THESIS Michigan Sm re University This is to certify that the thesis entitled V The Theory and Empirical Estimation of Currency Substitution presented by Steven Husted has been accepted towards fulfillment of the requirements for i l _Eh...lL_degreein_E.QQnomics ' L i l i RETURNING MATERIALS: 1 bVIESI_} Place in book drop to LJBRARJES remove this checkout from a“... your record. FINES will be charged if book is returned after the date w ‘ stamped below. $9 a" i 10 K030 I". ’ ‘4 - e ' f A iv ‘. I ~ . - K \I ‘ . " 1K. 7"" '. fl 3' “'4" ‘0’” '5" Wt“. c, “tr. . z ‘- __.A- - - .— _. o. “Ham—w. w. .. THE THEORY AND EMPIRICAL ESTIMATION OF CURRENCY SUBSTITUTION BY Steven Husted A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Economics 1980 ABSTRACT THE THEORY AND EMPIRICAL ESTIMATION OF CURRENCY SUBSTITUTION BY Steven Husted Currency substitution is said to exist if the following two conditions hold. First, transactors in an economy must hold, as a normal course of events, foreign currency balances. Second, the levels of foreign and domestic balances held in the economy must change in response to changes in other economic variables. That is, substitu- \/ tion between balances must occur on the demand side. The main purpose of this thesis is to develop a model which explains the degree to which foreign currencies are viewed as substitutes for domestic money balances. Our model differs from previous studies in several ways. First, it bases its formulation on the transactions demand for foreign as well as domestic balances. Second, incorporated into the model is a speculative component which allows for additional holdings of a currency if it is expected to appreciate vis a vis the domestic currency over the holding period. The model is then tested using quarterly data on the Canadian holdings of their own and United States dollars. Estimates of the elasticity of substitution in demand for these two currencies are found to be very low. Further, the size of these estimates dependsxuxnithe definition of the relative holding costs and the definition of foreign balances. Even after one allows for changes in relative holding costs, additional substitution into holdings of 0.8. dollars occurs as transactions levels in Canada rise. This finding yields substantial improvement in the explanatory power of the model. Considerable evidence is also found that the poten- tial for currency substitution is enhanced during fixed exchange rate periods. This result is in sharp contrast to previous studies and tends to support traditional assump- tions about the ability of flexible exchange rates to insulate the monetary policies of an economy. Questions of separability, lags in adjustment of asset balances, and technological innovations are also addressed in the empirical tests. The main conclusion from the study is that any model of currency substitution which does not model explicitly the role of transactions in the demand for foreign currencies is considerably biased. To my parents ii ACKNOWLEDGEMENTS One cannot spend five years in a graduate program without making a large number of friends and acquiring a long list of people who deserve recognition for their assistance along the way. First, I would like to thank my main thesis advisor, Lawrence Officer. A thesis student could not find a better major professor than Dr. Officer. His willingness to devote immediate attention to my work, his perceptive comments, editorial skills and general encouragement all made my task much easier. The other members of my committee, James Johannes, W. Paul Strassman, and Dennis Warner were also extremely helpful, both with editorial advice and in assistance with a number of troublesome issues. In addition several faculty members at Michigan State offered encouragement and advice. These peOple include Richard Anderson, Daniel Hamermesh, Mordechai Kreinin, and William Quinn. Some of the best advice and encouragement I received over the course of this project came from fellow graduate students. I would especially like to thank Mohsen Bahmani, John Fizel, Susan Pozo, Mike Thomson, and Ed Weber. Two iii 0.4 very special people helped enormously with this thesis and deserve special mention. Tom Husted helped me collect data and find several obscure articles. Marie Connolly read several of the early drafts, offered considerable advice and in general was willing to listen to my ideas even at the expense of her own work. Finally, Terie Snyder typed this manuscript. Her careful and expedient typing was sincerely appreciated. iv Table 10 11 12 LIST OF TABLES Summary of Evans-Laffer Results . . . . Summary of Brillembourg-Schadler Results. Summary of Miles Results. . . . . . . Estimation Results with Full Adjustment Model - Estimation Period 196211—19781V. Estimation Results: Full Adjustment Model Fixed Exchange Rate Period 1962111? 197011 . . . . . . . . . . . . . . . Estimation Results: Full Adjustment Model Flexible Exchange Rate Period 1970111— 19781V . . . . . . . . . . . .y. . . Estimation Results: Partial Adjustment Model - Full Estimation Period 196211- 19781V............... Estimation Results: Partial Adjustment Model - Fixed Exchange Rate Period 1962111 - 197011 - - . . - . . . . . Hatanaka Two Stage Estimates: Partial Adjustment Model Fixed Exchange Rate PeriOd o o o o o o o o o o o o o o 0 Estimation Results: Partial Adjustment Model - Flexible Exchange Rate Period 1970111 - 1978IV - - . - . . - - . - Separability Tests- - . - . - - - - - . F Statistics for Ho: Regressions are Equal in Fixed and Floating Sub Periods- - Page 37 39 42 116 127 129 132 136 140 141 145 147 TABLE OF CONTENTS LIST OF TABLES . . . . . . . . . . . . . . . . . . . Chapter ONE INTRODUCTION . . . . . . . . . . . . . . TWO SURVEY OF THE CURRENCY SUBSTITUTION LITERATURE . . . . . . . . . . . . . Introduction. . . . . . . . . . . . Origins . . . . . . . . . . . . . . The Theory of the Substitution Process. . . . . . . . . . . . . Macroeconomic Implication of Currency Substitution. . . . . . Empirical Tests and Measurement of Currency Substitution. . . . . . FOOTNOTES 0 O O O O O O O Q C O O O O Q 0 APPENDIX 1: Development and Use of the CES Function in the Near Money Literature. . . . . . . . . . . . . THREE A THEORETICAL MODEL OF CURRENCY SUBSTITUTION. . . . . . . . . . . . . Introduction. . . . . . . . . . . . The Wealth Constraint . . . . . . . Separation Rule . . . . . . . . . . Asset Partition . . . . . . . . . . The Model . . . . . . . . . . . . . The Cost Function . . . . . . . . . Technology for Money Services Production . . . . . . . . . . . Transactions Demand . . . . . . . Currency Substitution and Exchange Rate Regimes . . . . . . . . . . FOOTNOTES. . . . . . . . . . . . . . . . Vi Page Ulb 10 29 34 45 49 60 60 61 63 65 65 68 71 74 79 85 Chapter FOUR FIVE SIX APPENDIX 4: BIBLIOGRAPHY ESTIMATING EQUATIONS AND EMPIRICAL METHODOLOGY . . . . . . . . . . . . . . Introduction. . . . . . . . . . . . . Production Function for Money Services . . . . . . . . . . . . . The Empirical Model . . . . . . . . . Estimation Techniques . . . . . . . . Lagged Adjustment Models. . . . . . . FOOTNOTES. . . . . . . . . . . . . . . . . APPENDIX 2: On Approximation of the Distribution of Derived Parameters. . ESTIMATION RESULTS . . . . . . . . . . . . Introduction. . . . . . . . . . . . . The Data. . . . . . . . . . . . . . . Estimation of the Full Adjustment Model. . . . . . . . . . . . . . . Estimation of the Partial Adjustment Model. . . . . . . . . . . . . . . Other Issues. . . . . . . . . . . . . FOOTNOTES 0 O O O O O O O O O O O O O O O . APPENDIX 3: Exponent Rule for a Two-Input CBS Function. . . . . . . . . . . . . CONCLUSIONS. . . . . . . . . . . . . . . . THE NUMERICAL DATA. . . . . . . . . . . . vii Page 90 9O 91 94 98 104 108 110 111 111 113 115 131 142 149 154 157 160 163 CHAPTER ONE Introduction The monetary approach to the balance of payments has become, since its modern reformulation, a strong rival within the set of alternative theories of the balance of payments and exchange rate determination. Beginning simply as a statement about domestic money market disequilibrium leading to trade imbalances or exchange rate changes, it has been refined and extended to include additional assets (e.g. capital, bonds, equity, foreign exchange) framed in a simultaneous equilibrium model. This movement to greater realism in the paradigm has spawned one subset of models concerned with a phenomenon known as currency substitution (CS). Currency substitution is said to exist if the following two conditions hold. First, transactors in an ~/ economy must hold, as a normal course of events, foreign currency balances. It is not necessary for all actors in an economy to hold more than one currency. Indeed, it is likely that they will not. However, it is possible to identify certain subgroups within the economy who would have strong motives for holding foreign monies. These would include all individuals and firms regularly engaged 2 in foreign transactions. Thus, importers, exporters, multi- national corporations, frequent travelers, and border area residents all would be likely to maintain stocks of foreign monies in order to facilitate transactions. One can also imagine situations where speculative and precautionary motives would encourage holdings of foreign exchange. The second condition for the existence of CS is that the levels of foreign and domestic balances change in response to changes in other economic variables. That is, substitution must occur on the demand side. We emphasize the question of CS is not one of convertibility, or of the prevailing exchange rate regime. In fact, models employing various exchange rate systems have all been considered. Full convertibility is an implicit (or explicit) assumption of all of these models. Indeed, its existence affords an opportunity to rephrase the question to be considered. Given no interference in foreign exchange markets, why do some groups in an economy prefer to hold foreign exchange at all times rather than to convert local currency for it at the time of transactions? Further, what factors determine at the aggregate level changes in the composition of money portfolios? The main purpose of this thesis is to develOp a model which explains the degree to which foreign currencies are viewed as substitutes for domestic balances. This theory differs from previous theories in that it bases its formulation on the transactions demand for foreign as well 3 as domestic balances. Second, we incorporate into our model a speculative component which allows for additional holdings of foreign exchange with a change in the expected rate of depreciation in the exchange rate. Finally, we test our model using data on Canadian holdings of their own currency and U.S. dollars. Chapter Two presents an extensive review of the CS literature. Because this literature is so new, our survey is the first attempt to integrate the various papers on the tOpic. In Chapter Three we develop a model of CS. This model is used to derive a set of asset demand equations based on marginal productivity theory of monies as inputs in a production function for money services. In addition, we allow for a changing level of transactions in an economy to affect the currency mix. We also model factor augmenting technological progress as an additional determinate of money holdings. Chapters Four and Five concentrate on the empirical side of this thesis. Specifically in Chapter Four estima- tion equations are developed from the theoretical model. We also discuss estimation of model parameters which must be derived from the regression coefficients. Chapter Five presents the results of various attempts at estimating our model. In Chapter Six, our conclusions and suggestions for future research are provided. CHAPTER TWO Survey of the Currency Substitution Literature Introduction The purpose of this chapter is to review the literature on currency substitution (CS). The body of this literature is relatively small. Several important papers on this topic remain unpublished. Bilson(l979) is the only author to attempt to integrate any of the ideas on this tOpic. However, since his sole concern was the role of CS in exchange rate determination, he ignores other important aspects of the CS literature (eg. theory of the substitution process, empirical tests). Most papers on international portfolio balancing allow for CS, since foreign exchange appears as an asset in the portfolio. However, in these models, CS is not afforded any special attention. Therefore, this last set of papers will not be considered here. Although Eurodollar deposits are considered by many authors as an example of the CS process, we feel that the literature on the Eurodollar market is sufficiently peripheral to our interests to allow us to ignore this large body of literature. Specifically, the Eurodollar literature tends to focus on institutional arrangements (see McKinnon (1979)), and on liability management questions 5 of the banks which supply these foreign currency denominated deposits. In contrast, the CS literature is concerned with substitutability in demand between various currency balances. In the chapter below we consider the origins of the CS literature, the theory of the substitution process, macroeconomic implications, and finally empirical tests and measurements of CS. Origins The theoretical framework for the analysis of CS was laid in several independent articles written about the same time by former University of Chicago students, including Russell Boyer, Rudiger Dornbusch, Chau—Nan Chen, and others. These peOple have developed models which consider the question of an endogeneous demand for foreign balances. The central theme of their papers is the "store of value" function of money balances. Domestic and foreign balances are demanded because they pay a rate of return (through price deflation or exchange rate appreciation), and not because they may facilitate transactions differ- ently (so that one would need one "type" of money for one type of transaction and another type of money for another transaction). Various monies are held in aggregate asset portfolios and their relative shares are assumed to change as risk-return combinations change. The special twist that is provided to the analysis of Open economies by the assumption of CS is the possible presence of a perverse 6 change in real wealth holdings when the exchange rate changes. That is, changes in the exchange rate lead to an appreciation in the real value of one part of real wealth and a depreciation in the other. The net change in real wealth from any change in the exchange rate will then depend upon the relative shares of domestic and foreign denominated wealth (monies) in the economy's total real wealth. To complete the circle, the shares of domestic and foreign balances in total real wealth depend upon the degree to which these alternative assets are substitutes in demand. A question that should be asked (but often is not) is what is so special about money in these models? Would not bonds denominated in various currencies exhibit the same perverse wealth changes? The answers to these questions must lie with the other characteristics of monies Vis a vis interest bearing assets. These characteristics would include, of course, liquidity and the ability of money to facilitate transactions directly. Russell Boyer presents the first formal treatment of CS. His ideas appear as a chapter in a book edited by Putnam and Wilford (1978) on the monetary approach to the balance of payments. Boyer is concerned with the different solutions to the problem of determining the equilibrium exchange rate under the alternate assumptions of zero and perfect currency mobility (perfect substitutability). Using a simple three goods (two monies, and a composite good) and two countries model, he finds that if currencies 7 are immobile (and therefore not substitutable) between the two countries, then there exists a stable long run equilibrium point where excess demands for real