A EEF’OSII-RUN MODEL OF MEMBER-BANK BORROWING $35313 M 152 32533121; or PH. 0. MICHIGAN STATE UNIVERSITY PETER A; FGRM'SZES, JR. 1968 r‘w, Ja--W . ad ”‘4‘; i IIERAPV f I" _‘ I: Nliclugn' '-'-' H» E ii Univcm 3? ma ””3 1 _r_ WV up 90:31"? This is to certify that the thesis entitled A DEPOSIT-RUN MODEL OF WEBER-BANK BORROWING presented by Peter August Formuzis, Jr. has been accepted towards fulfillment of the requirements for PH.D. degree in Economics Major professor Date 72;: ////// 0-169 ABSTRACT A DEPOSIT-RUN MODEL OF MEMBER-BANK BORROWING By Peter A. Formuzis, Jr. The demand for member-bank borrowings from the Federal Reserve System is one of the links which connects the actions of the monetary authority to the stock of money. The purpose of this dissertation is to build and test a model that will identify and measure the variables that determine the volume of borrowings under uncertainty. The theoretical structure used to derive the demand fer borrowed reserves from the Federal Reserve System is based on a profit maximizing one-state optimal inventory model under uncertainty. Given a set of subjective probability distributions regarding reserve flows and interest rates, the bank derives the optimal quantity of reserves to hold that udll minimize expected losses. The demand for borrowed reserves is then derived from the difference between the actual and desired stock of re- serves. The model was tested with the monthly data for all member-banks from 1954 to 1967. All data on.member-bank reserves, borrowings, and interest rates were taken from the various issues of the Federal Reserve Bulletin. The estimation procedure employed a distributed lag scheme to derive expected values for the independent arguments while a multiple re- gression analysis was used to determine the coefficients and stability of the demand fUnction. The empirical results show that a demand fUnction for member-bank borrowings can be isolated from the monetary data. The most striking Peter.A. Formuzis, Jr. and important aspect of these results is that the strength and signifi- cance of the interest rate variables have undergone a secular decline fran 1954 to 1967. Evidence is provided that shows this behavior of the interest rate variables to be related to the rapid'growth in federal funds trading as an alternative method of reserve adjustment. Fran the viewpoint of monetary policy the evidence developed here supports the proposal that the non-price barriers to borrowing imposed by Regulation A be removed and that the Federal Reserve discomt rate be tied to the Treasury bill yield. The enactment of these recomnenda- tions should slow or eliminate the current erosion of discomting by federal funds and should make discomting a more accurate and potent tool of monetary policy. A DEPOSIT-RUN LDDBL OF MHWBER-BANK BORROWING By Peter August Formuzis, Jr. A THESIS Submitted to Michigan State University in.partia1 fUlfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Economics 1968 ACKNOWLEDGMENTS During the two years that this dissertation was in progress I have had the privilege to be associated with numerous individuals who have given generously of their time in improving both the style and content of the dissertation. IMy greatest debt is to the chairman of my committee, Thomas R. Saving, to whom I am unable to give an adequate expression of my full appreciation. Dr. Saving guided my work from the preposal stage to the final draft and in the process corrected imprecisions too numerous to recall and at least one serious error. I am, of course, responsible for all the shortcomings and errors that remain. I would also like to thank Bruce T..Allen, Karl Brunner, Samuel B. Chase, Jr., Edward J. Kane, Robert F. Lanzillotti, and Gurcharan S. Laumas for the assistance that they gave at various stages of my work. I feel the need to give a special acknowledgment to Robert F. Lanzillotti whose support and encouragement made the Ph.D. possible and to Karl Asmos whose clarity in using the tools of economic analysis al- ways represented a goal that I have sought vainly to emulate. Dorothy Burger typed the manuscript and I am.very much in her debt for correcting many errors in notation. Lastly, I would like to thank my wife, Jane, not only for her encouragement and inspiration but also for her assistance with numerous computational problems. Peter A. Formuzis, Jr. ii TABLE OF CONTENTS Page LISTOFTABLES......... ..... ......... iv LIST OF FIGURES ........... . ............. v INTRODUCTION . . . . . ............... . ...... 1 Chapter I. THE THEORETICAL MODEL OF MEMBER-BANK BORROWING ...... 4 The Uncertainty Characteristics. . . . . . . . . . . 12 The Treatment of vault Cash ........... . . 19 me Data 0 O O O O O O O I O O O O O O O 0 O O O O O 2 0 II. THE ESTIMATION PROCEDURE. . . . . . . . . . . . . . . . . 22 Distributed Lags and the Construction of Expectations 22 Construction of the Coefficient of variation . . . . 29 III. THE EMPIRICAL EVIDENCE. . . . . . . . . . . ....... 31 The Time Period. . . . . . . . ........... 31 The Regression Results ............... 33 A Survey of Research on Member-Bank Borrowing. . . . 46 The Early Period. . . ............. 46 The Later Period ................ 53 IV. .A SUMMING UP AND FINAL REMARKS ............. . 67 Summary. ................... . . . 67 Final Remarks .................... 70 A Recommendation .................. 73 APPENDIX - DERIVATIVE AND PROPERTIES OF THE EXPECTED LOSS FUNCTION ............................. 75 BIBLIOGRAPHY ........................... 79 iii LIST OF TABLES Table Page 1. MUltiple Regression Estimates of the Demand Functions for Borrowed Mbmber Bank Reserves 1954-60 and 1960-67 . . . . 34 2. Mbltiple Regression Estimates of the Demand Functions for Borrowed.MEmber Bank Reserves, 1960-67 . . . . . . . . . 3S 3. Effect of Federal Funds Activity on Reserve Adjustment PraCt ices O O O I O O O O O O I O O O O O O O O O O O O O 39 4. 'Multiple Regression Estimates of the Demand Functions for Borrowed Member Bank Reserves for TWo Year Periods . . . 