result of excess supply of goods and leftward shift of the FF curve until it reaches point B2. Interest rate declines initially as a result of open market purchase of home bonds. However, at point B1 real income starts to rise. Demand for home bonds declines and thus, its price falls. As price of home bonds declines interest rate shoots up. Indeterminate behavior of interest rate is more visible when the BB curve happens to be tangent to the IS curve at extrema, Figure 8—31. Even then the r occurrence of the catastrophe (or large swings), in this case only, depends on the sen- sitivity of real and financial IS markets to changes in open Y market operations. Figure 8-31 Vc. Intervention in Foreign Bond Market A government open market purchase of foreign bonds leads to a reduction of outstanding supply of foreign bonds held by home wealth holders and an expansion in the supply of real cash balances. Table three summarizes the effects of open market pur— chase of foreign bonds by government on the economy. 171 TABLE 3 EFFECTS OF OPEN MARKET PURCHASE OF FOREIGN BONDS ON REAL AND FINANCIAL MARKETS E—Y Plane r-Y Plane F°+ > M 3% > M IS curve shifts up no change shifts down no change FF curve shifts up shifts up* shifts up shifts up BB curve shifts up: shifts down: shifts up* shifts down* Information, given in Table three, is obtained from Table one. Table three provides the necessary information about the economy in both E—Y and r-Y planes. The comparative statics of exchange rate and real income can be illustrated in Figure 8—32. Figure 8-32 An open market purchase of foreign bonds does not cause shift in the BB curve initially. The home bond market shifts up and down by l/Fp (or (b/P)/(wbr)) in E—Y (or r-Y) plane, (equation 8-25). The * * foreign bonds market shifts up by the amount of ((b /P)+(E/P)(1-b ))/ 172 ((Fp/P)(l-b*)). (or ((b*/P) + (E/P)(l-b*))/wbr), in an E—Y, (or r-Y), plane. The IS curve shifts up (down) by the amount of (cE(r*-fl))/ (XE(P*/Ph)+(c(r*-fi)(FP/P)),(or(cE(r*-w))/(n-c(Bp/P))) in an E—Y (or r—Y) plane. An open market purchase of foreign bonds causes the economy to move from the equilibrium point A0 (or B0) to A2 (or B2). I skip the explanation of the equilibrium path from A.o (or B0) to A2 (or B2) since it is similar to the accounts given earlier. Thus, as a result of this operation exchange rate rises. But the direction of change in real income depends on real and foreign bonds markets sensitivity to open market operational policies of government. The flatter (i.e., the more elastic) the curves are, the more likely that real income would rise. Furthermore, the higher (lesser) the FF (IS) curve shifts up the more likely that real income would rise. The occurrence of catastrophe can be illustrated by Figure 8—33. Figure 8-33 Here, in this figure, open market purchase of bonds implies different responses by foreign bonds market. Suppose the economy was initially at point 1 in Figure 8—33. If the FF curve shifts up less than (equal to) the IS curve, E would 173 rise and real income would decline (real income would remain the same). The economy would move from point I to 3 (4). But if the FF curve shifts up more than the IS curve, the economy would move from equilib- rium point 1 to 6. Both E and Y would rise suddenly. The comparative statics r /FF 0 of interest rate is illustrated by Figure 8-34. An open market purchase of foreign bonds leaves the BB curve unchanged. The FF curve shifts up and the Figure 8-34 IS curve shifts down. Thus, the economy moves from the equilibrium point A0 to Al in Figure 8-34. Interest rate (real income) would rise (declines) as a result of open market purchase of foreign bonds. Vd. Fiscal Policy Government fiscal policy can either be bond or money financed. Table four depicts a money financed fiscal policy effects on real and financial‘markets. TABLE 4 EFFECTS OF MONEY FINANCED FISCAL POLICY E—Y Plane r—Y Plane G+ ==========> M+ G+ ==========> M+ IS curve shifts down no change shifts up no change FF.curve no change shifts up* no change shifts up* BB curve no change shifts down: no change shifts down* 174 The equilibrium conditions of real and financial markets are illustrated by points e0 in Figures 8—36 and 37. According to Table four, money financed fiscal policy causes the FF (or BB) curve to shift up (down) in both planes. The IS curve shifts up (down) in r-Y (E—Y) plane, Figure 8—37 and 36 