balances in both countries are zero.1 This of course is a standard result of the monetary approach to the balance of payments. The stability of the long run equilibrium is guaranteed under either a fixed rate system (where disequilibrium in a domestic money market would lead to goods flows which would alter official holdings of foreign money until equilibrium reappeared) and under floating rates (where domestic prices would adjust to restore zero excess demand for real balances.) As Boyer notes, however, once the assumption about zero currency substitution is relaxed, then it is possible that no unique equilibrium can be established for the desired holdings of real balances.2 Under flexible exchange rates regime, it is conceivable that price levels and the exchange rate may even be indeterminate. This result will be pursued further below. The basic indeterminary in the equilibrium exchange rate arises because currency substi- tution implies shifts in demands and supplies of real balances in both countries due to exogenous shocks while the monetary approach emphasizes supply shifts (through trade imbalances under fixed rates and price movements under flexible rates) as the vehicle for returning to equilibrium. Subsequent papers by Laffer (1976), Evans and 8 Laffer (1977), Girton and ROper (1976), and Kareken and Wallace (1978) elaborate on this question of equilibrium exchange rate indeterminacy and hence expand upon the ideas of Boyer. While Dornbusch has never formally considered the question of CS, several of his papers have provided a frame- work for analyzing the effect of CS on the macroeconomy. In a 1973 article Dornbusch presents an elegant model of the dynamic behavior of the balance of payments. Money in his model is treated as the only marketable asset, but residents are restricted to maintaining balances only in their own currency. Under these conditions, devaluation is found to be a largely monetary effect.3 Real balances change because prices change, and to the extent that this change in wealth affects expenditures, deValuation will be successful. Lapan and Enders (1978) extend the Dornbusch analysis by examining the "efficacy of a devaluation when residents of a country hold assets denominated in terms of the foreign unit of account."4 As the authors use the word assets instead of money, it is not clear whether this paper is an explicit example of the CS literature.5 Actually, because the authors specify a demand for wealth equation which is identical to Dornbusch's liquidity demand function6 and assume that the assets in their model pay no interest (or, the government taxes interest payments away),7 we shall continue to classify this paper 9 as a CS model. Inclusion of holdings of foreign denominated balances as part of domestic wealth leads to a set of cir- cumstances whereby the efficacy of devaluation on the balance of trade is assured. Specifically, the authors derive Marshall—Lerner type conditions (regarding prOpor— tions of total wealth held in the form of foreign currency in each country) that are sufficient to guarantee that a devaluation would be successful. Another article by Dornbusch (1976) provides the basis for the contribution of Calvo and Rodriguez. Their paper concerns itself with exchange rate dynamics in a world of CS and rational expectations. Other seminal papers on CS focus on specific issues, and emphasize the transactions demand for foreign and domestic balances. Chen (1973) employs a Keynesian, short run, underemployment model of a two country world in order to examine the efficacy of monetary and fiscal policies when CS exists. Chen hypothesized that foreign and domestic balances are used as inputs in a Cobb-Douglas type techno- logy in order to produce money services. Miles in several papers challenges Chen's specification that the elasticity of substitution in demand between domestic and foreign monies is unity and attempts to measure empirically this elasticity using a model suggested in the domestic near money literature.8 Miles finds that elasticities of substitution within several countries between foreign and domestic balances are significantly greater than unity 10 during periods of flexible exchange rates, and are not significantly different from zero during periods of fixed rates.9 King, Putnam and Wilford (1978) attempt to develop a theoretical model of the degree of currency substitution. They consider both transactions and speculative demands for foreign balances. Their model suggests that currencies become closer substitutes as the sets of goods and financial assets that these currencies jointly command grows. Hence "it is the integration of world markets for goods and financial assets that allows different currencies to perform similar monetary services and thus provide the institutional framework within which currency substitution is possible."10 The Theory of the Substitution Process One of the problems in organizing this literature around a central theme is that there is no agreement as to how to specify the actual substitution process. One possible solution is provided by Per Meinich in his comment on a paper by CooPer COOper asks the following question: "But if the public can hold foreign money, what role should that play in the money stock equation?" I would answer that question by introducing separate supply, demand, and market clearing equations for foreign money in the model. Several of the papers follow this approach to some extent. Most ignore the supply side by assuming long run conditions (i.e. infinitely elastic supplies of foreign exchange). Which is to say that in the long run the process of CS 11 itself will have no effect on the exchange rate. Others posit a short run perfectly inelastic supplies of foreign monies within a country so that intra-country exchange rates between the various monies must respond in order to guarantee that citizens are content to hold existing stocks of assets. The earliest paper on CS offers a simple model of the substitution process. In his view that foreign monies should be treated as financial assets, Boyer describes the meaning of substitutability as follows: As financial assets, the relevant variable that...is crucial to determining the amount of each held is the rate of return on that asset as compared with other substitutable assets. For assets without pecuniary yield, their relative rate of return is just the rate of appreciation of one in terms of the other. Therefore, according to Boyer, expectations of an apprecia- tion in value of any one currency will lead to accumulation of assets denominated in that currency and short sales of asset denominated in the other. The degree of substitution between any two currencies depends upon the degree to which currencies are viewed as substitutes (which depends upon the nature of expectations of changes in the exchange rate.)13 Laffer's model is virtually identical to that of Boyer's. He writes one money i is defined as a substitute for another money j if an (percentage) increase in the supply of j, AMSj > 0, leads to an (percen- tage) increase in the price of goods in terms of money i, AP - > 0. Money i is a perfect substitute forgmdney j if MSj > 0=> AP . = O. 911 12 And finally, money i is a perfect substitute for money 3 1f AMSj > 0 = APgli (APgli > 0). The mechanism at work is as follows. Under perfectly fixed 14 rates, monies are perfect substitutes on the supply side. This is because an increase in the money supply in country A leads to an excess supply of money in A and therefore a deficit in the balance of trade. This leads to monetary expansion in B (through a trade surplus) and hence an increase in the price level of goods denominated in B's currency. Similar results obtain if monies are perfect substitutes in demand under flexible exchange rates and can be traded internationally. That is, decreases in the supply of money in A lead to increased demands for holdings in A for B's currency. The only way for equilibrium in B's money market to remain (as demand in B rises), would be for prices in B to fall by an equal percentage (since nominal supply of B's money is unchanged.) Laffer does not specify the characteristics of the various monies which would make them close substitutes in demand. Presumably, however, they must be related to the ability in one country for actors to "carry on their business using foreign money balances."15 This implies that "non-substitutability of a country's money for another's occurs when each country's excess demand function (for money balances) is strictly independent of the other's."16 The most complete model of the processes described by Boyer and Laffer can be found in the paper by Girton l3 and Reper. They introduce a portfolio balance model which V/ considers a three asset world: two traded currencies and a fixed capital stock. They use this model to consider the currency substitution process in one country. The authors define the following variables: L. = real demand for the jth currency j = d 3 (domestic) and f (foreign) E nominal demand for currency j deflated by the price level of a common set of goods. Fk = real demand for capital rj = real interest rate for jth asset j = d,f ij = nominal interest rate for jth asset j = d,f P. = price level of a common set of goods in J currency j j = d,f pj = a log Pj/dt j = d,f 6 = Pd - Pf = differential inflation rate W = real wealth pj* = anticipated rate of inflation rj* = ij - pj* = anticipated real interest rate 6* = differential in the expected inflation rate The real demand for each of the assets in the national portfolio is hypothesized to be a function of the national stock of real wealth and the expected real rates of return of all of the assets.17 Setting available asset supplies equal to demands then summation across equations determines the assets markets equilibrium condition (i.e. the wealth constraint for the economy.) This condition is described below as (after several simplifying assumptions)18 14 (2.1) Ld(6*, 9*, W) + Lf (5*, 8;, W) + Fk(§3, 8;, W) = W The term 6* represents the anticipated differential in the price of a numeraire good (or common set of goods which either currency commands) denominated in the two currencies. This can be viewed as the expected rate of change in an internal exchange rate: E = 5% (or a change in an exchange rate derived from purchasing power parity of traded goods prices). Thus, 6* is a measure of expected relative holding costs (in terms of changes in relative purchasing power) of the two currencies. P3 represents the "own" opportunity cost of holding currency j. Partial derivatives of (2.1) with respect to its various arguments take on the following interpretation. 3Lj/36* (j = d,f) is the substitution effect between the two currencies. That is, the substitution effect measures the change in demand for real balances of one currency when the relative holding cost of that asset is anticipated to rise vis a vis the other type of money. The hypothesized signs of these terms, hence, are 3Ld/35* < 0 and 3Lf/36*>0. Further, 3Lj/3p*d (<0) is the "own" effect on demand for real balances denominated in currency j (j = d,f) of changes in anticipated inflation rates. Total differentiation of (2.1) yields the following constraints on the partial derivatives: 3 (2.1a) Ld/36* + aLf/35* = o 15 That is, the sum of the substitution effects between the two currencies (holding real wealth constant) must be zero. From this we see that decreases in the real demand for one currency due to changes in the anticipated differential rates of inflation will be exactly offset by increases in demand for the other currency. The second and third constraints implied by the model are: 3 (2.1b) Ld/aéa + aFk/aég = o 3 (2.1c) Lf/aég + aFk/afig = 0 These conditions refer to the substitution between real balances and capital due to an "own" change in the expected rate of inflation. Finally, the fourth constraint is 3 (2.1a) Ld/aw + 3Lf/aw + 3Fk/aw = 1 This implies that changes in real asset demands brought on by a change in wealth must exhaust that change in wealth. To reiterate, the variables that cause portfolio holdings of currencies to change are changes in the expected inflation rates and changes in the level of real wealth. How are these expectations formed? Girton and ROper consider two cases. First, they assume perfect foresight. This allows them to replace the arguments in (2.1) which contain expectations terms with their actual values. When this is done, a dynamic equation 16 describing the movement of prices can be posited. These price movements will be affected strongly by the degree of currency substitutability.19 Alternatively, expectations can be generated according to an adaptive expectations process. This assumption leads the authors to formulate a system of differential equations to explain price changes. Again the degree of substitutability between currencies influences the prOperties of the system.20 All of this discussion leads us to point out that while the authors consider the substitution process, via equations (2.1) and (2.1a-2.ld) they ignore the important question of what determines the degree of substitutability between currencies. This criticism applies to the next paper as well. Calvo and Rodriguez (1977) design a two good, two asset model of a "small" Open economy under flexible exchange rates. Their two assets are again domestic and foreign monies (hence CS). Further, actors in this economy are assumed to hold rational eXpectations about future events. The two goods in this model are both produced locally and are composites labeled traded (t) and home (h) goods. The price of the traded good is given in terms of foreign exchange (which is exogenous and here it is assumed to be unity), so that the "small" country price is equal to the exchange rate. This allows the authors to define a "real" exchange rate, E = e/ph(which is nOthing more than the domestic relative price ratio Pt/ph). 17 According to the authors, CS occurs because of changes in expected asset returns. That is, the ratio of domestic to foreign currency held by the country's residents is a function of the difference in the rates of return of * for domestic and the two monies. These are ~65 and é* - pd foreign balances respectively. Where fir = expected change in domestic price level e'k expected change in the official exchange rate The differential expected return then is é*, the anticipated change in the official exchange rate.21 Lapan and Enders are not concerned with substitution processes, since they provide no explicit CS relation. The authors begin their analysis with two CS type variables, m and mf, where m equals the proportion of total wealth held by domestic residents in the form of foreign assets (exchange) and mf is an analogous variable for foreigners. These ratios are presented as fixed parameters. A clue as to how the values of these variables are determined is given in a footnote within the paper. Specifically, the ratios are treated as constants because residents of the two countries are "assumed to have static expectations" about changes in the exchange rate.22 Hence, we have the implicit assumption that CS takes place due to changes in expected rates of return (which has been modeled elsewhere as expected changes in the exchange rate). Since the authors are concerned with the effects of foreign currency 18 denominated assets in domestic portfolios under a fixed exchange rate regime, the assumption of static expectations is not too disconcerting. However, it is clearly incorrect to assume static expectations and fixed values of m and mf under a flexible rate regime (as the authors do in their appendix).23 Further, if trade has been unbalanced for some time under fixed rates, then residents of both countries may expect a reallignment of currencies. This of course leads to CS (and further pressure on the currency of the deficit country). Thus, the lack of endogenously determined portfolio of assets weakens this paper. The papers we have discussed so far deal with at most two countries and two currencies. Two papers extend the CS model to the more general N country N currency case. In an unpublished paper, Evans and Laffer explore the role of CS in the cosmOpolitan demand for the several national currencies of the world. The authors specify a set of liquidity demand equations for the world's n + 1 countries (under flexible exchange rates), viz: (2.2) mot—pot = ao+ Boyot-Grlt-ert- ~- 0 where o elasticity Of substitution H II the intensity Of integration Of global goods and capital markets While the level Of I is assumed tO be given at any point in time, it will change when various market and political variables change. Specifically, the authors describe this process as follows: 24 (2.3b) I = h(T, C, ¢, W) h1 < 0, h2 < 0, h3 < 0, h4 > 0 where T = barriers to trade C = capital controls ¢ = transportation costs w = information availability The authors conclude that "the ultimate institutional rigidity making for nonsubstitutability (is) the accept— ability On the part Of sellers of goods and assets Of only the domestic currency in their marketing Operations."32 While it is the degree of integration of world markets which determines the degree Of substitutability between monies, the actual share Of foreign balances in desired money services depends upon the existing exchange rate regime. That is, if exchange rates are expected tO change during holding periods real capital gains or losses will also be anticipated. Thus we find the authors defining the following relation: (2.3c) ¢ = j (E°,V/I) j1 < 0, j2 < 0 where E° = the expected exchange rate, in units Of domestic currency per unit Of foreign currency, relative to the current spot rate V/I = the uncertainty associated with exchange rate expectations conditional on the institutional structure which determines the degree Of currency substitutability.. 