41 S. Compensating variation in Member Bank Borrowing for Changes in the Monetary Base from January, 1918 to December, 1929. O O O O O O O O O O O O O O O O O O O I O O O O O O 48 6. Goldfeld-Kane Estimates of the Demand Function fer Borrowed Reserves, 1953-1963 .......... . . . . . . . . . 62 iv LIST OF FIGURES Figure Figure l. Exponential Weighting Streams ............... 28 2. Expansion Path for Borrowings . . . . . . . . . ...... 55 3. Relation of Member-Bank Borrowings to Least-Cost Spread, July, 1953 through December, 1966 . . . . . . . . . . . . 56 INTRODUCTION The link between the monetary authority and the stock of money depends in large measure on the particular reaction of comercial banks to variables that affect their behavior. Economists' understanding of these links between Federal Reserve policy and bank behavior has quick- ened of late due to two major factors. The first has been the develop- ment of the theory of choice under uncertainty and the second is the widespread availability and use of the high-speed electronic computer.1 The theory of choice under uncertainty made its first entry into the theory of bank operation with an article by F. Y. Edgeworth on the mathematical theory of banking in 1888.2 Edgeworth built a model, to be presented in more detail in Chapter 1, where the only uncertain factor facing the bank was the amount of daily cash deposits and with- drawals defined by a normally distributed density function. From this information Edgeworth was able to derive the bank's demand for cash given some probability of not being able to meet the cash demands of the bank's depositors. Little was done to extend EdgeWorth's analysis into a more representative theory of bank behavior until after the pub- lication of the seminal article on inventory policy under uncertainty by Kenneth.Arrow, Theodore Harris, and Jacob Marchak, in 1951.3 1Some of the relevant contributions on the theory of choice under uncertainty are J. Tobin, "Liquidity Preference as Behavior Toward Risk," Review of Economic Studies, XXV (February, 1958); H. Markowitz, Portfolio Sélection (New York: John Wiley 6 Sons, 1959); D. Farrar, The Investment DEC1$Ion Under Uncertainty, Ford Foundation Doctoral Dissertation (Pren- tICe-Hall,“1962). 2F. Y. Edgeworth, "The Mathematical Theory of Banking," Journal of the Royal Statistical Society, LI (1888), pp. 113-27. 3K. Arrow, T. Harris, and J. Marchak, "Optimal Inventory Policy," Econometrica, XIX (1951), pp. 250-72. 1 Since 1951, a number of articles and books.have emerged encorp- crating these developments into the theory of bank operation.4 George ‘Morrison has called the synthesis of these developments ". . . the ex- isting formal theory of banking," and has stated its basic assumptions in the following way: (a) banks maximize expected profits (or minimize expected losses), (b) banks construct probability distributions of gains and losses from investment in assets, and (c) following the lead of Edgeworth, profit maximization takes place subject to a specified distribution of cash drains during the planning period. This existing formal theory of banking will be used as the base on which to build and test a model that will identify and.measure the vari- ables that determine the volume of member bank borrowing from the Federal Reserve under uncertainty. The theoretical model and.method of estima- tion will be derived largely from the works of Samuel Karlin, Phillip Cagen, and Milton Friedman.6 In brief, the model will view the amount of reserves that a bank has available for cash withdrawals and other clearing drains as the inventory of the bank. The hypothesis will then require the bank to choose a stock of reserves at the beginning of the 4S. Karlin, "One State Inventory MOdels with uncertainty,” in K. Arrow, S. Karlin, and H. Scarf (eds.), Studies in the Mathematical Theory of Inventory and Production (Stanford, Calif}: Stanford Univer- sity Press, 1958), pp. l09-34; R. C. Porter, "A.Mbdel of Bank Portfolio Selection," Yale Economic Essays, I (1961), pp. 322-59; D. Orr and W. G. Mellon, "Stthastic Reserve Losses and Expansion of Bank Credit,” Ameri- can Economic Review, LI (September, 1961), pp. 614-23; G. Morrison, Lig- uIdity Preferences of Commercial Banks (Chicago: University of Chicago Press, 1966). SG. Merrison, op. cit., p. 8. 6Karlin, . cit., pp. 109-34; P. Cagen, "The Monetary Dynamics of Hyperinflation,” 1n M. Friedman (ed.) Studies in the Quantity Theory of Money (Chicago: University of Chicago Press, 1958), pp. 25-117; M. Fried- man, A Theory of the Consumption Function (Princeton: Princeton Univer- sity Pfess,I1957). period so as to minimize expected losses from holding reserves. If the existing stock of reserves at the beginning of the period is less than the optiJmIm stock, the bank will equate actual and desired holding of reserves by either borrowing reserves from the Federal Reserve Bank, making some portfolio adjustment, or both. The model will be tested with monthly time-series data for member banks from 1954—67. The statistical methods used will‘be a distributed lag hypothesis to estimate expected values for bank reserves and interest rates, and a multiple regression analysis to determine the coefficients and the stability of the demand function for borrowed reserves. The plan of the dissertation is as follows. The first chapter will develop the theoretical model; the second chapter will develop the es- timation procedure and describe the data to be used; the third chapter will present the empirical evidence supplied by the econometric model and its relation to other empirical work performed on member-bank bor- rowing; and the fourth chapter will summarize the model and the conclu- sions from the empirical tests. CHAPTER I THE THEORETICAL MODEL OF MEMBER-BANK BORROWING F. Y. Edgeworth was the first economist to view a bank's demand for cash reserves to hold as a.prob1em of holding an optimal stock of inventory under uncertainty.1 In his model he assumed a large number of depositors, each of whose