respectively. Figure 8-36 Figure 8-37 These figures clearly indicate that real income and exchange rate rise. But interest rate declines. An important point is that the equilibrium point el was basi— cally determined by real and foreign (home) bonds markets in E-Y (r—Y) plane, Figure 8-36 (8—37). The catastrophe nature of the model (large swings in the economic variables) can be illustrated by Figures 8-38 and 39. E Figure 8—38 Figure 8-39 175 Suppose the economy is initially at equilibrium point A in Figures 8-38 and 39. An expansionary fiscal policy financed by printing money leads to changes in the original locations of the IS, FF, and BB curves. ‘As explained before, the IS curve shifts down (or up) in the E-Y (or r-Y) plane, Figure 8-38 (or 39). The FF curve shifts up in both E-Y and r-Y planes. The BB curve shifts down in both E—Y and r-Y planes. Therefore, the economy jumps suddenly from the equilibrium point A to B in Figures 8-38 and 39. Figures 8-40 and 41 trace out the equilibrium path of exchange rate, real income, and interest rate as a result of continuous fiscal policy changes. E (or Y) r i \H L / \ Figure 8—40 G financed Figure 8-41 G financed by M by M Table five depicts the effects of a bond financed fiscal policy on the economy. TABLE 5 EFFECTS OF BOND FINANCED FISCAL POLICY E-Y Plane r—Y Plane G4} ==========> BC»; or Bf G‘f‘ =========> Bcwlr or B’f IS curve shifts down shifts down shifts up shifts up FF curve no change shifts up* no change shifts up* BB curve no change shifts up* no change shifts up 176 Again I assume that the necessary condition of stability is met (i.e., Z <0) and consider a down (up)eward sloping IS curve in a r—Y (E-Y) plane. Equilibrium conditions of the economy are specified by points eO in Figure 8—43. E r I Figure 8-42 Figure 8—43 According to Table five, bond financed fiscal policy causes both the BB and FF curves to shift up. The IS curve shifts down (up) in E—Y (r-Y) plane. Figures 8-42 and 43 clearly indicate that bond financed deficit would raise exchange rate, real income and reduce interest rate. The catastrophe nature of the model can be illustrated by Figures 8—44 and 8-45. E Figure 8-44 Figure 8—45 177 Suppose the economy is initially at the equilibrium point A in Figures 8-44 and 45. An expansionary fiscal policy financed by open market operations in the home bonds market leads to changes in the original locations of the IS, FF, and BB curves. As explained by Table five, the IS curve shifts down (up) in an E-Y (r-Y) plane. The FF curve shifts up in both E-Y and r-Y planes. The BB curve shifts up in both the E—Y and r-Y planes. A continuous bond financed fiscal policy leads to the occurrence of a large swings in the equi- librium values of exchange rate, real income, and interest rate, Figures 8-44 and 8—45. The equilibrium path of exchange rate, real income, and interest rate can be illustrated by Figures 8—46 and 8-47. E (or Y) r J! / K Figure 8—46 Financed by Figure 8—47 Financed by bond (Bc+) bond (B°+) 178 CHAPTER 8—-Footnotes lFoley (1975). 2In an E—Y plane, the dynamic behavior of exchange rate and real income are explicitly depicted by equations (8-18 and 19) and Figures 8-3 and 4. The dynamic behavior of interest rate is not explicitly examined in such a plane. This is due to the fact that interest rate can not be presented in such a plane, Figure 8-6. As a result, the dynamic behavior of the model is primarily dictated by the IS and FF curves. The BB curve is reacting passively to the rest of the model. This issue will be examined further in section V of this chapter. 3The picture changes as the economy is represented by a r-Y plane. The dynamic behavior of real income and interest rate rather than the exchange rate can only be shown in a r-Y plane. This can be accomplished through equations (8-19 and 20) and Figures 8—9 and 8. The dynamic behavior of exchange rate is not explicitly examined in such a plane, Figure 8-11, even though a market clearing condition for foreign bonds is derived in Figure 8-10. As a result, the dynamic behavior of the model is dictated primarily by the IS and BB curves. The foreign bonds market is reacting passively to the rest of the economy. I examine this issue further in section V. 4Under a flexible exchange rate regime, the balance of pay— ments (BOP) is zero. The current account imbalances, x(Y,P*/Ph), are equal to foreign asset flows, F, equation (7-67): x(Y,EP*/Ph) = EF=E(FP+FC) (7-67) In a BOP (flow) theory of exchange—rate determination, the equilibrium values of exchange rate is determined by the BOP constraint, equation (7—67). But in a portfolio—balance model of exchange—rate determina- tion (applied in this chapter), the equilibrium value of exchange rate is determined when stock demand for foreign bonds equal their (stock) supply, equation (8—3): * b (r—n,r*-N,Y)w = EFp/P (8-3) To use the BOP constraint, equation (7-67) as a structural equation for our model leads to the overdetermination of exchange rate. CHAPTER 9 Conclusion In this thesis, I develop and examine a real and financial model of real income, exchange rate, and interest rate determina— tion. Different versions of this model have been investigated. The purpose of such models were to deal with the subject of exchange rate and other variables' fluctuations within the mathematical framework of catastrophe theory. It was basically the existence of large swings and sudden jumps in the equilibrium values of exchange rate that warrant this dissertation. I searched the real sector of the economy rather than expec— tations formation (as explained by De Grauwe 1983) and/or monetary sector (as explained by Tobin 1979) of the economy as a major con- tributing factor in explaining the catastrophic nature of the ex— change rate. Tobin (1979) can be a starting point in building a monetary model of catastrophe theory. Appendix A examines this case briefly. De Grauwe (1983) develops an expectational formation model of catastrophe theory. Appendix B discusses such a model. The characteristics of this model were examined in Chapter one. Therefore, I restrain myself on this issue and pay closer attention to the consequences of this thesis. First of all, catastrophes are defined as sudden or 179 180 discontinuous jumps (or swings) in the equilibrium values of a dynamic system when the parameters of such a system are changing continuously. Probably real income and interest rate are the two major factors among many, which can explain the equilibrium values of exchange rate. Any factor (or factors) that can cause catastrophes in real income and interest rate could do the same for exchange rate. Therefore, I searched the real sector of the economy and theories of income fluctuations in order to pinpoint the sources of catastrophes. The sigmoid-shape investment function developed and modified in Chapter two was treated as the major source in the development of in- come fluctuations and turning points. This has been confirmed by Ichimura (1954, 1955) and Varian (1977, 1979). The sigmoid-shape investment function was originally developed by Kaldor and applied by Ichimura and varian to catastrOphe theory in a closed economy. This investment function was considered as a function of real income and the stock of capital. Chapter two provided the theoretical groundwork to modify this investment function. The interest rate was added to the sigmoid-shape investment function as an additional variable in the line of existing literature on investment theory. In Chapter three, in line with the modified version of sigmoid-shape investment function, I derived a non-linear IS curve. Chapter fourexplained the essentials of catastrophe theory. A Keynesian macroeconomic analysis of a closed economy was presented in Chapter five. The equilibrium values of real income and interest rates were explained in an IS-LM model. It was shown there that continuous changing financial policies could lead to the l8l occurrence of catastrophes in this model. This was a significant departure from the existing literature on the application of catas— trophe theory to a closed economy model. Both Ichimura and Varian could explain the catastrophes of real income through a continuous change in the stock of capital. But in my model, I was able to explain large swings in the equilibrium values of real income in terms of continuous fiscal and monetary policy changes. The stock of capital was a third source of catastrophe. Besides, in my model, not only the equilibrium values of real income, but also that of the interest rate was explained. With the brief examination of aggregate demand and supply I finished the first part of the thesis (i.e., closed economy). In the second part of the thesis, I examined catastrophes in an open economy. Chapter six presented a brief review of the