25 Hence we find as authors have previous suggested that expectations Of domestic currency depreciation lead to increased holdings of foreign monies. Further increased uncertainty about the future value of domestic currency given the structure Of world markets also leads to increased holdings Of the less risky foreign monies. While King, Putnam, and Wilford model the degree Of CS as being determined largely by institutional factors, Chen and later Miles adOpt a more traditional approach to the role Of foreign monies in money services. Specifically, they consider the different monies to be inputs in a production function for money services and then use standard production theory to establish hypotheses about the CS process. We consider the Chen paper first. Chen builds a two country, two currency, Keynesian under-employment model with perfectly elastic supplies Of output at constant prices. He assumes both currencies are allowed tO flow freely across borders in ordertx>insure that the internal exchange rate conforms to the external(official) exchange rate. He then specifies a Cobb—Douglas production function for the desired level Of money services with domes- tic and foreign monies as inputs. The desired levelcnfmoney servides is assumed to be a function of total output and the interest rate. Once this desired level is determined, the ratio Of domestic to foreign balances is determined through the minimization Of Opportunity costs in holding these balances subject tO achieving the desired level Of services. Mathematically this becomes: 26 F . Y a l-a M _ = (2.4) min chd + cf f st. /v Md (er) _ . . .th where cj — Opportunity cost Of holding the j currency j = d,f _ . .th Mj - nominal stock Of j currency Y = domestic output V = velocity e = currency exchange rate = price Of foreign currency in terms Of the domestic currency The first order conditions for the minimization Of the Lagrangian function Of (2.4) will yield the following results: Ma (2.4a) EM; = ‘I:E E; where a the fraction of total holding costs of all money balances accounted for by the holding Of domestic currency. Chen assumes that 1/2 < a i 1 so that money services in the home country are always domestic currency intensive.33 Chen defines the ratio Of Opportunity holding costs as follows: cf r - (it + é*) 34 (2.4b) E— : r _ i d d where r = real interest rate ij = nominal interest rate on jth balance \V//E* = expected rate Of change in e He then assumes that money balances yield no nominal returns (i.e. id = if = 0) and that é* = 0 (the long run solution). 27 Therefore cf/cd = 1 in equilibrium (since real rates Of interest are equal world wide). This specification displays homotheticity in money services and interest rates since neither affect the Opportunity costs Of holding balances Of either currency. If Md is increased while Mf is held constant, then there will be an equivalent percentage increase (depreciation) in e. TO the extent that changes in e lead to changes in e*, CS occurs. The underlying Cobb Douglas production function insures smooth substitution since the elasticity Of substitution is constrained by this technology tO be unity.35 Note, CS will continue tO occur until equilibrium has been restored (cf/cd = l). . \J/éhen doe not focus in on the link between e and é*. V// In fact, there is no equation to explain how expectations are determined. Miles in several papers is critical Of Chen's paper, but not for this reason. Rather, he considers the value Of the elasticity Of substitution in demand to be an empirical question rather than a given, predetermined parameter Of the model. Miles considers the problem Of maximizing the output of money services faced with an existing asset constraint. He defines a CBS production function Of money services (Of which Chen's Cobb-Douglas formulation is just a special case) with domestic and foreign balances considered as inputs in the production process. This production function is given below as (2.5) 28 _ { ‘P 'P}u1/p (2.5) MS — Bde + Bf(er) where B and Bf weights reflecting the efficiency Of domestic and foreign balances in the production Of money services. These parameters are assumed to be fixed and exogenous tO the model. d p = substitution parameter between domestic and foreign balances The asset constraint below (2.5a) represents the size Of a one period loan necessary to produce the desired level Of money services. Specifically, (2.5a) M" = Md(l + id) + e M (1 + if). f This implies that in the notation Of Chen the Opportunity costs Of holding the various monies can be defined as: II g.) + l-' (2.5b) cd Substitution Of (2.5b) into (2.5a) and then maximization Of (2.5) with respect tO (2.5a) yields the following first order conditions for constrained maximization: 1 1 (2.50) 34—3—— = (22,1:5 <33) 1+p e f f Cd We see from differentiation Of the log-linear version Of (2.5c), that in the case Of a CBS production function, the .elasticity Of substitution in demand 29 r M ‘ Bln (_Q_) eM f . . l . c 13 given by the term 1:; . Hence, only in the Mn (C—f) L d J special case that p = 0 will the production Of money services exhibit a Cobb-Douglas technology. Hence, Miles presents a direct vehicle for measuring the degree Of substitutability in demand between any two monies. Macroeconomic Implications Of Currency Substitution Much Of the CS literature is devoted tO the implica- tions Of the presence of CS on various macroeconomic variables or policies. Several papers are concerned with the determination Of equilibrium exchange rates in a world Of flexible rates. Other papers consider the role Of CS in the efficacy Of various macroeconomic policies. The general policy conclusion Of most Of this literature is that as long as monies are going tO be substitutes on the demandside, then the Optimal international monetary system ought to be one Of perfectly fixed exchange rates. We consider each Of these issues in the section below. An argument that is Often presented for the Optimality Of flexible exchange rates as an international monetary system centers around the idea that flexible exchange rates insulate economies from the economic policies Of the rest Of the world. Expansionary policies abroad lead to changes in assets prices (including a depreciation Of foreign exchange rates) until existing asset stocks are again 30 willingly held. Adjustment is very rapid and therefore nO net flows Of Official foreign reserves appear. This result is in sharp contrast tO the fixed exchange rate regime whereby Official flows Of reserves occur because economies find it impossible to insulate themselves from the policies Of other countries. The literature on CS suggests that where private demand for foreign money exists; there are no capital controls tO prevent international flows Of private monies; and these monies are perfect substitutes in demand, then "there is nO economic difference between fixed or floating exchange rates'.’.36 This is because under perfectly fixed rates monies are perfect substitutes on the supply side. If we alter our assumptions so that currencies become perfect substitutes On the demand side, then we witness the same economic phenomena. For instance, consider two countries, A and B, whose monetary authorities attempt independent monetary policies. Suppose these authorities increase the money supply in A by X% and authorities in B reduce the money supply by an equal X%, then initially there will be an excess supply of money in A and an equal excess demand for money in B. With flexible exchange rates, prices would rise by X% in A, fall by X% in B and A's currency would depreciate by 2X%. (We assume throughout a simple quantity theory Of money and purchasing power parity). If currencies are perfect substitutes in demand however, excess demand for money by residents Of B can be Offset by increased holdings Of balances in A. In fact, in this 31 . 1,2 v; example, the excess supply Of funds created by expansionary policies in A would be completely eliminated by increased demand from country B. In this case, there would be no effect on prices in either country (because a fall in the money supply Of B is Offset by a fall in demand) or on the exchange rate. Therefore, "the failure of the exchange rate tO change because Of the close substitutability Of money negates the policy effects Of the monetary authorities control over the money supply."37 The notion that monetary authorities have no control (in the case Of perfect substitutability)over the supply Of money within their country leads to the conclusion that the exchange rate is no longer determinate. This result is reached in several papers (Laffer, Evans and Laffer, Boyer, Miles, Girton and ROper). Kareken and Wallace agree with this and point out that the only way for monetary authorities to preserve some control over the impact of their policies is to impose portfolio controls or to manage exchange rates.38 Obviously, currencies, in general, are not perfect substitutes in demand. Hence, the exchange rate is then determinate. But, exchange rates may deviate from equili- brium values for long periods Of time because forces which would lead to a return to equilibrium values (i.e. differ- ential interest rates) are weaker under any degree Of CS.39 The dynamic path Of exchange rates under CS is considered in the paper by Girton and Roper. Girton and A VJ ‘ 32 ROper conclude that the exchange rate could be unstable with high degrees Of substitutability. Adjustment paths tO equilibrium may or may not be cyclical depending upon how inflationary expectations are formed.40 As Brillembourg and Schadler point out, in a world V” Of more than two currencies, these monies might be either substitutes or complements in demand. Hence, "this inter- dependence among currencies can produce quite interesting patterns of exchange rate movements. For example, policy changes in one country may induce an appreciation Of another country‘s currency, which in turn may induce a third _ currency tO appreciate With it and fourth tO depreciate."4l Currency substitutition under fixed exchange rates leads to equally perverse results. Miles concludes that because Of CS, devaluation will not improve a country's balance Of payments position by as much as traditional theory indicates. This is because devaluations may serve tO increase the perceived risk Of continued holdings Of the now devalued currency. Both domestic and foreign holders~ Of this currency will switch to balances in other countries. thus "exacerbating the excess supply Of money rather than relieving it."42 Lapan and Enders focus on the wealth aspects Of devaluation. They tOO find that when foreign balances are held, the efficacy Of devaluation may be undermined. They point out that devaluation works through changes in real wealth. Devaluation lowers real wealth because domestic 33 balances command fewer goods in world markets. Hence, consumption falls in the devaluing country leading to an improvement in the balance Of payments. When foreign currency balances are included in asset portfolios, the effect of a devaluation on levels Of real wealth in the devaluing country are now unclear (because holders Of foreign balances achieve capital gains due to the devalua- tion). The authors arrive at a condition which must Obtain in order to guarantee the efficacy Of devaluation.43 Because Chen begins with a different assumption about the shape Of the aggregate supply curve, (specifically, he assumes a short run less than vertical aggregate supply curve so that output will be affected by changes in aggregate demand), he finds that CS implies an hybrid currency system between a world Of fixed rates and one Of perfectly flexible rates (but currency immobility). In particular, he discovers a "Rybczynski" effect whereby increased supplies Of a currency in one country will lead tO increased holdings Of both currencies in that country (in a two country world). Hence, "we Obtain the surprising result that an increasing stock Of the first currency with the second currency held constant, may lower the second country‘s income."44 As we have seen, the implications Of CS are that under flexible exchange regimes, a country‘s exchange rate may be indeterminate or unstable, her monetary authorities may lose control over domestic money holdings and hence 34 monetary policy is ineffective. Under fixed rates, devalua- tion may be ineffective. The policy conclusion is clear and Oft repeated: so long as (and to the degree that) the private market treats monies as substitutes, the Optimal exchange rate regime is one Of fixed rates. As Karaken and Wallace point out, for a feasible international monetary system, governments must make a choice. They could choose not to have capital controls...But that regime is politically feasible only when budget policies are coordinated...We believe there is a stronger case for coordination than we have made here. Indeed, there is a persuasive case for continuing budget balance in all countries and for what is then feasible--cooperatively maintained (fixed and not adjustable) exchange rates.45 We continue to stress that the results Of CS depend upon the degree to which currencies are viewed as substitutes in demand. We pursue empirical measures Of this degree in the section below. Before leaving this section, we point out that we are interested in this thesis in pursuing the modeling Of the CS process and measurement Of the degree Of substitutability. The macroeconomic implications Of CS we leave unchallenged. Empirical Tests and Measurement Of CS Several models have been presented in the literature to test for the presence Of currency substitution and the degree Of substitutability in demand. Evans and Laffer develop a model Of exchange rate changes based upon purchasing power parity and rational 35 expectations. This equation is presented below as: (2.6) Aeit = ai - bi Amit + bO Arnot + ci Ayit - co Ayot + vit where Ae. = % change in the exchange rate between it . . country 1 (1 = l,...,n) and country 0 Amit = % change in the nominal money supply Of country i (i = l,...,n) Amot = % change in the nominal money supply Of country 0 “ Ayit = % change in the real output