deposits in some part may be withdrawn dur- ing the next period. The amount withdrawn is a random variable X', nor- mally distributed, with mean u, and standard deviation 0. The bank then chooses some probability p of not being able to meet all cash withdrawals. In terms of a cumulative-distribution function, the probability of not being able to meet all cash withdrawals is the area to the left of any specified value on the density function. For the assumed density function, the normal deviate Z is given by X-u/o so that if X is the amount of cash on hand, then X = u + 02. The actual cash is a random variable X-X' so that X-X' = u-X' + 20 whose expectation is 20. This quantity is then the safety allowance needed by the bank against cash withdrawals given the above specifications of the particular density function and the bank's chosen probability p of not being able to meet the'withdrawals.2 Edgeworth's model is an incomplete specification of the actual in- ventory problem.facing the bank in that it neglects other clearing drains and other cost and revenue considerations. we will now extend Edgeworth's model so as to obtain a more complete description of an hypothesis that derives the equilibrium quantity of reserves to hold in inventory against 1F. Y. Edgeworth, op. cit., pp. 113-27. 2The following presentation of Edgeworthfs mathematical banking model is taken from Arrow's chapter on the "Historical Background," in K..Arrow, S. Karlin, and H. Scarf, op. cit., p. 7. 4 cash withdrawals and clearing drains in an uncertain world. This de- velOpment will depend on at least the four following assumptions. (1) The bank maximizes expected utility which is a function of expected pro- fits; (2) there are only two earning assets which the bank can purchase. These are represented by loans and securities, where all loans and sec- urities are homogeneous and have a single rate of return fer each; (3) supplements to the actual stock of reserves at the beginning of the per- iod can only come from the Federal Reserve through borrowing or through the sale of securities;3 (4) the time horizon of the model is only one period. The total amount of reserves available for cash withdrawals and clearing drains that the bank has at the beginning of the period is (y) which.may have been supplemented by borrowing (B) from the Federal Re- serve at a cost (c), where c = a + rdB. (a) represents some fixed cost of borrowing and (rd) can be taken as the Federal Reserve discount rate. The amount of reserves on hand before (B) was added to inventory was then x = y - B. During the period, demands on inventory can come from three sources. These are cash and deposit withdrawals, loan demand by the nonbank public, and the bank's own demand for securities. If the sum of these three demands is (2), then two possibilities arise. The first is if y - z > 0. In this case an opportunity cost will be incurred which depends on the loans and securities that would have been held if y - z = 0. In order to determine the appropriate opportunity cost, it is necessary to know what portion of y - z:>0 would have been in loans and what portion of y - z > 0 would have been in securities, assuming 3This assumption implies the absence of the Federal funds market as an additional source of reserves. 6 that their rates of return are different. This matter of portfolio choice will be handled by the introduction of a portfolio parameter (Q). (Q) is the proportion of reserves that would have been allocated toward loans. A.simp1ifying assumption being made here is that the portfolio parameter (Q) is taken as datum and is constant for all relative and absolute levels of interest rates. This neglects changing portfolios between loans and securities but is justified on the basis of exposi- tional simplicity and will neither be further explored or tested in the remainder of the dissertation. From the above infermation we will take the opportunity cost of not purchasing loans and securities as being represented by the term.nm{Q(y-z)] + rs[(l-Q)(y-z)], where (rm) is the market rate of interest of loans and (rs) is the interest rate on secu- rities. Also if y - z > 0, some portion (k) of total demands (2) went toward earning assets while (l-k) took the form of cash withdrawals and certain clearing drains. The bank will then earn a positive gain (negative loss) on (kz) divided between loans and securities. This will be represented by the term - m(ka) - rs[(l-Q)kz]. The second possibility arises when 2 - y > 0. In this case, a pen- alty cost is incurred which is represented by the function p(z-y). The penalty function (p) can be thought of as being composed of two parts. The first part is composed of the additional borrowing necessary from the Federal Reserve in order to meet some legal minimum of average re- serves to be held over the accounting period. The second cost included in the penalty function is that some additional cost may be imposed by depositors who had expected a loan and had included this'expectation as a.part of the expected marginal revenue of demand deposit money and time deposit bonds. If we assume that individuals were in equilibrium at the 7 old expected marginal revenues of cash, demand deposits, and time depos- its, they cannot be in equilibrium any longer and thus will reduce their deposits in accord with the new lower marginal revenue of deposit money until new cash-demand deposit and cash-time deposit ratios are achieved at Which individuals are again indifferent to additional increments of either type of money.4 gain) will be earned fran the loans demanded and securities purchased. Also, if z - y > 0, a negative loss (positive This will be represented by the term -rm[Qky] - rs[(l-Q)ky] . If we combine the information in the two preceding paragraphs we can write the expected loss function of holding an inventory of reserves (y), where ¢(z,rm,rs) is a contimious joint probability den- sity function and where all functions are continuous and at least twice differentiable as mm] = a + rdcy-x) + IZIZIZIrmtocy-zn + rs[(l-Q)(y-2)] - rm[ka] - r5[(1'Q)kZ]}¢(z,rm,rs)dzdrmdrs + gin: {P(z-y) - rlekY] - rs[(1-Q)kv]}¢(2.rm.rs)dzdrmdrs (1.1) 4For an interesting analysis of this particular way of looking at the cash-deposit ratio, see B. P. Pesek and T. R. Saving, Mone Wealth and Economic Theogy (New York: The MacMillian Cmpany, 1967i, pp. 97-99. 