existing literature on alternative theories of exchange rate determination. It was concluded that the portfolio—balance theory has superiority in explaining exchange-rate behavior. At this stage, I had built two major building blocks in order to examine consequences of financial policy changes in an open economy. The first building block was the IS and LM model that I developed in Chapter three. The second building block.was the port- folio—balance model of exchange-rate determination which was speci- fied in Chapter six. In Chapter seven, I used these two building blocks to con- struct a real and financial model of income and exchange-rate deter— mination. The goods market clearing condition in conjunction with the portfolio-selection equilibrium condition determined the 182 equilibrium values of real income and exchange rate. Interest parity condition was assumed throughout this chapter. As a result of this assumption and in conjunction with small country assumption home interest rate was constant and equal to foreign interest rate (corrected for inflationary expectations). I examined the equilibrium behavior of exchange rate and real income in this model with a static, rational expectations, infla— tionary, and dynamic framework. The stability conditions of these models were carefully examined. It was proved that as long as the economy remained on the conventional down (up)dward sloping parts of the IS curve in a r-Y (E-Y) plane, the system is stable. As a result of any small perturbation the system returned to its original position. However, the system displayed unstable behavior on the unconventional upward (down)~ward sloping part of the IS curve in a r-Y (E-Y) plane. The displays of both stable and unstable behavior in my model are the major keys enabling me to explain discontinuous changes (i.e., catastrophes) in the equilibrium values of exchange rate and real in— come. It was shown that continuous changing financial policies can move the system from one set of stable equilibrium positions (surface) to another set of stable equilibrium positions (surface) suddenly and rapidly. There exists a locus (surface) of unstable equilibria between the two stable surfaces in which the economy could not sustain itself. This was the crux of this thesis. The occurrence of catas- trophes in exchange rate is beyond any doubts. Exchange-rates fluc- tuate continuously in small amplitude daily. This has been explained 183 by the existing literature in the field. However, along this small amplitude, there sometimes happens sudden discontinuous changes in the values of exchange rate. This thesis was bent to explain such changes in the framework of dynamic theory of catastrophe. Catastrophes happened when our model moved from one stable equilibrium surface to another stable surface as a result of con- tinuous financial policy changes. However, the result obtained when the system did not change its stable equilibrium surface did not contradict the results obtained from the existing literature on exchange-rate behavior. The same can be argued about Chapter eight. In this chapter I relaxed the assumption of interest parity condition. As a result, home and foreign bonds were no longer perfect substitutes. This made me able to study the behavior of interest rate side by side with real income and exchange rate. Due to the mathematical intractability, the case of stability analysis was harder in Chapter eight. However, the general conclu— sions of this chapter inspired the same results as Chapter seven. The last issue which is probably the first question raised by an intelligrant critic is the extent of the truth of this thesis in a world of empirical investigation. I do not believe, at this stage, that the lack of empirical investigation can harm the significant theoretical contributions of this thesis. Catastrophes (or dis- continuities) are very, very new phenomenon in the foreign exchange market (exchange rate) literature. When I started this research, there was no work on this area. Only recently De Grauwe (1983) attempted to explain exchange—rate behavior within a catastrophic 184 framework. My dissertation is the first full-scaled effort in building theoretical grounds of such a model. I do not assert that this work has developed and finished the whole picture of an exchange- rate behavior analysis within a catastrophe theory framework. Rather it can be the beginnings of new efforts and researches to contest and test the theoretical derivations of this thesis. 