Of country i Ayot = % change in the real output Of country 0 Avit = stochastic error term The hypothesis to be tested is that if currencies are not substitutes in demand, then changes in any Of the right hand side variables will lead to a proportionate change in the exchange rate. In other words, if we assume zero substitut- ability, then the probability limits Of bi’ b , c , c all O i O equal unity. On the other hand, the presence Of perfect CS implies that exchange rates are indeterminate but tend not to change regardless of various policies. Hence, under the I alternate assumption Of CS, the probability limits Of bi' b 46 0 ci, and co are all zero in value. Evans and Laffer estimate equation (2.6) using monthly data for Canada, France, Germany, Italy, Japan and the U.K. The Observation period is from January 1968 to December 1975. The United States Was taken to be the numeraire country. SO that the equations estimated for the 36 six countries represents explanations Of changes in U.S. bilateral exchange rates over the period 1968-1975. The authors' results which were Obtained through ordinary least squares regressions on (2.6) are presented in Table 1 below. They conclude that because no coefficient is larger than .383 in any Of the six equations, during the period of analysis substantial CS must have occurred. Measures of the degree Of substitutability have been attempted in two different models. Brillembourg and Schadler are concerned with determining whether currencies are net substitutes or complements in demand. These authors derive a system Of exchange rate equations where the right hand side variables are rates Of return on own and (n-1) substitute-monetary assets, viz. e = a + o.r. + v k O 1 l 1 ll M5 (2.7) i k where e = logarithm Of the exchange rate between country k (k = 2,...,n) and the U.S. ri = £1 + BiYi + YT = return on the ith currency which is assumed to be a linear function Of the forward premium (fi) Of that currency vis a vis the U.S. dollar, that country's real output (Yi) and a time trend (T). vk = error term A comment should be made here regarding the term which reflects the expected rate Of return upon holding balances in currency k, rk. The authors assume that foreign balances yield two types Of returns. The first is a pecuniary return which is proxied by the forward premium (fk). The TABLE 1 37 Summary of Evans — Laffer Results Country i 1 b0 1 c0 R F D.W. Canada .001 -.002 .117 -.000 .011 .01 .1 1.61 (.89) (-.04) (.59) (-.01) (.27) France .001 .132 .149 .020 .082 .04 1.0 1.78 (.46) (1.10) (1.22) (1.38) (.82) Germany .005 .149 .059 .076 .032 .04 1.0 1.56 (1.75) (1.27) (.56) (1.55) (.25) Italy -.001 .120 .190 -.008 -.074 .09 2.2 1.97 (-.21) (1.73) (2.67) (-.87) (—1.l4) Japan .001 -.027 -.001 -.002 .158 .06 1.4 1.86 (.58) (-.42) (-.02) (-.08) (2.23) U.K. -.001 .383 .107 .007 .135 .11 1.3 1.56 (-.26) (2.12) (.96) (.15) (1.19) "t" statistic for the parentheses. test that the coefficient equals zero in 38 second return represents a non pecuniary return which is tied to the volume Of transactions in the ith currency. This return is proxied by a function Of real income and a time trend.47 Using monthly data from eight countries over the period March 1973 through June 1978, the authors estimate the system Of equations in (2.7) with a full information maximum likelihood procedure. Their results are presented below as Table 2. The coefficients (oi) represent semi elasticities on the rates of return on currencies. Semi elasticities on own rates Of return (i.e. the first eight diagonal elements Of Table 2) are assumed tO be positive. In seven out Of eight cases the authors found positive (and Often significant) "own" semi-elasticities. The only non- negative "own" semi elasticity was for the Japanese Yen. This value was insignificantly different from zero. The authors also had trouble measuring with precision the semi- elasticities Of substitute currencies-~the Off or cross- diagonal elements. Only about one fifth Of the estimates Of the Off-diagonal elements were significantly different from zero at the 95% confidence level, but the authors take comfort in the fact that many of the t statistics Of these "insignificant" coefficients were greater than unity. The estimated parameters suggest the following interrelation— ships The continental European currencies exhibit strong complementarity, with half the cross semielasticities being significantly different from zero. Both the U.S. dollar and the Canadian 39 mwumu owcmnuxo m>wuumommn mum mumsuo Ham .HO>OH .mowumfiumuw u ucomouowu mononuamumo ca mumnaba .N .umaaov m: can mfi> m mH> momma m: msu mo acufiumwoa ecu we manmfium> ucowcooov .H In)- ...... Amo.Ho Aoo.Ho fi~m.o Aao.ao Amm.o ANH.~V Aoo.v A-.V Am~.~V uuuuuuuuuu moo. ooq.H moo. sum. mmo.- omm.) who. Hmo. Hom.u umHHoo mo AoH.mV Aom.mo Aoe.v Amm.v Aoo.v Aeo.o AoH.Ho Aoo.ao Awm.Ho AmH.HV em. Hmo.~ oHo.H oNo.m- ooo. Nao.) Noo. moo. moo. ooH. mao. moo. so» mmmamamo n~m.~o Ao~.mo Anu.ao Aao.ao Ase.o Awm.Ho Ano.o Aoo.o Amm.Ho Amo.v Aoa.v mno.o Noo.u e~o.ou emo.~ ooo. Noa. moo. who. NmN. moH.) 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H..H o~.e t _mo .o <\z :o Aor _ _o .n <2 .w <2 .4 «N n .e so 2 ___oh .n _ w m: A<2o A<2o __- Snuo>om Aim + :o + 2o ox. <2 . <2 . . . . - . o o .n m o mm H q om m -o n m m: oaoom A . . . to + :o + 2o ohzox E <2 <2 . t H muo>em _ .m .m .o mm N .q we n so .n H w m: ~ma wop.mwa m mo ozlo:c_o .~ N~.o ._ ox. o~.m ox.~ - ___o~ _ + a w 2ooo . a + I m 22o Aom.mo o.olo >_ma «wmzwzw m m: ob..ux_.o . . . . s . Illldl .~ sq a ._ oi as 2 ea N >_oo yo. + _ m 22o os_wut Ano_om_omoz ov Ameaomaocom av whootezc c. c_.cu omco eqaewum> ooeboazcrcxo .2.: m c 3c =O_»ee zo_c:ouooeo ocoocoeoc n a;= 0. For other less liquid assets (e.g., consumer durables, the capital stock, long 3X. term government debt, etc.) —3% y: 0. These last two assumptions imply that if the level Of transactions demand rises, the private sector will adjust their portfolios as to hold a higher proportion Of its nominal wealth in the form Of liquid (money and near-money) assets. We assume that whenever substitution between asset 62 types occurs, the following wealth constraint must hold: N (3.2) )xi = w i=1 Substitution Of the N demand equations into the wealth constraint and partial differentiation with respect to the various functional arguments yields, N 3X. (3.3a) X 8F£ = 0 for any j. i=1 j N axi (3.3b) .2 ~75? = o 1=l N Bxi (3.3c) 2 SW = 1 . , 1—1 Relations (3.3a) and (3.3b) imply that trading assets at one point in time (due to changes in rates Of return or transactions demand) cannot change the value Of wealth held by the nonbank public. Equation (3.3c) means that nO new assets can be created in this economy, hence, any increase in private nonbank wealth must be distributed among the N assets. One measure Of the degree Of substitution is the cross price elasticity Of demand. In our model, the degree Of substitutability between assets Xi and xj may be described by the cross rate Of return elasticity, viz: 8X, r. 3 n = 1 . _l . XOXO .320 X- 1 j j 1 63 In a world of many assets and many rates Of return, the analysis Of substitution in demand within a particular subset Of assets may be unnecessarily complicated if every asset demand equation must be specified and studied. It may be possible to separate the various assets into subsets based upon certain common characteristics, e.g., liquidity, return, riskiness, maturity, etc. If we assume that the assets in these subgroups are held in efficient combinations to their ability to produce desired levels Of services, then we can consider the degree Of substitutability between the assets within these subsets.4 Separation Rulg We begin by assuming that wealth holdings provide both pecuniary and nonpecuniary services to any economy. That is, we should consider both pecuniary attributes such as liquidity, return, riskiness as well as nonopecuniary attributes like status and security that wealth delivers when defining a wealth function. Consider the following production function for wealth services: (3.4) ws = W(Xl, x2,....,xN) If we can partition the elements Of W into subsets, then it is possible tO consider a two stage procedure whereby wealth services are maximized. That is, in the first stage relative input shares are Optimized within each subset and then, in the second stage Optimal holdings are found by 64 holding input intensities fixed within subsets and Optimizing across subsets. The conditions necessary for such a two step process are well known. They involve the notion Of separability of production (or utility) functions.5 These conditions will be discussed below.6 We begin by considering the set of all asset inputs X = {X1,X2,....,XN} . Then, we partition X into M (M i N) ). mutually exclusive and exhaustive subsets (51’ 82"""SN The production function W(X) is said to be weakly separable with respect to a partition (Sl'SZ""”SM) if the marginal rate Of substitution, Wi(X)/Wj(X), (where Wi(X) = aW/axi) between two assets 1 and j from Sk does not depend upon the quantities Of assets held in subsets other than Sk' Mathematically. we have: 3 8W(X) ( 8X2 axi /§§%¥L) = O for all i,jeS J k and i t Sk . It is possible to show that the condition Of weak separability with respect to a partition (Sl'SZ"""SM) is necessary and sufficient for the function W(X) tO be Of the 2 form W(X1,X ,....,XM), where X1 is a function Of the elements Of Si i = l,....,M only. This last result is known as the fundamental theorem Of weak separability.8 From the above, it is Obvious that if the weak law Of separability holds for the partition we have chosen, then we can consider the slope Of an isoquant (hence, the degree Of substitutability) between two or more members Of an asset 65 subset without regard to the quantities Of assets held in other subsets Of the set Of assets.9 Asset Partition We choose to partition the set Of all assets, 1 and 82. Let Sl contain the stocks Of domestic and foreign currency denomin- X = (XI, x2,....,xn} into two subsets, S ated money balances held by the private sector Of the economy. We choose this partition rule because these balances share the common characteristic that they may be used directlywowithout conversion——to pay for all (or at least certain) transactions. We assume that assets in 82 must be converted into money balances before transactions can be affected.10 Thus, we see that if our petition rule is correct, then we can consider the question Of how efficient combina- tions Of the elements of S1 are chosen in order to provide money services. That is, we can specify a production function for money services with the elements Of S as 1 factor inputs and identify the degree Of substitutability between these elements without regard to the levels Of other assets held in the economy. The Model The model we prOpose differs from existing models in CS and near money (NM) literature in several important ways. Specifically, we consider the dual role that transactions 66 play in the demand for the several forms Of money held in the private sector. This dual role arises when one considers that changes in the level Of transactions within an economy will affect both the efficiency in the use Of any Of these balances and the relative holding costs Of these currencies. yéecond, we incorporate a different Objective function for the Optimization problem faced by the economy. We posit x? that residents—-rather than seeking to maximize the level of monetary services or the utility Of monetary servicesl} subject to a fixed level Of holding costs--act to minimize the dual Of the problem. That is, we assume that money holding costs are minimized subject tO an existing technology (production function) for the production Of the desired level Of services. This emphasis on cost minimization allows us tO explore in depth the nature Of costs inherent in the maintenance Of the various money balances. Again,\ \ our analysis Of the holding costs Of both domestic and 3 foreign monies goes beyond previous work in either the NM! / or CS literature. We begin by assuming a multi—countr world where each country has a monetary authority that issues its own money supply. Residents Of each country are assumed to holdlnono \/* negative amounts Of thenae!era1_gurrangl§§1__QQB§id§£iES ............. tion problem described above as follows: (3.5) min X. i 1 ll M10 1 Cielixi Soto. is. (elxllelixi) i=1'.Q-‘\Q(Q 67 v; where ci = ‘expected holding costs/(in percent) for currency 1 i = l,....,Q X. = nominal holdings Of currency 1 by residents Of country 1, denominated in units Of origin. i=l'OOOO’Q e1i = exchange rate = country l's currency price for 1 unit Of country i's currency 1 = 2,...,Q (e11 = 1) MS = desired level Of money services12 M = production function for money services. This minimization problem may be solved through the Lagrangian constrained minimization technique. Specifically, we form equation (3.6) and take partial derivatives Of that equation with respect tO the Q monetary assets (Xi) and the Lagrangian multiplier (A). A necessary (or first order) condition for constrained minimization is that each Of the partial derivatives equal zero.13 We have: sofi 3 Q . BL (3.6a) —————— = c - 1M = 0 BellX1 l 1 BB (3.6b) —————— = c - 1M = 0 aelzx2 2 2 BL (3.6q) ————— = c .. AM = 0 BeIQXQ Q Q (3 6r) BE = M(e X e X . e X ) - MS = 0 ' ° 31 ll 1’ 12 2"' " 10 Q (NOTE: Mo = J!!— i = l'ooat'Q) 1 aelixi 68 Division Of (3.6a) by any Of the other q—l first order condition equations, (3.6b) — (3.6q), yields the familiar result that for cost minimization, balances should be held relative tO levels Of domestic money (X1) such that the ratio Of their marginal products (in the production Of money services) equals the ratio Of their expected costs. The Cost Function There are several costs in holding money balances Of any sort. These can be divided into Opportunity costs and actual (or perceived) costs Of transactions and exposure tO risk. It is clear that the Opportunity cost in holding money is the interest foregone on the next best alternative asset. There is no consensus on how this should be modeled. Two interest rates that have been used in the domestic near money literature are the rates OHDShort term treasury bills and(%h long term (3v5 year) government bonds. The other costs mentioned above are less easy to model and hence have been largely ignored in both the domestic near money and the CS literature. Actual costs in maintaining domestic balances include checking account charges, minimum balance fees, accounting costs, etc. Foreign balance accounts are likely tO be even more costly. In particular, they may be subject to special taxes. Further, it is likely that these balances must be maintained in banks located in major financial centers or in 69 overseas accounts. Hence, the cost Of communicating long distances must be added to any standard service charges. Finally, there are likely tO be substantial costs to Obtaining the requisite information necessary for the efficient use Of these balances.l4 Added to this are the political risks which must be borne by anyone taking an Open position in foreign exchange. These risks could be as extreme as government confiscation Of foreign owned accounts, the freezing of foreign assets during times Of political turmoil, or "currency reforms” which could leave foreign balances essentially worthless. More likely, political risk might entail the imposition Of exchange controls after an Open position in the "controlled“ currency has been assumed. This could lead to the prevention .Of repatriation Of funds or the payment Of exorbitant black market exchange rates and thus, substantial capital losses. While these may seem extreme cases, 3px government imposed impediment to exchange convertibility can be construed as an element Of political risk and the degree Of likelihood that such actions would be taken gpgpp to be factored into the costs of holding foreign balances. The greatest cost faced by a money diversifier is the possibility that between the time foreign balances are Obtained and the time they are utilized for transactions a change in the exchange rate will have occurred. This will alter the purchasing power Of the desired level Of money balances and must entail a component Of the cost Of holding 70 foreign balances.15 Specifically, if foreign balances were to depreciate vis a vis domestic balances while they are held then the bearer faces increased total costs (in terms Of the number Of domestic goods his balances command) because Of the additional assets that must be liquidated in order tO maintain a constant level Of transactions. If, on the other hand, these foreign balances are expected to appreciate in the short run, the holder can reduce the size Of his domestic money holdings given any desired level Of transactions. We introduce these ideas into our model by defining equations for the holding costs Of each Of the several monies: (3.7a) cl = r1 + t1 = - * (3.7b) ci ri + ti + oi e1i where r. = monetary yield on the apprOpriate alternative 1 asset i = l,....,Q. \/ti = transactions costs (as a percentage Of total holdings) Of maintaining balances in currency 1 i = 1,....,Q. Q. = expected losses relative to total holdings due V// 3 tO political risk Of foreign balances j=2'OQOQ’QO é..