8 The problem now is to find the optinum amount of borrowing B where B = y - x for all (x) given, such that E[L(y)] is at a minimum for all y > 0. This requires that the first derivative of E[L(y)] vanish and second derivative be greater than zero. 5 The derivative of (l . 1) set equal to zero is :Bucyn = rd + I:f:{rm[Q(y-z)] + rSICI-QMy-Z) - erkz - rs(1-Q)kz - P(z-y) - rmoky - rscl-QIkyu(rm.r5.y)drmdrs + @213er + rSCI-QN ¢(rmrsz)drmdrsdz + I;I:I:{-k[er + rs(1'Q)] - P}¢(rm,rs,z)drmdrsdz = o (1.2) The terms in (l . 2) represent the various expected marginal revenues and marginal costs from holding an inventory of reserves. If we expand and rearrange (1.2) by placing all the expected marginal cost terms on the left side of the equation and the expected marginal revenue terms on the right side of the equation we can state the profit maximizing condition in the more familiar "marginal cost equals marginal revenue" form. This rearrangement of terms is shown as 5One must be cautious in taking the above derivative by use of the conmon form of the Fundamental Theorem of Calculus since (y) appears as a limit of integration. See Appendix A for this computation along with a discussion of certain internal relations of the function. 9 rd + IZI:{-rs(1-Q)kz + nmcl-QIky + rs(1-Q)ky}¢(rm.rs.y)dnmdrs + IZIZI:{-erk - Ts(1'Q)k}¢(rm,rs,z)drmdrsdz + f;f:f:{-ker - krs(l—Q)} ¢crm.rs.z)drmdrsdz = IZIZIancy-x) + rs(1-Q)(y-Z) - P(z-y)}¢(rm.rs.y)dnmdrs + [:[:f:{-P}¢(rm,rs,z)drmdrsdz + f;f:f:-P¢(rm,rs,z)dtmdrsdz (1-3) As long as the minimized value of (1.1) yields a value of the optimum amount of reserves (y) such that y > x, then y - x = B which is the Optimum amount of borrowing that will minimize the expected losses of the bank. Thus far the model has been developed without providing empirical counterpart for the reserves variable (y) and (x). The (x) quantity of reserves is that quantity of reserves on hand at the beginning of the period before any borrowing has taken place that is available to meet both expected and unexpected clearing drains. .As a first approx- imation, unborrowed reserves would Serve as an appropriate variable. Using the following accounting identities between total reserves (R), borrowed reserves B, unborrowed reserves (Ru), required reserves RR, and excess reserves RE, we can write RU=RR+RE-B Ru=R-B 10 Ru is the total amount of reserves supplied to banks which are not with- in the control of banks. Ru includes currency, the gold stock, Treasury balances, and the Federal Reserve holdings of securities. RR, RE’ and B are factors which are under control of banks. However, given the nature of the deposit run model specified here, the sum of total unborrowed reserves is overstated since not all of required reserves is available to meet deposit runs. Only 83 1/2 and 88 cents are available along with 4 cents for time deposits. The unborrowed reserves figure has been reduced by a weighted average of these ratios so as to convert the reserves variables (x) and (y) into numbers that represent avail- able reserves. However, in order to employ standard bank terminology, "6 the reserves variable will be termed "unborrowed reserves. Borrowing will be taken to be a function of those factors summarized in Ru, or the amount of unborrowed reserves.7 If we now return to the expected loss fUnction we can say that borrowed reserves depend on unborrowed reserves, the market rate of interest on loans, the rate of interest on securities, the Federal Reserve discount rate, and the level of total deposits (D) since this will affect the size of total demands on reserves (2). If we now call out these variables and place them in a general function form, we have, B = B(Ru,rm,rs,rd,D) (1.4) 6I am indebted to Thomas R. Saving for pointing out that total un- borrowed reserves is not the appropriate independent variable. 7The amount of unborrowed reserves was taken to be one of the inde- pendent variables in two other pieces of research on.member bank borrow- ing. See R. C. Turner, Member Bank Borrowin (Columbus: The Ohio State University, 1938); and S. . 0 1e an . J. Kane, op. cit. . 11 and by assuning (1.4) to be homogeneous of degree one in Ru and D, we have ’ AB = B(A&,,rm,rs,rd,>.D) (1.5) and by setting A = l/D we obtain IE; 3 f(§prmsrssrd) (106) The three interest rates in (1.6) will be interpreted in the following manner. The market rate of interest will be considered to represent the net yield on loans, or the marginal revenue on loans . . In order for the bank to make a decision on whether to borrow in re- sponse to a givenrm, it must also have some measure of the relevant marginal cost, which in the case of borrowed reserves will be taken as the Federal Reserve discount rate. In order to take account of this profit maximizing behavior we will use the algebraic spread between the two rates rather than each rate separately, or (rm - rd) .8 The security interest rate will be considered as the yield on that asset that the bank views as the closest substitute for reserves in the sense that security sales represent a reasonable alternative for borrow- ing on the part of the bank. The particular securities involved are short-term bills since they are preferable to longer term bonds as a 8A. J. Meigs has experimented with both the algebraic spread and the ratio of the two rates discussed here in his study ofthe demand for free reserves . His empirical evidence shows both the algebraic difference and the ratio to perform well, while such not being the, case when the rates were used individually. See Meigs, 9. cit., pp. 95-102. 