185 APPENDIX A An investigation of equilibrium in a financial market was depicted in the beginning of Chapter three, where Figure 3-2 showed an upward sloping LM curve. Our objective in this appendix is not to elaborate further this issue, rather to clarify the condition under which the sign of the slope of the LM curve may change. By showing out that at certain levels of income the slope of LM curve changes, we may have an additional case for application of catastrophe theory, even though it is not explained in the main body of the thesis. The theoretical groundwork of this appendix is based on pioneering works developed by Blinder, Solow, and Tobin.1 Following Tobin (1979), The LM curve is given by equation (Arl): M/P = m(r,Y,w) (A-l) where (am/8r) = mr<:0, (am/BY) = my2>0, and (am/aw) = mw> 0. w denotes the demand for wealth, which is equal to its supply. a = w(r,Y) (A—Z) where (aw/3r) = wr:>0, and (aw/BY) = wy:>0. Wealth changes by saving and by government budgetary policies. If w=B/P+M/P+K, then s=w=é/P+fi/P+x=D+I or s = .=B/P+M/P+R=D+(I—6K) (A-3) 186 where 6 denotes depreciation per unit time S denotes saving D denotes government deficit B denotes the proportion of government deficit financed by selling bonds M denotes the proportion of government deficit financed by printing high powered money To derive the slope of the LM curve I substitute (A—2) into (A—1) and differentiate totally: 0 = mr(3r/8Y)+my-i-mw(wy(3r/3Y)-l-wr) (A—A) Rearranging (Ar4) yields -(my+mwmy) = (mr+mwmr)(3r/3Y) (ArSa) or (Br/BY) = '— ($y41fiwfiy) /(ar+$W$r) (A—Sb) The numerator of the right-hand side fraction is negative, while the denominator is of ambiguous sign. (where my, mw, wy, and wr are all greater than zero, while mr is less than zero). The sign of the slope of LM curve (A—5) depends on the sign of the denominator. We have a standard LM curve if the denominator, —++ ++ - 0 . O . <-mo mr+mww is negative. This may hold if mer r The LM curve might possess a non-conventional down sloping curve, (Br/3Y‘0, or m w =>—m (A-6) r w r w r r After an extensive mathematical calculations, Tobin argues that for this condition to be met: " the elasticity of wealth with respect to the in- terest rate would have to be much larger than the substitution elasticity of demand for money. 187 Tobins's belief in this condition grow thinner as he considers capital accumulation in his model. I became interested in the outcome of Tobin's paper even though he himself had some reservations about the empirical validity of his theory. This paper is the only attempt in the investigation of a downward sloping LM curve. As it was mentioned before, the possibility of changing signs in the slope of LM curve can contribute to our own investigation of discontinuous changes in an economic system when parameters are changing slowly and continuously. However, the explained IS curve is sufficient to generate catastrophes. 188 APPENDIX A--Footnotes lSee Blinder and Solow (1973, 1974), Tobin and Buiter (1976) and Tobin (1979). 2Tobin (1979). p. 224. 189 APPENDIX B In this appendix I present an exposition of De Grauwe (1983). The first building block in De Grauwe (1983) is the following equilibrium condition in domestic money market: h-p = ~Ar+9y (B—l) where all variables are indicated as logarithms: h = logarithm of the nomial money stock p = logarithm of the home price level y = logarithm of real income De Grauwe assumes further two restrictive assumptions of interest parity and purchasing power parity conditions. The interest parity condition is given as: * r = r + u (B-Z) where r = the log of one plus the domestic interest rate * O r = the log of one plus the foreign 1nterest rate u = the expected rate of depreciation of the home currency The purchasing power parity condition can be given as: 5 = 5-p* (B—3) where E = the equilibrium value of exchange rate in the long run, expressed in logarithm E = the equilibrium value of home price level in the long run, expressed in logarithm the log of foreign price level '0 ll 190 De Grauwe assumes further that when the current exchange rate sur- passes its long run value (e>E) an "expansionary" process is set in motion in the goods market. Real income rises above its long run value and the rate of price increases hastens. The reverse holds true when e