* = expected rate Of change in the domestic 13 currency price Of country j‘s currency 3 = 2,....,Q. Therefore, the ratio Of holding costs Of the jth currency relative tO domestic balances is then defined as: . + . — ° .* c rj tJ + Qj e 13 (3.3) .4! = C1 ‘1 + t1 71 We assume that domestic balances are free from political risk and hence, omit that term in equation (3.7a). The inclusion of the expected rate Of change in the exchange rate reflects the fact that the realization Of capital gains on foreign balances clearly reduces the cost Of maintaining these 16 accounts. Technology for Money Services Production We assume that the primary reason for maintaining balances in several currencies is tO affect various domestic and foreign transactions. That is, from the point Of View Of certain actors within the economy it may be cheaper (in terms Of both money and time) tO maintain these several accounts than to convert domestic assets (money or other less liquid assets) into foreign money whenever foreign goods or services are acquired. ~Thus, just as producers find it expedient to maintain inventories Of their output, agents in our economy are assumed to maintain foreign balances. The decision as tO the level Of each currency type to be held depends upon relative holding costs Of that currency vis a vis domestic (or third currency) balances 223 the degree to which these balances are viewed as substitutes. That is, our Optimality condition implies tangency Of the isocost line with an isoquant representing the level Of desired money services. The shape Of this isoquant reflects the nature Of the technology available to the economy for 72 conversion Of domestic and foreign balances into the output Of money services. The nature Of this technology is largely an empirical question for each economy. It is possible to describe various scenarios, however, regarding possible technologies. If we assume that domestic and foreign trade are conducted in domestic and foreign currencies, solely, and none Of these monies is viewed as a substitute for another then the level Of holding Of each is a function solely Of the level Of transactions to be conducted with that currency. Relative holding costs would play no part in money holdings decisions. This is a world described by a fixed prOportions X X production function for MS (e.g., MS = min{~—, e —3-,...., X cl 12c2 e1Q 52 }). In a world with only two currencies, the above Q function is represented by right angle isoquants in the input space. Efficient combinations Of the two currencies would occur at the vertices Of these isoquants and remain there for all finite, nonnzero cost ratios. It is also possible to imagine a technology whereby changes in relative cost ratios lead to equal percentage changes in money balance ratios in the Opposite direction. This type of smooth substitution between inputs is defined by the Cobvaouglas function 0!. a a 2 x)Q _ 1 MS — xl (e12x2) ....(e1Q Q Q where 0 < “1 < l i = l,....,Q and Z a. = 1.17 For the . 1 1=l two input case, both the fixed proportions and the 73 Cobmeouglas production functions can be identified by their elasticity Of substitution18 (0) since they are special cases Of the more general constant elasticity Of substitution (CES) production function. In order to consider the degree Of substitutability between any two inputs Of a multioinput production function, we introduce the concept Of HickSvAllen (HA) direct partial elasticity Of substitution:19 X. 1 3 1n (T) o.. = 3 . 13 c. a 1n (—1) Ci We can think Of the meaning Of Cij in the following manner. Hold the level Of money services constant and fix all other inputs. This allows us to determine a curve in the Xi, Xj space. Along that curve, oij measures the elasticity Of Xi/Xj with respect to the cost ratio cj/ci. Replacing the functional notation Of M in the minimization problem described in (3.5) with a multi«input CES production function yields: . Q —— “'91 "32 (3.9) m1n 1:1 Cielixi s.t. MS = {Ble + 82(e12X2) _ -1/p Q +0 0 o o 8Q (elQXQ) } efficiency (distribution) parameter associated with the ith money balance i = l,....,Q where Bi _substitution parameter associated with the jth \/ 1 money balance 1 = l,....,Q '0 ll overall substitution parameter r <1 n 74 / ,a )2 Following Dhrymes and Kurz,29 we can calculate a value for o.., viz. 1] v"(3.10) 0.. = 1 13 -p. / 1 + 83'"ij 3 + . - . (l + pi) (03 pl) ”pi Bipi i in the case where pi = pj i = 2,....,Q, then (3.10) reduces to the standard CES form described by: (3 11) o = o = l > 0 if p = 9 V i = 2 Q , 0 ij 1 + p i i [0000’ o \d If, however, pi # pjfor some i then it is possible for certain (but not all) Oij to take on values anywhere on the real number line. The interpretation one should applytxathesej valuesiesthat inputs (Xi,Xj) are net substitutes(complements)i in production as Oij is > 0 (<0). .It'is possible to show, I but beyond the scope Of this study, that notoallminpntslinma- : productionmfipngtionwcan be netmcomplements (as defined 3 ;.. above).21 Transactions Demand The criterion we used in separating the set Of assets X into S1 and S2 was based on the assumption that various forms Of money are held in order to pay for the goods. While we dO not rule out1the speculative motive (desire for realization Of capital gains) for holding foreign balances we emphasize in this section the role that tranSo actions demand plays in determining the Optimal levels of 75 the various balances. If we assume that the demand for the several forms Of money depends upon the level Of‘transactions faced by the v“ holder, then it is likely the transactions elasticity Of demand for each Of these balances will differ. If this is so, then the money services function of equation (3.9) is incorrect. Specifically, the function, as written, presumes that the relative input shares((8i) remain fixed for ally levels Of money services. But, if transactions elasticities differ, then as transactions levels change, these parameters should also change reflecting substitution among respective currency holdings irrespective Of changes in relative holding costs. In other words, increased domestic transactions may 1 lead asset holders to §Wit9h REP Of foreign balances even if 1-1 ' ~0..v relative holding costs have remained unchanged. There are several reasons for thinking that this process occurs. \Eirst, as total transactions in an economy rise, the likelihood Of external trade increases. With increased trade comes the possibility Of economiesixicurrency conversion costs through the maintenance Of foreign balances. Also, it is likely that as trade expands, domestic banks will seek to recapture some Of the business lost as domestic balances are converted to foreign by Offering to service foreign denominated accounts. This process serves to lower transactions costs and may Offer the holder marginally larger rates of return in the form Of interest payments than those Of foreign banks.22 In addition, rising levels Of 76 transactions (and incomes) mean that more and more people are able to purchase the requisite information for the efficient use Of their liquid balances. Because Of all these factors, as well as the conclusion Of various international agreements which increase currency convertibility and lower trade barriers the sc0pe23 for CS is broadened.24 oi=E7737ii x . In addition to the likelihood that relative balances [ {I will change (i.e., CS will occur) with varying levels Of transactiOns, it is also poésible that factor augmenting; technological changes will occur in the banking and financial sectors Of the economy over time. These improvements, which are largely associated with advances in computerization of accounting functions and communications facilities, need not have affected the efficiency Of the several monies in the provision Of money services to the same degree or at the same time. That is, for example, innovations in the banking industry Of one country may increase the efficiency (there? fore lower the requisite level) Of balances in that currency. Since innovations may be country specific (responding to local legal restrictions or requirements) and are dependent upon the prevailing technology they need not be adOpted across countries or currencies simultaneously or affect holdings in the same fashion. To incorporate these ideas into our model we focus on the impagtothat the level Of transactions and the\degree_ Of teghpigal innovation have in determining thedistribution of monies as inputs in the money service function. __pm 77 Specifically, the parameters Bi(i = 1,....,Q) should not be fixed 25but should be functions Of both the level Of trans— actions and Of time, viz ._ —ei (3.12) Bi = BiTt exp (—sit) where 8i = constant term. i = l,....,Q ‘Ei.> 0 Tt = level Of transactions at time t 6. = transactions related efficiency parameter growth rates 1 = l,....,Q Bi > 0 s. = rate of factor augmented technological change v i = l,....,Q si > 0 The specification suggested by equation (3.12) is unique to our model and represents-eas we will demonstrate—ea signifi- cant refinement Of the CS literature. Substitution Of equation (3.12) into relation (3.9) allows us to write the final form of our theoretical model as: I“ 0 Q —— _— (3.13) m1n igl Cielixi s.t. MS = {Bth e -o 1 exp ('31t)x1 1+.... -l/p + EéTt Q exp (—th) [elQXQ] Q} In order to solve this problem we form equation (3.14), viz: 3 1 Q —. '61 "91 - ( .1415 = i C'elixi' M(Bth exp (-slt)Xl +....+ 1—1 1 -9 -p '1/0 __} — Q - Q — MS BQTt exp ( th) [eIQXQ] ) Following the Lagrangian constrained minimization technique, we partially differentiate 3.14 with respect to the local currency values of the Q currencies and the Lagrangian 78 multiplier (1). Setting these terms equal to zero yields the first order conditions for cost minimization. These conditions are written below, -1 -e — —1 85' _ —— _ 1 R P(3 .l4a) 532-1- - ANS BlT exp (—slt)Xl -- C]. _e' ' —-l _ l -p.-l _ 3.141) Fgé“x“ = AMS 8.T exp (—s.t)X. l ‘ Ci 11 i 1 1 1 313' . (30141:) 53‘.— = 0 l = 2’....'Q Division Of equations (3.14a) by (3.141), determines the thimal relative balances Of home.and ithwcurrepgyfiheldmby {gsidents_inpgounpfywl;_ Several terms cancel when this Operation is carried out. Simplification Of the resultant ratio and substitution Of equation (3.8) for relative costs yields: 1 8.-8 '— l X B I+p /.-f"‘*.l S."S (3.15) e 1x = (i). It'thwal expufL 1m 11.1 31, .x r '91 ~ . —— o-"o r. + t. + Q. r e .* l+pl —3-—l- . 1 1 1 11 (e X ) 1+p. r + t 11 i l 1 1 On the basis Of this equation we assert the following testable hypotheses: l. The Optimal relative holdings Of various currencies within the asset portfolio Of an economy depend upon the relative costs Of holding these currencies. An important component Of these relative costs is the expected rate Of change Of the exchange rate, * , e1i . 2. Even if relative costs remain constant, relative balances may change (hence CS occurs) due tO changes in the level Of transactions or to differences in the rate 79 Of technological change in home or foreign financial industries. Currency Substitution and Exchange Rate Regimes The previous chapter pointed out the implications Of the presence Of CS for the efficacy Of monetary policy under different exchange rate regimes. Briefly, under fixed rates, the presence Of foreign currency denominated assets in the set Of all assets means that perverse changes in domestic wealth holdings would occur with any change in the exchange rate. (Likewise, under a free float, CS implies that national monetary policies are not fully insulated from each other—was had been hypothesized. While CS has differ— ent macroeconomic effects under different exchange rate regimes, these alternative systems have differing impacts on the relative holding cost ratios Of the various monies. Specifically, under a rigidly fixgd exchange rate system éli* + 0. Hence, this term vanishes from equation (3.8) (so long as Central banks are 100% expected to be able to maintain parities). The ratio Of costs between the ith currency balance and domestic balances becomes: ci ri + ti + 0i (3.16) ——- = c1 rl + tl If, in addition, we assume that 01 = 0 and that t1 and ti are constant, thgn,changes in relativeholding costs would depend really on (relative , interest rates” in the 12199 _ Count tries. In other words, changes in the relative holdings Of .~._.-_‘p-—. . "on. 80 {these two currencies due to changes in relative holding costs (would, in this case, depend upon the ability and willingness (of domestic and foreign monetary authorities tO maintain 1 \ c iinterest rate differentials. : c 1 In the long run, under fixed exchange rates and x.perfect capital mobility, there is a tendency for interest rate convergence.27 If this occurs, and there are no changes in transactions costs or political risk, then the ratio Of holding costs for any ratio Of currencies becomes invariant. Hence, from equation (3.15), if relative balances are to change (i.e., if CS is tO occur) under fixed rates, in the long run, given these additional assumptions, it must be due to changes in the level Of transactions (either within or between countries) or because Of technological innovations in the use Of one or both Of the two monies. Under floating exchange rates, the situation is different. The exchange rate Varies and thus, éli* # 0. Further, if floating rates afford even some degree Of insulation from the monetary policies Of other countries, then divergent internal and external Objectives may be pursued in each countrywéincreasing the likelihood of long term deviations in interest rates. The result Of all Of this is that relative holding costs will vary as monetary policies are implemented in either country which alter rates Of return or exchange rates. As costs vary, relative balances will change even if transactions levels or rates Of technical change remain constant. 81 In order to pursue the question Of CS under floating rates, we must define HOWEExpectapiggwgfiwexchangemrateeiim _ghangeslare1£Q§m§§;WM0ne model of expectations is derived from the "interest rate parity condition" defined below f 28 (3 17) e11 e11 = r1 ’ r1 ° e . l + r. 11 1 where eii = forward exchange rate = domestic currency price Of one unit Of country i's currency to be delivered at a future date 1 = 2,....,Q. If domestic and foreign assets are perfect substitutes, supplies Of arbitrage funds are infinitely elastic, and there are no transactions costs, then equation (3.17) would hold exactly. If it did not, then under these conditions, the existence Of riskless profits would induce capital flows until (3.17) is Obtained. In the real world, this relation never holds. It serves onlyto approximate conditions in spot and forward currency markets. Given that (3.17) is a reasonable but imperfect r flection Of foreign exchange market conditions, we make \flie simplest assumption possible about the formation Of exchange rate expectations, viz = efi where e1i* is the expected spot rate for country i‘s currency one period hence i==2.....Q. (3.18) e \/ 9: 11 That is, equation (3.18) asserts that today's forward rate is expected to prevail as tomorrow's spot rate. Insertion Of (3.18) into (3.17) yields a relation that defines the expected rate Of change in the exchange rate. This relation 82 can be written in several ways: e .