12 method of obtaining reserves because the price variance is less on short- term bills than on long-term bonds. Just as in the case of loans, the bank in making a decision on whether to borrow or sell securities to ob- tain additional reserves must take into account the marginal cost of borrowing relative to the marginal cost of security sales in terms of interest returns lost so that the algebraic difference (rs - r d) is the relevant variable. If the algebraic difference terms (rm - rd) and (r5 - rd) are now substituted for the three interest rates in (1.6) we obtain 1%.: g $51-er - rd),(rs - rd) (1.7) where the signs of the relevant partial derivatives are 3 93 a g: < 0’ 3(rm-rd) > 0’ Bug-rd) > 0 (1.8) D THE UNCERTAINTY CHARACTERISTICS The general ftmction (1.7) that describes the relationship between the volume of borrowing and its independent arguments was develOped from a profit maximizing inventory model under uncertainty. However, the particular manner in which the presence of uncertainty affects (1.7) and in what way it differs from a more simple canplete certainty formulation has not been made clear. In this section we will describe the develop- ment of a model under the condition of complete certainty and compare its implications with the implications fran the uncertainty model. This SET" .- c} O. OH qt: IE t—C 13 will also enable us to precisely formulate the manner in which uncer- tainty will be taken into account. Under complete certainty the bank knows exactly the amount of re- serves that will be forthcoming during the period, the exact quantity of the various demands on those reserves and the exact interest rates on loans, securities, and discounts that will prevail. This would re- quire the optimal inventory relationship described in equation (1.1) to read LCY) = a + rd(y-x) + Ig'IrmIQIy-xn + rs[(1-Q) (y-zII - erkz _ rs(l-Q)kz}dz + f;{p(z-y) - erky - rs(l-Q)ky}dz (1.9) The significant difference between (1.1) and (1.9) is the absence in the latter of the density function ¢(z,rm,rs) since all these variables are known with certainty. Under these conditions a bank will borrow reserves for only two reasons.9 The first is that the bank may have some desired rate of asset acquisition that cannot be achieved because its reserves at cer- tain points of time are less than is desired even though when the en- tire period is taken into account, reserves are sufficient. This motive for borrowing is analogous to that of consumers selling bonds in order to make their actual consumption paths equal to their desired ones. 9This discussion is mainly inspired from the analysis of consumer borrowing presented by M. Friedman, op. cit., pp. 7-19. 14 The second motive for borrowing derives from the profit maximizing behavior of the bank in taking advantage of discrepancies between mar- ket interest rates and the discount rate. This motive for borrowing is made up of two sub-hypotheses. The first sub-hypothesis is that the bank will raise the additional reserves by choosing the least-cost com- bination of security sales and borrowing; the second is that the bank borrows in order to purchase market income streams that possess yields greater than the discount rate. This second sub-hypothesis has been dubbed the ”profit-theory" of borrowing and has been prescribed by the Federal Reserve's Regulation A. This provision and its effect on bank borrowing will be discussed more fully in Chapter 111 when the empirical evidence is presented. The introduction of uncertainty into the loss function (1.9) will yield the expected loss function (1.1) where the variables no longer known with certainty are the demand for reserves, the market rate of interest, and the Treasury bill rate. These variables are described by the density function ¢(z,rm,rs). This particular density function was chosen over the alternative approach of describing each density function separately, since one would not expect the separate density functions to be independent. The major difference between the certainty and uncertainty formu- lations is that the bank will have to hold emergency reserves as a buffer inventory in order to meet unexpected clearing drains and cash withdraw- als. It was this motive for holding reserves that was of prime interest to Edgeworth in his derivation of the safety allowance. Edgeworth as- sumed the density function of cash withdrawals to be normally distributed so that the distribution of the random variable could be fully specified 15 by its mean and variance. In earlier work on the theory of uncertainty, Markowitz and Tobin have hypothesised that the variance can be viewed as a measure of risk with the degree of risk increasing as the variance increases.10 With regard to the bank borrowing problem, let us suppose that the level of unborrowed reserves is constant but that the variance of those reserves becomes greater. This will increase the probability that the bank will find itself with insufficient reserves at some point during the period. In order to hold a constant probability of being able to meet total drains, the bank will have to increase the loss-mini- mizing quantity of reserves. Likewise, if the variance of reserves should decrease, the bank will find that the old loss minimizing quan- tity will decrease the probability of not being able to meet total drains and thereby will reduce the loss-minimizing quantity. In order to take account of the effect of increases in the fluc- tuation of unborrowed reserves on the bank's demand for borrowed re- serves, we will include the coefficient of variation (standard devia- tion divided by the mean) of unborrowed reserves as an independent variable in the demand function. The reason for choosing the coeffi- cient of variation instead of the variance as a risk measure is that equal variances will not represent equal levels of risk if the expected value of unborrowed reserves should change. The coefficient of varia- tion will standardize the risk variable for changing levels of unborrow- ed reserves. The "emergency" motive seems, on an a priori basis, to be of signi- ficant value in the banking model since there exists a near zero marginal 10H.‘Markowtiz, op. cit., J. Tobin, op. cit., pp. 65-86. 16 cost of deposit withdrawals thus reducing to a minimum the resistance of consumer portfolio switches from deposits to cash. The presence of uncertainty will tend to make the loss-minimizing quantity of reserves held greater than under perfect certainty since the presence of the buffer inventory motive is not offset by reductions in the other two motives. The presence of uncertainty with regard to interest rates does not suggest whether the loss-minimizing quantity of reserves will be greater or less than under perfect certainty. It seems reasonable, however, on an a priori basis to consider the bank as holding expecta- tions of interest rates as well as reserve flows and that it would be rather forced at a theoretical level of analysis to claim that interest rates are known with certainty while reserve flows are not. It has been assumed that the bank holds subjective probability dis- tributions of the flow of unborrowed reserves and the values of the var- ious interest rates. This uncertainty notion implies that the bank holds some expectation of the value of each variable which may be different from their actual values. Unfortunately, the magnitudes that represent these expected values are not directly observable. The nature of the problem therefore is to conceptualize a hypothesis that will transform the actual observations in a manner consistent with the theoretical model so that they will be useful in interpreting the empirical evidence. The same general problem described above was encountered by Milton Friedman and Philip Cagen in their studies of the consumption behavior and money demand in that the central problem was to take observable mag- nitudes and convert them into other magnitudes that more closely coincide 17 11 with the theoretical model. The Friedman approach which we will follow here, consists of separating any measured or actual observation into a permanent and a transitory component.12 In terms of unborrowed reserves, the permanent component is derived from all those factors which the bank considers as determining the level of unborrowed reserves. Among those determinants would be the payment schedules of the depositing public, demand for loans, securities, cash, special economic conditions and the bank's expectation of Federal Reserve Open Market operations or other policy changes. This permanent component of unborrowed reserves would be analogous to the "expected" value of a probability distribu- tion.13 The transitory component is composed of all those factors which make the actual level of unborrowed reserves different from what was expected. It is composed of those "other” factors which the bank does not take into account because it is believed that they are largely unimportant or that they are so random that there is no way to forecast their presence or degree of severity. If we now apply the same proce- dure of separating all independent arguments in (1.7) into permanent and transitory components, we have (1.10) C48” cram Clczj 11M; Friedman, op. cit.; P. Cagen, op. cit. 12M; Friedman, op. cit., pp. 21-22. 13Ibid., p. 21. 18 (rm - rd) = (rm - rd)p + (rm - rd)t (1.11) (rS - rd) = (r5 - rd)p + (rs - rd)t (1.12) where the subscripts (p) and (t) indicate permanent and transitory com- ponents respectively. If we now substitute the permanent terms for each argument in place of its measured argument into (1.7) and add the coefficient of variation of unborrowed reserves VRu we have B Ru 15 = h[ D—p’(rm ' rd)p’(rs ' rd)p9VRu] (1.13) where the signs of the relevant partial derivatives are ah ah ah ah ' a D131"; < 3(rm rd)p > 23(rS rd)p ' BVRu The general demand function given by (1.13) represents the function to be fit to the time-series data by use of the estimation.procedure set down in Chapter II. 19 THE TREATMENT OF VAULT CASH The demand function developed here does not discriminate between reserves held as vault cash and reserves held as balances at the Federal Reserve Bank. In the expected loss function (1.1) it was assumed that some portion (l-k) of total demands (2) went for cash withdrawals. The aggregate density function ¢(z,rm,rs) then includes the bank's estimate of these withdrawals. The portion of total demands on reserves (2) that come from cash withdrawals can be viewed as determining the bank's de- mand for vault cash while the reserve demands from loans, security pur- chases, and other clearing drains determine the demand for Reserve bal- ances held at the Federal Reserve Bank. The procedure of aggregating these separate demands for reserves is clearly more appropriate after November 1960 than in the period previous to that date. This is so due to the fact that in July 1959 the Board of Governors was authorized to treat vault cash as a part of member bank reserves, whereas before vault cash could not be counted as reserves. In December 1959, the Federal Reserve allowed member banks to count a portion of vault cash as reserves. On August 25, 1960, and on September 1, 1960, the percentages of vault cash that counted toward reserves was increased, and in November 1960, all vault cash was counted as a part of member bank reserves.14 The addition to reserves of this change in the law was by no means negligible in that vault cash amounted to approximately 2.5 billion dollars, which was about 10 percent of the level of unborrowed reserves. With respect to member-bank borrowing, an important impact of this change was to make reserve balances at the Federal Reserve and vault cash perfect substitutes for one another with regard to the need to 14M. Friedman and A. J. Schwartz, A.Mbnetary History of the United States (Princeton University Press: Princeton, N. J., 1963), p. 447n. 20 meet the Federal Reserve requirements ratios on demand and time deposits, and thus their separate demand function.more closely associated with one another. The model has been constructed on the basis of total unborrowed reserves and thus for the earlier period the data are applicable only to that portion of borrowed