* 1 e . e . — . 11 11 rl r (3.19) é .* = = ' = ______£ 1 r - r. 11 e1i 11 l + ri l 1 Any Of the last three terms on the right hand side may be used in our model, the question Of which is most appropriate seems largely an empirical one.29 For the time being, we will focus on the latter two terms because we can analyze directly the CS implications of monetary policies which alter relative interest rates. Recall the cost ratio for the home country currency and the currency Of country 1 defined by equation (3.8). c. ri + ti + 0i - é * 11 (3.8) ~3— = C1 r1 + t1 For simplicity, we assume that 0i = 0 and that t1 and ti are constants that can be ignored. Replacing the expectations term with the definitions based on interest rate differen~ tials in relation (3.19) allows us to define two alternative cost ratios which are solely functions Of interest rates, viz r - r. 1 r —(-—--i) (3 20a) :1. = 1 l + ri = ri(l + ri) - (r1 - ri) = c1 r1 r1 Tl + r1) r.2 + 2ri — r1 (A1+2). r1 (1 + r2) and c r - (r1 — ri) 2r — r1 Cizob)-—i t i 83 Consider the impact Of contractionary monetary policy on the part Of the home country monetary authority through increases in domestic interest rates. Not only with relative Oppor- tunity costs Of domestic and foreign balances be altered, but expectations about changes in future exchange rates will change, tOO. Consider the following example, based on differentiation Of equation (3.20a). A ceteris paribus increase in the domestic interest rate will affect the relative cost ratio vis a vis country i‘s currency as follows: c i 3(51) wri (ri + 2) —ri r. 1 (3.21) a: = 2 =——2.-——1—(1+r) 0 i r1(1+r.)2 1 i (1+r.) 1 1 and Ci ri r1 3(C—) 3(?) 3(F) 3(e .*) (3 22a) ___1 = ___l_.+ ___l_ . .___l£__ ' Br. Br. Br. Br. 1 1 1 1 Again this result is intuitive. An increase in ii leads not only tO an increase in the relative Opportunity cost Of holding currency 1 vis a vis domestic balances but to the expectation Of an appreciation in the exchange rate. The conclusion, then, is straight-forward. Restrictive monetary policies relative to other economies in the home country will lower the relative costs Of maintaining foreign balances and may induce, ceteris paribus, larger shares for foreign balances in the money component Of wealth, and vice versa . FOOTNOTES CHAPTER THREE 1Substitutability between similar assets has been the subject Of many papers. Feige and Pearce (1977) provide an extensive survey on various studies that attempt to measure the substitutability in demand for various domestic (U.S.) liquid (near money) assets. A subset Of this litera— ture was begun when Chetty (1969) proposed a CES type utility function for money services and attempted to measure the elasticity Of substitution between money and various domestic near monies (e.g., time deposits, savings and loan shares, etc.). See Appendix 1 (end Of Chapter 2). None Of the near money (NM) literature cited in the Feige-Pearce survey considers the role that foreign balances may play in determining Optimal money balances. Miles (1978), employing the technique Of Chetty, is the first published author to attempt tO measure the elasticity Of substitution between domestic and foreign balances. We will comment extensively on his methodology and his results in the empirical chapter Of this thesis. 2Two goods are gross substitutes if an increase in the price of one gOOd leads tO increased consumption Of the second. Both income and substitution effects are taken into account in the relation between the two goods. This implies 8Xi 3Xi 5E7’ < 0 for all 1 # j and 5:7- > 0 for 1 = j. J J 3 ”1 Note, absolute values are used here, since 55- < 0 for all j # i. We define the following rule: j Xi and Xj are strong (weak) substitutes in demand as Inxile (<1). 4It is also possible to consider the degree Of substitutability between subsets Of assets. Since currensyi .substitution_is limited tgmghe_analysislofisubstitutabilitym betweensspegifigwhighly_lignid1assetsifle_leaxg_1hi§1999§ti°n for later research. 85 86 5Moroney and Wilbratte (1976) suggest the issue Of separability in the NM context. However, they do not provide a formal treatment Of the concept. 6The following section borrows freely from a discussion of separability Of production functions in E.R. Berndt and L.R. Christensen (1972). Their paper is an excellent summary Of various issues critical to production theory, e.g., separability, multifactor elasticities Of substitution, translog functions, etc. 7The proof Of this statement is found in Leontief, W.W., in "A Note on the Interrelation Of Subsets Of Indepen- dent Variables Of a Continuous Function with Continuous First Derivatives," Bulletin Of the American Mathematical Society, 53, 1947. 8A condition Of stron separability also can be defined. Berndt and Chr1stensen, Op. cit., write The production function F(X) is said tO be strongly separable with respect tO the partition ,N 2,....,Nr] if the marginal rate Of ghst1tution Fi(X)/Fj(X) between two inputs from different subsets NS and Nt, respectively, does not depend upon the quantities Of the inputs outside NS and Nt- (pg. 11) Strong separability with respect to a partition [N1,....,Nr ] is necessary and sufficient for the lproduction function F(X) to be Of the form F(X1+ X2 +....+Xr) where xr is a function Of the elements Of Nr only. (pg. 12) Weak separability is both a necessary and sufficient assumption for our analysis to continue. 9Actually, we can also exclude the level Of wealth from our analysis. That is, since we are interested in the degree Of substitutability between any two monetary assets, we begin by taking logarithms Of the asset demand equations and subtracting one from the other. In the process, the wealth term vanishes. From equation (3.1) we have: Xi = W - Xi(r1,r2,....,rn,T) i = l,....,n Therefore 1n Xi = 1n w + 1n {Xi(ri,r2,....,T)} and hence 87 1n Xi-1n Xj == 1n {Xi(r1,r2,....,T}-ln {Xj (r1,r2,....,T)} + 1n W - ln W. Obviously, the last two terms cancel. 10This is equivalent to assuming that the following partitioning rule is used: 0X. 3T 2 and specifying that all transactions must be conducted with domestic or foreign money balances. llNote, because monetary services may include nonpecuniary attributes such as convenience neither the utility nor the level Of monetary services is an observable variable. Hence, because we cannot label the particular isoquant that represents the desired level of money services we impose the condition that with respect to the output Of money services all isoquants are identical. This condition occurs if and only if the production function is homothetic with respect to the level Of output. 12We make no assumptions about how the desired levels of money services are determined. One reasonable hypothesis is that a portfolio balancing model (a la Tobin Markowitz) which incorporates risk and return, could be used to explain the relative levels of all assets in the wealth holdings Of the economy. SO long as the function M is homothetic with respect to the level of MS and the elements Of 51 are functionally separable from the remaining elements in the asset set X, then the two step Optimization procedure is valid. X. is 6 S1 iff > 0 otherwise, X. e S V. . 1 1 1 13 that the Bordered Hessian matrix be positive definite. 14There is a reason to believe that for some foreign currencies, the level of transactions costs decreases as transactions increase. This could occur as domestic banks became more familiar with this currency and began to Offer banking services for these balances. 15We assume that wealth holders in our economy are risk neutral. 16The term 911* is subtracted in the ith cost equation because 911* > (<) 0 as the home currency is expected to depreciate (appreciate) against currency 1. 17This implies constant returns to scale. The second order condition for cost minimization is .p, 88 18 X1 £n(-—) x2 0 = —— > 0 c2 - £n(E—) 1 Specifically, the production function is fixed proportions if and only if 0 = 0. It is Cobb-Douglas if and only if 0 = 1. Finally, X1 and X2 are said to be perfect substi- tutes as o+w. 19 There are, in fact, several alternative measures of the partial elasticity of substitution. For several ‘ descriptions of these see E.R. Berndt and L. Christensen, Op. cit., pp. 9-10, C.E. Ferguson, The Neoclassical Theory Of Production and Distribution, Cambridge Press, 1969, pp. 107-111 and R.M. Solow, (1967) PP. 41-46. Both V.K. Chetty, Op. cit. and Dhrymes and Kurz (1964), pp. 287-315, employ the Hicks-Allen direct partial elasticity measure. 20Dhrymes and Kurz, p. 290. 21See R.G.D. Allen, Mathematical Analysis for Economists, London: MacMillan & CO., 1953, pp. 503-509. 22This would certainly be the case for foreign owned U.S. dollar demand deposits in the United States prior to the invention of interest bearing draft accounts. 23We wish to differentiate the notion of the sc0pe of CS from the size Of the HA direct partial elasticity. The sc0pe for CS refers to the number Of times or ways in a particular period that CS can be efficiently utilized. The size of the elasticity of substitution is determined by the nature Of the money services technology faced by residents Of the economy. 24King, Putnam, and Wilford (1979) present a theoretical model Of currency substitution where the propor- tion Of monetary services provided by foreign money is an increasing function Of the integration Of world goods and capital markets. 25Moroney and Wilbratte, Op. cit., introduce income* terms as proxy for transactions in the efficiency parameters._ Again, their analysis is part Of the domestic near moneymi literatugs- 26Lieberman (1977) argues forcefully that both a transactions term and a time trend must be included in any theoretical or empirical specification Of the demand for money. His analysis, however, does not employ a CES specification as we suggest and he does not consider foreign 89 balances. Brillembourg and Schadler (1979, pp. 534-535) suggest that the nonpecuniary return for any currency can be described by a linear function of real income and a time trend. 27For a discussion of this point as a part of the "global monetarist" approach to the analysis of the balance of payments, see Kreinin and Officer (1978, p. 13). r - r. 28The term _l____$. 1 + ri and since 1 + r. is usually very close tO unity it is Often approximated as rl - ri. 29Note, however, that each term to the right of the second in equation (3.19) represents a slightly more restrictive assumption. is known as the interest agio CHAPTER FOUR Estimating Equations and Empirical Methodology Introduction In the last chapter, we develOped a theoretical model of currency substitution (CS). We posited that agents in any economy determine their desired holdings of various assets according to a two step process. First, they Optimize their holdings according to some Objective function within the various subsets of assets and then they Optimize between these asset subsets. Our analysis focuses on one particular asset subset, the set Of domestic and foreign monies. We suggest that agents hold foreign and domestic balances because they may directly facilitate transactions. Nonetheless, these balances are costly to hold. Costs would include returns available on alternative assets (Opportunity costs), transactions costs (both in maintaining balances and switching between them), and the various risks (political and exchange rate) faced by the holders. Therefore, it is our hypothesis that agents act to minimize the holding costs Of these balances subject to Obtaining a desired level Of "money services" from them. In the chapter below, we develop a set of asset 90 91 demand equations for the Q-l foreign balances from our theoretical model. We discuss how these equations can be estimated based on several alternative hypotheses about the speed Of adjustment and the nature Of substitutability between the various monies. We also demonstrate how estimates of the several parameters Of the model can be obtained from our empirical work. Production Function for Money Services Recall from the last chapter that we chose to describe the technology for the production Of aggregate money services from the holdings Of Q currencies as a generalized CES production function where the various monies serve as inputs in the production function. This is written below as, »/' -l/o { -01 -92 — Q (4.1) MS — (lel + 82(e12X2) + ....+ BQ(e10XQ) } where el. = domestic currency (1) price of one unit of 3 country j's currency j = 2,....,Q Xi = nominal holdings Of currency i by residents of Country 1 denominated in units of origin i=1’oooo’Q 81 = efficiency (or share) parameter associated with the ith money balance 1 = 1,....,Q pi = substitution parameter associated with the ith money balance i ; l,.-.1¢Qe p = overall substitution parameter We rule out non positive holdings Of any money balances by assuming the following inequality constraints. Bi,xi3;0 _. 'x‘ H‘». 4 - \ \ 92 (i = l,....,Q). Further, we assume that markets for foreign balances are nicely behaved so that exchange rates are always positive £313.:19 j = 2,....,Q). In addition to these assumptions we place certain restrictions on the substitution coefficients Of the model. Specifically, in order to guarantee that money services are produced within the economic region (i.e. to insure cost minimization) and to rule out inadmissible behavior (i.e. "negative returns to scale") we impppppthefollowing wgonstraints: pi > -1 and pi(and p) must be of the same r~w sign.1 The advantage of using the generalized CES functional form is that it places very few constraints on the under- lying technology. Rather, the degree Of substitutability between any two assets (Xi,Xj) is a byproduct Of the estimation process. In particular a measure Of the degree Of substitutability between Xi and Xj, the Hicks-Allen direct partial elasticity of substitution (dlj) can be calculated from the estimated parameters of the model. Specifically, we define Gij as follows: d ln [xi/Xi] (4.2) oi. = , . 3 aMS/ax. d 1” 3M5 axi This term can be calculated using the following formula,2 _ l (4.23) Oij - -pj B.p. (ele.) (1+p.) + (p.-p.)/1+—3——l - ———-2—_ 1 j 1 B.p. p. 1 1 (elixi) 1 3' I‘ 93 If we assume that there is a Sgpspangflgngfigguglwpagtial. \/__. " plasticityrbetweenmallpairemofias sets ,, , 5.199,. ”constant. returns pomscalethen equation iix21 canbe simplified greatly. In this case, pi = p. = p for all i,j=l,....,Q and (4.2ayf 3 reduces to (4.2b) for all pairs Of assets, (4.2b) Oij = 1 + p = 0 Another feature of the generalized CES function is that is allows us to incorporate into our model various assumptions about the role that the volume of transactions plays in the CS process. As we have indicated previously, there are a priori reasons for believing that changing volumes Of transactions affect Optimum holdings Of the various balances differently. That is, it is an empirical question whether the transactions elasticities Of demand for the different monies are identical. Likewise, it remains an empirical question whether there have been differential rates of technological change in the utiliza- tion Of the various monies tO produce money services. Our model addresses this last question as well. . In addition to testing these hypotheses, we want 6 model which examines the following issues: 1. Are all monies equally substitutable between each other and especially with respect to holdings of domestic balances? This would imply that Qi_= pjwfor all i # j. Or, are some foreign balances weaker substitutesmforw> domesticibalancesiii.e. pj #pl for some 3) 2. What impact do expectations about exchange rate changes play on the degree Of 94 substitutability between assets? In addition, do we witness different behavior regarding the CS process under different exchange rate regimeS? ((3- Are domestic and foreign balances separable in “J demand from other near money assets? 