reserves which was used to increase bal- ances at the Federal Reserve. Some borrowing presumably was inspired by the need for additional vault cash for which the model is unable to estimate since the independent variable is void of vault cash. In or- der to compensate for this difference, we have added the figures for vault cash to unborrowed reserves so as to make an estimate for unbor- rowed reserves that will more closely coincide with the model. THE DATA .All basic data, with the exception of vault cash estimates, were taken from the various issues of the Federal Reserve Bulletin from 1954 to 1967. All data, again with the exception of vault cash, rep- resent daily average observations for one month.periods for the aggre- gate of all member banks. Daily average figures for vault cash did not exist before Nevember 1958, and as a substitute we have taken the series developed by Milton Friedman and Anna Schwartz in theirIMonetagy History of the united States. The deposit level was taken as the sum of demand plus time deposits adjusted for interbank deposits. The market rate of interest rm_was taken as the 4-6 month prime commercial paper rate which is an average of daily offering rates of dealers. 21 The security rate rs, was taken as the market yield on U. S. Gov- ernment 3-month bills. The discount rate rd, was taken as the rate applicable to advances and discounts under sections 13 and 13a which represent the rate charged if borrowing is secured by eligible paper or by U. S. Government obli- gations. With regard to changes in the discount rate, any change that occurred before the 15th of the month was recorded as applying to that month, while any change after the 15th was recorded as applying to the following month. CHAPTER II THE ESTIMATION PROCEDURE The theoretical model developed in the last chapter must be made statistically operational if it is to be tested against the data on' member-bank borrowing. This chapter will set down the method by which the bank arrives at particular values for "permanent" or "expected" un- borrowed reserves, as well as some of the economic and psychological hypotheses that underlie the method. DISTRIBUTED LACS AND THE CONSTRUCTION OF EXPECTATIONS Irving Fisher was the first to hypothesize that an effect of a change of one variable upon another may be produced during the passage of time in a manner such that the effect is divided up or distributed over that interval.1 In Fisher's words: The reason for distributing the lag is that the full effect of each P' (a rate of change of a price level index item) is ex- tremely unlikely to be felt at only one instant, such as seven months later, and not felt at any other time either earlier or later than this seven months . . . It is far more probable that the influence began at once, showing itself in the very next month . . . and that it then gradually increased to a maximum a few months later and thereafter tapered pff indefinitely according to the probability distribution. Analytically, the general hypothesis suggested by Fisher can be written as Y = ont + wIXt-1 + szt-2 + . . . + wnxt_n =.{ wiXt_i (2.1) 1Irving Fisher, "Our Unstable Dollar and the So-Called Business Cycle," Journal of the American Statistical Association, vol. XX, June, 1925, pp. 179-202. 21bid., p. 184. """ 22 23 where Xt-i is the measured value of the independent variable and ”i is the appropriate weight for a particular point in the time horizon of n periods.3 The particular economic and psychological rational of a dis- tributed lag reaction as given in (2.1) with regard to bank behavior is two-fold in that it can be considered to be composed of psychologi- cal and institutional factors. A consideration of these factors will yield a basis about which a more formal and empirically determinable hypothesis can be constructed. Therefore, let us consider each of these factors separately. The most straight forward method of computing an expected value of a variable would be to find the simple arithmetic mean of past levels of that variable. This method lacks appeal as a psychological proposi- tion because it assumes that the memory path of the banker is such that it remains of undiminished intensity with respect to the passage of time until a certain point where it instantly vanishes! In terms of (2.1) this would imply that mo = ml = . . . = “n which carries with it the same introspective objection that Fisher had with the case of only one mi different from zero. A more introspectively pleasing hypothesis put forward in the field of psychology is that of Jost's Law of Exponen- tial Forgetting which states that if two observations are of equal strength but of different age, the older diminishes leSS‘With time.4 A.memory pattern compatible with Jost's Law would be some type of 3This discussion is for the most part taken from the contributions of L. Koyck, Distributed Lags and Investment Analysis (North-Holland Publishing Co., 1954); M. Nerlove, "Distributed Lags and Demand Analysis for Agricultural and Other Commodities," Agriculture Handbook 141, U. S. Department of Agriculture, 1958. 4H.A. Simon, ”A Note on Jost's Law and Exponential Forgetting," Psychometrika, XXXI (December, 1966), p. 505. 24 at. If a banker's mem- exponential decay function, say, of the form e ory were of this exponential form, the affect on him of an observation would fade away as it recedes into the past. Under institutional factors that would produce a distributed lag, one might consider such things as changes in the pace of economic ac- tivity, natural disasters, changes in tax rates, bank runs, etc. Given any of the above conditions it would be reasonable to assume that the effect of the particular situation would be greater the closer is the event to the present. An exponential decay pattern is also applicable here if we can suppose the dynamic adjustment rate of the system to be directly proportional to the degree of divergence of the system from equilibrium. As an example, suppose a series of bank runs produces an excess demand for excess reserves by some amount (E). Assuming the ad- justment time to equilibrium where all excess demands are zero to be multi-period, let the degree of adjustment during period one be (A) where O < A.