4. How quickly do economic agents react to changes in exogenous variables? In answering these and other related questions, we seek to develOp a complete and consistent model Of CS. The Empirical Model The estimating questions of our empirical model are derived from the first order conditions (FOC) for cost minimization subject tO achieving a desired level Of output of money services. This problem is described below: ‘0 X l (4.3) m1n C X +-C e X +u... 81 l l l 2 12 2 +CQeIQXQ s.t. MS = -1/0 p - 2 Q +....+ 8Q(elQXQ) } + 82(e12X2) where: C1 = per unit holding cost Of currency i,i=l,....,Q In addition, we wish to incorporate the roles played by the level Of economic activity (transactions) and technolo- gical change in achieving the Optimal mix Of currency holdings. This we do by allowing the efficiency parameters in equation (4.1) to vary according to the following relation: _ -0i (4.4) 81 = BiTt exp {-sit} i=l,....,Q where 8. = fixed efficiency parameter for the 132 money balance 1 = 1,....,Q 95 Tt = level of transactions at time t. 61 = transactions efficiency parameter growth rate for the 13p currency balance, i=1,....,Q. 6. > 0. 1— s. = rate of growth Of factor argument technolo- gical change for input 1 (i=l,....,Q). si :_0. exp = exponential Operator. Upon substitution Of the relations defined in 4.4 into equation 4.1 the minimization problem described by 4.3 is solved algebraically using the Lagrangian constrained Optimization technique. That is, we form equation 4.5 (below) and then take partial derivatives of that equation with respect to the desired domestic value of the Q monetary assets (eliXi) and the Lagrangian multiplier (A).4 A necessary (or first order) condition for constrained Optimization is that each Of these partial derivatives equal zero. Specifically, we have -0 _ _ — 1 -9 '1/0 ‘ p _ -p exp (-slt)Xl l+. . . .+ BQTt Q exp (-th) (eleQ) Q) _ p.5- } where the first order conditions for constrained minimiza- tion are: _ -0 -p -1 3L 1 l 1 _ -1 _ _ (4.5a) 5X— - C p MS Bth exp ( slt)p1Xl - 0 aL A ‘1 2 ”9"1 Bra-27‘: - C2 " ‘0- MS BZTt exp ('Szt) 02(912X2) (4.5b) (4.5g) EE———— = c - 4 as B T Q exp (-s t)p 3e x p Q Q 10 Q eleQ) = 0 3L _ (4.51:) .37 - 0 ( Dividing the first equatiOn 4.5a above by any Of the other ) )id V ' (.'T:m“‘jk '9 firstgqxgoc, then taking logrithms and solving for ln(eljxjfi (j=2,....,Q) yields a demand equation for the jth foreign ) money balance, viz. (4.6) 1n (e..x 978'» w (e - .) ———£—— 1n 3 3 x4 l 13 3 j — _\.l + 1“‘p p181 l+pl sl-s. . x, " 1 {’c + _ll+pj®+ 1——+pj 1n.-l+pj In). Our estimating equations are derived from the set of Q-l foreign money demand equations defined by 4.5. The jpp equation Of this system is presented below:5 . .. . = . + .. + . + . + . (4 7) ln (eljxj) YQJ Y1] lnTl: 7231:— 733 lnx1 :43 1n 'J . 1 c. .1. -= [Cl] + ut j 2,....,Q 97 where: u is a stochastic error term (assumed % N(0,oi)) t .0 = __1._1.[En_82.] , = [1:31] . 1 + . fl ‘4' 3. 1 + . 3 DJ 0181 3 D] Y = 3:1 = - ._].'__. 13 1 + p. \ Y4j 1 + o. J J j- Y = 31 - s 2] l + pj Standard relations from economic theory and the restriCtions which we imposed provide us with the signs Of several Of the terms defined above. One expects, for instance, that the utilization Of any input would vary inversely with the price Of that input. Here, we expect that holdings Of any monetary asset will be inversely related, ceteris paribus, to the holding costs of that asset. Therefore, Y4j is negative. This, Of course, establishes a theoretical reason for our initial restriC2 tion that pj must be greater than minus unity. Specifically, we see that if: _ _ l . . Y4j - I_:—E; < 0 1mpl1es 1 +50 > 0 or i . > -1 0:) 3:2: - 0'0 From the restriction that o and the pi (i=l,.....Q) have 98 the same sign, we assert that YOj will be positive and so W111 Y3j (if p) -l). However, the Signs of Ylj and Y2j are indeterminate and depend upon the differentials in the transactions elasticities of demand for and the rates of technological progress in the use of the various balances. Estimation Techniques So far we have not discussed methods for estimating our model. Clearly if all of the right side (r.h.s.) variables are exogenous (or predetermined) and the stochastic error terms are contemporaneously uncorrelated, then each equation of the system defined by equation 4.7 can be estimated using ordinary least squares (OLS). However, from the derivation of the estimation equation, it is clear that the holdings of domestic balances (X1) should not be treated as exogenous to the system. Since these balances are not exogenous, we introduce the possibility of simultaneous equations bias. That is, an element of the r.h.s. variables will be correlated with the stochastic error term. Using OLS to estimate the equations of such a system will yield biased and inconsistent estimates of the regression coefficients. It is necessary therefore to use an estimation technique which purges the r.h.s. of its endoge- nous elements. Several estimation methods have been suggested in the literature. Chetty solves his system of FCC for the demand for domestic balances function (again, this is in 99 the context of the domestic near money literature). Included as arguments of this function are all of the interest rates in the model as well as the level of national income. Expansion of this function through its Taylor series allows one to estimate each near money balance demand equation using two stage least squares. Bisignano uses a slightly different technique to obtain nonrandom estimates of domestic balances. That is, he calculates an independent measure of money demand as a function of real permanent and transitory income, real interest rates, and a measure of the implicit yield on money balances. These independent (of the model) estimates are then substituted for domestic balances as a r.h.s. variable in the other asset demand equations.7 The method we would choose is to estimate each zSLS foreign money demand equation using two stage least squares. This means that in the first stage domestic money balances will be regressed on all the exogenous variables in the system of demand equations. Included in the set of exogenous variables are all of the Q holding costs (ci, i=l,....,Q), the logarithm of the level of transactions, and a time trend. In the second stage of the estimation process the predicted holdings of domestic balances are inserted into the foreign money demand equations replacing their actual values and OLS is applied to the resultant equations. Two stage least squares procedure yields estimates 100 of the coefficients of equation 4.7. These can be used to obtain estimates of some of the underlying parameters of the model. That is, from the following relation, we obtain estimates of the j (j=2,....,Q) substitution parameters: = _ 1 Y43' ‘ 1 $“5j - (l + Y4j)/Y4j A implies pj Since the estimated values (Ej) are non-linear functions of the ;4j' the variances of the Sj must be calculated according to some approximation technique. The method we have chosen was first suggested by Klein. (See appendixZ). Here, for instance, we approximate the variance of pj as follows: A l A var . m var ( .) 0J “ [Y4jz] Y43 Each estimation of the Q-l foreign money demand equations yields an independent estimate for 91’ the substitution parameter associated with domestic money balances. This is, in each equation estimated, pl = - Y3j/Y4j - 1 Again, because the 61 are non-linear functions of the estimated regression coefficients, we must approximate the variances of these estimators. The formula is derived, as above, from Klein's technique and is presented below: 101 A var (pl) % [Viilvar (Y3) + [$32/Y44] var (Y4) -2 [Y3/Y43] COV (Y31Y4) Because we have Q-l estimates of p1, it is too much to expect (given the stochastic nature of the process) that all of the estimates will be identical. This leaves the problem of which estimate to report. One solution to this dilemma is to incorporate the information from all of the model equations by constructing an estimate of $1 from a weighted average of the Q-l point estimates of 31' The weights would be proportional to the inverse of the estimated variances for 81 from each equation; would sum to unity; and would thereby give extra weight to the most precise estimates.8 It is impossible to identify estimators for the 6i and the si (i=l,....,Q). Nonetheless, we can solve for estimates of the differences between 6 (the transactions 1 related growth rate in the efficiency parameter 81) and its counterpart ej for any of the j foreign balances(j=2"..,Q). This estimate is derived below: 91 - ej = - Y1j/Y4j 3:21'0'0IQ Following the method described above, the estimated variance for this estimator is approximated by the following term: 102 A A 1 A 2 A 4 A var - . x—-—' . . . . 61 63 % Y4j2 var (Ylj) + [Y1] /'y43 ] var (Y43) -2 I; /§ 3] cov <§ § > lj 4j lj' 4i We are interested in measuring this term because it allows us to test the hypothesis that changes in transactions levels affect the demand for the various money balances differently. A rejection of the null hypothesis that the term 61 2 ej = zero would establish some validity to our argument that this is the way the CS process works. Similarly, we can examine the question of differen- tial rates of technological change in the use of these balances through testing the value of the estimator derived S - S o — v o ' The variance of this estimate is approximated by: A A A 2 A 2 A 4 var (s1 - sj) g (1/Y4j ) var (Yzj) + [Yzj /Y4j ] A A A 3 A A var (Y4j) -2 [Yzj/Y4j ] COV (YszY4j) Finally, the use of two stage least squares guarantees that the estimates of the equation coefficients are consistent. Further, they are also efficient relative to all other single equation estimation techniques.9 If all monies are viewed as identical substitutes for domestic balances and if we assume constant returns to scale (ie. pi = pj = p for all i f j, then the problem 103 described above can be handled with a simple transformation of the estimating equation. Specifically, we impose the constraint that the substitution parameters are equal. This implies a theoretical value of unity for the Y3j of each of the j equations (j = 2,....,Q). Empirically this constraint is imposed by setting the coefficient (Y3j) to unity in each equation and rearranging terms so that the lOgarithm of domestic balances is moved to the left hand side of the equation. This yields a constrained estimating equation for a homothetic CES technology of: (4.8) in eijxj - ln X1 = AOj + Alleth + Azjt4-A3j ln Si ( ) + 6 Cl t where at is a random error term Bl Aoj - 0 1D ('B—j A23. - 0 ($1 " Sj) A . = o (8 - e.) A . = - —l— = —o l] l j 33 l+p and where each pair of assets has a Hicks-Allen partial elasticity of substitution defined below as: o 1%3 Provided the constraint is valid,10 OLS on equation 4.8 is the appropriate single equation estimation technique since the remaining r.h.s. variables are all assumed exogenous to the system.11 Additional parameters of the underlying production function for money services can be estimated from the 104 regression coefficients: 0 = ‘XlL" _ l A3j A -111. 91 - 9. = 'T—l J A3j A -A . S " S. = 72-1 l J A3j Again, the variances of these constructed estimates may be derived using the Klein approximation technique, described above and in the appendix to this chapter.12 Lagged Adjustment Models So far, we have assumed that desired balances are determined and achieved within one period. That is economic agents fully adjust their various money assets to changes in exogenous variables within each period. This assumption may be too restrictive. Rather, it may take several periods for full adjustment to new equilibrium values to occur. If so, our model (implied by either equations 4.7 or 4.8) is mis-specified, and estimates derived from it would be biased due to omitted r.h.s. variables. One way to model lags in the CS process is to specify a standard partial adjustment model. We can describe this partial adjustment process for the jth foreign balance as follows: 105 'b 6. (4.9) eljtxjt = eljtxjt 3 eljt-l xjt-l eljt-l th-l where el'tx't = the domestic currency value of currency 3 3 j in time period t j = 2,....,Q ’0 e1. X. = the desired domestic currency value of 3t 3t holdings of currency j in time period t. j=2’oooo’Q 6. = rate of adjustment of holdings of currency j 0 g 6. < 1 If we take natural logrithms of both sides of 4.9, then collect terms involving previous period holdings of the jth foreign balance, we obtain ’1: (4.10) ln (eljtxjt) - éjlileljtxjt + (l - Sj)]J1e1jt-l th-l :1 X. with the estimation equation ijt 3t develOped previously (equation 4.7) yields an alternative Replacing the term lne estimating equation which allows us to measure the speed of adjustment, viz. (4.11) In eljtxjt = “Oj + n1 1n Tt + nzjt + n3 1n X1 1 _l + "4j n C1 + n5] 1n eljt-l th-l + vt h r . = 6. . w e e “03 J{70]} '1le = 6'{Ylj} 106 “Zj = 6j{Y2j} "3: = 5j{Y3j} "4j = 53{Y4j} “Sj = 1 - oj v = stochastic error term t Because the simultaneous equations bias described for equation 4.7 still remains in the model defined by 4.11, this partial adjustment model must be estimated using a systems estimator. Again, in this case, we choose to employ the most efficient single equation systems estimator — two stage least squares. Note, the addition of a lagged value of the endogenous variable as a r.h.s. variable adds no new econometric problems so long as the error terms of each demand equation are assumed to have the usual nice proper- ties.13 A similar partial adjustment scheme can be thought of in terms of our restricted-homothetic CES model. Again, we begin with the following relation: R. '13. Kj (4.12) _J_t_ = (_LL) R. R. jt-l jt-l el'tx' where R.t = ——%——l— = the ratio of holdings of foreign 3 lt balance j to domestic balances at time t. fijt = the desired ratio offoreign to domestic balances at time t. K. = the adjustment coefficient for ratio j J (j=2I'°°°IQ) 107 After taking logrithms, rearranging terms, and substitution ’0 of equation 4.8 for R. , we have jt el'tx't Si . ——JL—4l—— = . + . + .+ . (4 l3) ln xlt $03 wljleTt $23 $3] ln (Cl) e . X. 3 lt-l where wOj = KjAOj wt = stochastic error . = K.A . 11’13 J 13 ‘i’ . = K.A . 23 J 23 . = K. . W33 3A3] w4j = 1 ‘ Kj Since we assume that the stochastic errors are nicely l4 behaved, 4.13 cagubfi&sstimated using OLS. waa Cc'nstmw1 wwfi" “Mt Equations 4.7, 4. 8, L 11 and 4. Tprovide us with \/h..——* 1%?“ M \fi-‘i IJ’I‘" the basic estimation equations for r model. Each imposes different constraints on the underlying process. Each can be used to test different hypotheses about currency substitution. The next chapter details the results of this process. FOOTNOTES 1For a more complete discussion of these restrictions see Dhrymes and Kurz (1964, pp. 292—293). 2Dhrymes and Kurz, pg. 290. 3This formulation is similar to the varying efficiency parameter model developed by Moroney and Wilbratte (1976). See pp. 186-187. 4The Lagrangian multiplier (A) can be interpreted as the marginal cost at the equilibrium point. See Dhrymes and Kurz, pg. 293. 5It is possible to write the jth first order condition for cost minimization (i.e. equation 4.5) in the form Cj = ggé exp (uj). Taking logarithms of both sides yields 3' an additive error term. Equation 4.7 is formed in a similar fashion. We assume that the source of error originates from imperfect knowledge about factor prices or imperfections in the cost minimization process. 6Chetty (1969) pp. 276-277. Actually, Chetty follows the approach first suggested of Dhrymes and Kurz. See Dhrymes and Kurz, pp. 293-295. 7Bisignano (1974) pg. 12 and pp. 36-39. 8This is the procedure followed by Dhrymes and Kurz, Chetty and Moroney and Wilbratte. 9See Schmidt (1976), pp.lSl-153 (for consistency) and pp. 164-165 (for efficiency). Note, each equation of 4.7 is overidentified since Q-2 cost terms are excluded and an equality constraint on the coefficients of the included cost terms is also imposed. Obviously, we must also assume that the errors are not comtenporaneously correlated across equav tions. If this assumption failed, thea a technique that employ- ed this information such as three stage least squares or full information maximum likelihood would be more efficient. 10In the case where there are only two minies in the production function for money services, the concept of a CES technology requires that 91 = Dz. This point is proven in the appendix to the next chapter. 108 109 11We assume, again, that there is no contemporaneous correlation across equations. 12Again, we have Q-l estimates of D. This necessi- tates a procedure such as that described above for constructing a weighted estimate of p. 13Violations of this assumption would include auto- correlated errors. If this occurs, maximum likelihood (or some other efficient estimation technique) estimation is called for. 14We assume that the errors are neither autocorren lated nor contemporaneously correlated across equations. APPENDIX 2 On Approximation of the Distribution of Derived Parameters APPENDIX 2 On Approximation of the Distribution of Derived Parameters E = Let mxl f(6) nxl Suppose 8 is consistent and /T(6 - 6) + N(0, 0) Then 5 = f(e) is consistent and mg - a) -> me. DwD') 3Eil.........3§l ael . aen _ ' ' - éé where D — . - - — d6 agm ° 85m 361 aen E l . = - l + . . xamp e pJ ( Y4J)/Y4J 8p.2 1 2 1 where D = (3A ) = (*_—§) = '”_—4 Y4] Y43' Y4j 110 CHAPTER FIVE Estimation Results Introduction Using quarterly data on private, nonbank Canadian holdings of their own and United States dollars between 1962 and 1978, we tested the models developed in the previous chapter. We chose Canadian-United States data for several reasons. First, the data are good and very detailed. In addition, most of the data are available over the last twenty years. This allows us to examine the effects of altering exchange rate regimes (the data divide almost exactly in half between periods when the Canadian dollar was fixed and when it floated) upon the CS process. Second, there is some evidence to suggest that the CS process is limited, for the residents of Canada, to a decision between holding their own andtui dollars. That is, over the past twenty years virtually 100 percent of the foreign currency liabilities of Canadian banks to private, non-bank Canadians has been denominated in U.S. dollars. While this is not conclusive evidence (since Canadians could hold foreign balances in third countries) it suggests that Canadian bankers felt that there was insufficient demand for deposit liabilities in these hypothetical third lll 112 currencies and therefore did not supply them.1 A more hueristic argument that CS in Canada is limited to Canadian and U.S. dollars would center around the close political ties and economic interdependence of the two countries. Third, if the proposition we have just suggested is true, then estimation of our models is somewhat easier. Specifically, as we show in Appendix 3 (at the end of this chapter), if there are gnly_twg inputs in a generalized CES production function, then the substitution parameters (91' i=l,2) must be constrained to be equal in order to preserve the idea that along any isoquant of money services there is a constant elasticity of substitution. Therefore, in the two currency input case, we can limit our analysis to the constrained CES estimating equations defined in the last chapter by equation 4.8 (full adjustment) and equation 4.13 (partial adjustment). Finally, the case of Canadian-U.S. currency substi- tution has been considered in several papers by Miles (1978A, 1978B, 1979). As we have pointed out, we feel that there is a possibility that his model is mis—specified. Hence, the results of his empirical work provide a conven- ient basis for comparison with the results we obtain.2 In this chapter, we present the results of our empirical work. We hope to demonstrate that our models provide a plausible solution to the problem of measuring the degree of currency substitution. Further, as our results will indicate, we feel there is considerable 113 evidence for a transactions demand based theory of the CS process. This chapter continues with a description of the data used in our study. Then the estimation results from the full adjustment model are presented. These results are compared to those of Miles. Next, the partial adjustment model is considered. Finally, we investigate questions of separability and discuss other unresolved issues. The Data In order to test our empirical model and derive a measure of the degree of substitutability in demand between currencies, data were gathered from a variety of published sources. The data are presented in Appendix 4. None of the data used in our study were adjusted for seasonal variation. Much of the data on Canadian money holdings (both Canadian and U.S. dollars), interest rates, and exchange rates were taken from various issues of the Bank of Canada Review and Statistical Summary. In addition, several time series on Canadian economic activity were found in issues of the Canadian Statistical Review and National Income and Expenditure Accounts published by Statistics Canada. Information on Canadian holdings of U.S. dollar accounts in the United States was located in the Treasury Bulletin of the United States Department of the Treasury. The specific time series used in our study are described more completely below. 114 Canadian holdings of their own currency were derived from the time series "currency and privately held deposits" estimated monthly by the Bank of Canada. This time series includes currency, non-interest bearing demand deposits, and time deposits. In excludes deposits in government owned accounts. A time series of Canadian holdings of United States dollars was constructed using information from several sources. This series was formed by adding U.S. dollar liabilities of Canadian banks to private non-bank Canadians (less "swapped" deposits outstanding)3.to the Canadian dollar value of short term liabilities of the U.S. banking system payable in U.S. dollars to all non-bank, non-official Canadians. We used as measures of opportunity costs several different interest rates. These included the weighted averages of tender rates on three month Canadian and U.S. treasury bills. These rates were obtained from Bank of Canada publications. Consequently, the U.S. rates had been adjusted to conform to Canadian reporting standards (i.e. the U.S. rates were converted to reflect an equivalent yield basis to Canadian rates).4 Data on other/interest and "swapped" deposit rates also come from various issues of the Bank of Canada Review and from International Financial Statistics (published by the International Monetary Fund). 115 Two different measures of Canadian economic activity (i.e. transactions) were located. One series was construc- ted from monthly observations on the value of checks cashed in Canadian clearing centers. The other transactions series was quarterly observations of Canadian gross national product valued at market prices. Observations on spot and three-month forward exchange rates (expressed as Canadian dollars per one U.S. dollar) were derived from monthly closing values of the different rates and in several cases from the closing three-month forward spread. Estimation of the Full Adjustment Model The full adjustment constrained CES model (equation 4.8) - which we denote as Model 1 - is presented below xl ,fi. , C2 (4.8) 1n = A +A 111T A -.t-:;,+ A 1n — + u elZXZ 01 ll _t’j 21 1 31 C1 t It was estimated with quarterly data for the period 19621I- 1978IV using three different definitions of the relative cost ratio (2%) . Table 4 presents the results from these experiments. Cost ratios 1 and 2 (defined below) represent attempts to incorporate into the holding costs of foreign balances a term representing expected changes in the exchange rate.5 Specifically, we define the following two cost ratios as: 116 .s>s. 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VJ: It... .75???» cha-PHHCS H555. .7.ch Han: voHIIHuxu I 2 onH¢ 0H2loc I z oaozh C. - :>. z - _ .,.-H......-.-.w-\... . .. 2...... ”nil-.0. 9:33ch 22 2:3 eouIHauHuu m. ._ .cc..c:ee as. xc tt_v ecu; .2x.» :2. :o ¢¢H2¢Hs¢> asceth22s xixxxH cue tutgH :62: =CHHc—tuuoU HQHHI¢ ac wastrel s 0H cHsh .n .2o.c:e¢ HcctH >L¢:.cuc x:_¢: Hes—H20 4503 ace—acavo HH< .N mmmvmmMI-Nuarmumw. autism HcoHuuHulum .ouusom .H ”muhoz ----- --- --- Ham. «H. Hnnc.H ccHawo.H HoN.HH --- --- HooH.H --------------- Ne. oH ~.x~H oHH. ~n¢.| Hex. cox. «Ho. so.H nun-u I-a-n own. n ----- ..coo.c --- .ec..c ...ec.. .a..oo.. .eH.H. .eooo.H --- ¢.c-.v --- xcco.- ----- ae.HH o.H~H cHH. =c~. are. HnH. «ex. Hx.H ane.| Ill-a axe. n ..ce..c --- .ANc.. .ep..v ..ooo.. ...oco.. H.H.HH --- «Anuo.v «Hac.v see.- ----- cao.- a..c. o.c~_ a... .m.. nee. .m.. as». an.H --- oeo.- mo.H n .e:... .ce..c .oa.cc .c.c.c ...c=.. .a..oo.. .n..H. Heoo.c «H.0H. c c..c... ~n.- mac. aa.Ho ~H.N x.ma NHH. soc. Hex. nnH. sex. ~x~. OHO. can.) Hx.n n --------------- asmn.v Annc.v cc-mc.v Acnc.v Ina-t nun-a Aan.H --------------- ch. <.~HH cHH. nnH.: sex. who. x~o. Ono. Ina-a nun-n cam. ~ ----- .eno.c --- .Na..v ..eeo.. .a.eeo. . ..o~o. . ..eooo.. --- ..o.~.o ----- oeo.- --- new. m.n~H eHH. cuH. max. neH. sax. mac. oHoo.n nun-n new. H ccHHFH. H ----- HH~.HH HeeH.H ccHnoc.v «canxo.v Acso.v Inn-u cH<~o.H ccAne.H an.- ----- an.H- ~HN. ~.n~H HHH. QHH. sax. HnH. ecu. ~99. nun-1 an.n sa.H N .cw._c .e_. _. .c.~e. .ce..v «Ancc.v ...nco.. H-o.. .ooo.. «.an.H a.n..H. H=.- Hm. x.oH- HaH. <.oa ~HH. ~ce. ncx. xHH. Nox. eHc. xcc. n-.u a<.n ~ --------------- Hemn.v H~mo.H cc.~mo.v HWNo.H --- --- HenH.H --------------- cam. x._x~oH r _anoH Try—LL.— H—C— arac— utm— ——HH..— l —..u—Z~= a—HQEJQHH—.—H< —P._ THE.— uma—HHRNB oHC—uPJB—dlm .... .. .4: h r n H n 2 < 133 using the COSTRATIO l and the full specification of the model, we see that only about 12% of the difference between the desired and actual relative balances is eliminated within a single period. These results seem even more surprising when one considers that assets markets are usually assumed to be very quick to clear.18 In the case of Model 2, the estimated coefficients of the transactions term (column 2),19 the time trend, (column 3), and the relative holding costs (column 4) are now all functions of the rate of adjustment, K. Therefore, in order to derive coefficients which have the same inter- pretation as those derived from estimating Model 1, it is necessary to deflate each estimated by the estimated value of K. Again, we find that because the parameters we construct are nonlinear functions of the estimated coeffi- cients, we must approximate the variances of thesepmrameters according to some approximation technique. We find, for instance, that depending upon the particular definition of the cost ratio, the estimated elasticities of substitution (column 11) are either very low (when COSTRATIO l or COSTRATIO 2 is used) or very high (corresponding to the use of COSTRATIO 3). In all cases, A we were unable to find values of O significantly different from zero at the 95% level.20 The interpretation of the estimated coefficients is that they represent short run elasticities, since they are all multiplied by the adjustment speed (K). The 134 parameters obtained by dividing these coefficients by K are then estimates of the corresponding variables long run elasticities. Following this interpretation the column O provides estimates of the long run elasticities of substi- tution. We see that these long run elasticities are (in most cases) approximately twice the size of the estimates of 0 from Model 1. This suggests that in the long run, the percentage change in relative balances due to a one percent change in relative costs, will be almost double the short run change. In this model, we again see the strong influence of the level of transactions on the determination of relative balances. In particular, the short run transactions elasticities - determined by the estimated coefficients of the transactions terms (column 2) - were all significantly below zero at the 95% level. This confirms our findings under the previous model that Canadians increased their holdings of U.S. dollars during periods of rising trans- actions. Estimates of w(=O(GU -9CANH(column 14), the long S run transactions elasticity vary substantially, according to whether or not a time trend is included as a right hand side variable. In all cases, where the trend was omitted the estimates of m were significant at the 95% level and between zero and minus one. When the trend is included, then the w become statistically insignificant (although the t values are all greater than one) but below minus two 135 in value. We are unable to explain these results, but the answer must lie in the strong collinearity between the level of transactions and the time trend. Because we were unable to estimate the coefficient of the relative holding costs term with very much accuracy in any of the specifications,we could not identify statis- tically significant estimates of the transactions parameter differentials (GUS-'6 AN)(column 12) or the technological C change differential (sUS - SC of w substantiate the role of the transactions term in our AN)(column 3). The estimates specification. However, since we were unable to verify the existence of a difference in the rates of technological change, we excluded the time trend from estimates during the different subperiods. Finally, we note that, in general, the specification of Model 2 seemed to be slightly worse than Model 1 in explaining the demand for relative balances. Specifically, in every case, the standard error of the regression was higher for equations in Model 2 then the corresponding equations from Model 1. Other goodness of fit measures (R2,F) were also lower when Model 2 was estimated relative to Model 1. Values of the Durbin-h statistic kept us from rejecting in every case the hypothesis of nonautocorrelated errors. We turn now to the results of estimating Model 2 during the fixed and flexible rate periods. Table 8 presents the regression results from the fixed rate period. Two different specifications of equation Ho>wH N92 02H an stuHqumHm «t Ho>aH Nno o:u Ha acouHuHcmnm « mascara HaauH auacHauo uch: pagan—Haw use acoHua:vo HH< .n aumogucouaa 2H macsua auspcaHm .N 136 230:2: husmaou... .augsm HauHuaHuoum "mason .H “mm—2232 (all: ----- an.xnv ccnnxo.v Anx=.v A~<.nv ((((( Hmsu.v III-a Inn-I ~H.¢Ho c.5n amH.l aas. ~HH. «ca. baa. ex.H- (cu-1 mam. n ««Am¢c.v «tam~°.v «Ao~.nv «tancH.v «tancH.H «Awe.uv «canac.v «sAc<.HH oHH.u mna.n no.x x.vHH en.H mum. afio. «cm. xao. ~c.n cmo.: «x.a n --- ----- 3.34 2.5.3 2.53 13.: ----- 3.x; III:- (1111 nx~.- a.o~ «as as assessazmsn .. ~o>o~ Nmo as accouuuswun « mozszucouue cu cucuuo vuovcmum .N E%MmemmiNHmmmmmm .NuoElsm ucoquaqusum .ouusom .n "nuke: .«A~=~.o .«xme.~o .«Aee~.o «gase.~c mcc. cc.s nm~.| ~o.- m «camc—.v «4A¢e=.v ««A~o~.v «aaum.~v ace. and. bew.| cs.- N «taco—.v «camsc.v ¢¢Ao-.~ «aaoo.~v nus. wad. noo.l ee.n~ — Ace Ame va Adv C—hmH N Ho>oH Nmm um ucmuwmwswwm ma um uamowmficwwm «« * pwuufiEo Noe cowum>uompo .m mononucmuma cw muouum thatcmum .N swumaasm musmmouH .mmeEDm Hmowumwumum "wousom .H nnnnnnnnnn no.mHHv «*Amwo.v Ammo.v AHm.Hv IIIII Amc~.v llllllllll w.mq m.OHH mam. mum. waa. «mm. omo. kuaa I HHHONmH pofiuom comm mwcmfioxm mawamam I Hmvoz ucmsumsmp< Hmwuumm ~..H "muasmmm sowumawumm OH m4m uv m AmsHm> uv wh<¢m<2m czoqm wEmem so oHemH Nod um ucmonstHm as Hw>mH Nma um ucmusmaawam « monsoon ummmH Hsmchuo nuwz pmumEHuwm mcoHumsom HHm .N 1"" moHumHumum HmHocmch HmcoHumspmucH .BmH>om mcmcmo mo xcmm "woousom .H "wwwoz OH. 050. «H. 050. «H. 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