< E. The amount of excess reserves at the beginning of period 2 is now E-A and assuming the degree of adjustment to be pro- portional to the amount of excess demand, the second period adjustment will be less than A. This process continues with the pressure to ad- just diminishing as the system moves to equilibrium. Changes in Federal Reserve policy produced through open market operations is a logical candidate for a reaction to be described by a distributed lag. Changes in Open Market policy are not made public but rather are known only after the net purchases or net sales of secu- rities have been made over a sufficiently long interval such that the bank can make a judgement about the policy. Since the precise opposite may be suggested by the evidence in any given week, the bank must form its expectation by taking into account past observations. Since Open 25 IMarket policy is reviewed every two weeks, the bank could be supposed to give observations that lie back in time less, but perhaps not zero, importance than observations that are closer to the present. From the foregoing discussion it is clear that a distributed lag model will allow one to construct an expectation of a variable on the basis of actual levels of that variable in the past. In a similar man- ner an individual would alter his expectation by some degree if the ac- tual value of that variable turned out to be different from what was expected. we can make this hypothesis analytically explicit by making the expected value of a variable depend on aweighted average of past values by use of an adjustment model that has been.used by Cagen, Fried- man, Morrison, Koyck, Nerlove, and others. This adjustment equation can be written in continuous form.as 31x t 3 B[Xt ' ECX)t] .(2'2) where E(x)t is the expectation of xt and B is a scalar. Equation (2.2) shows that the change in the expected value of a variable is proportion- ally related to the difference between the actual value of a variable and its expectation. 8, the coefficient of expectation gives the speed of adjustment of expected to actual observations. An assumption to be made from this point on is that for a given time series of a particular length the value of B is constant. This however, will not prevent us from splitting up a time series in different ways to discover whether the value of 8 actually changes. Equation (2.2) is first-order linear 26 differential equation whose solution with the constant tenm equal to zero is E(x) = e8t [EBxyesydy (2.3) The assumption that the constant term is zero implies that at some point in the past the actual values were constant. (2.3) can be expressed as t 3 xye >dy [T 1500 = T g_ (2.4) 8 Equation (2.4) shows the expected value of a variable to be a weighted average of actual past observations with geometrically declining weights given by the term.e8y. Note that this weighting stream is compatible with that suggested by Jost's Law of memory decay mentioned earlier. Another property of this particular method of forming the expecta- tion of a variable is that the sum of the weights over the interval [-T,t] is equal to the denominatory of (2.4).6 Since only discontinuous data is available, a discrete approxi- mation.must be used in place of (2.4). .A form developed by Cagen and 27 also obtained by Nerlove is T E(x) = (1 - e'B) Z x 1:0 .e‘Bi t_1 (2.5) where 0 is current time. This approximation is derived from solving the differential equation given in (2.2) with the weighting pattern 7 The weighting pattern (1 - e'8)e'Bi shown in (2.5) given by est. will yield a different value for each predetermined B . The range of 8 values chosen for this study were .1 < B < l where each 8 differs by .05 from the one directly above it. (The time horizon for each 8 was taken where the sum of the weights to three decimal places is unity. 6This result is seen by the following: t t Syd = 11' y d = 1 BY = 1. 8t - BC'T) = [Te Y B,_Te (B y) fie B[6 e ] -T 18-[6812 _ eB('T+t-t)] = %_[est _ e'B(t+T)eBt] = E:_t_(1 _ e)-B(T+t) -6(T+t) as long as -T is Chosen so that the term.e can be neglected. 7P. Cagen, op. cit., p. 39. 28 Some representative weighting streams for various 8 weights are shown in Figure l. WEight 1 .90 * .70" .500 .30" .10 > . A n A A A J A L ‘ V V f v 1 3 s 7 9 11 13 15 17 1:9 Time EXPONENTIAL WEIGHTING STREAMS Figure 1 The value of 8 selected for a particular variable will be that value of 8 that generates a series of permanent terms which minimizes the variance of the simple regression of borrowed reserves per deposits, B/D, on the permanent independent variable, i.e., maximized the R2 .8 It will then be assumed that the value of B that maximized the R2 will ”best" represent the distributed lag weighting scheme of the bank. .All time series of the independent variables will be converted into "best" permanent time series as explained above. Each "best" series will then be placed into a linear multiple regression to estimate the general de- mand function for borrowed reserves given by (1.13). 8For an econometric development of this approach, see Ibid., pp. 92-93. 29 Unfortunately, the above procedure will produce an econometric prob- lem of nulticollinearity in that the two interest rate variables are highly intercorrelated. Since it is difficult, if not impossible, to estimate the separate effects of each variable when the degree of inter- correlation is high, we will report the empirical evidence in two forms. One form will include permanent unborrowed reserves and the market rate- discount rate differential and the other form will include permanent un- borrowed reserves and the bill rate-discount rate differential. CONI'RUCI‘ION OF THE COEFFICIENT OF VARIATION The model requires that we have an estimate of the coefficient of variation of unborrowed reserves to coincide with each expected or per- manent value of the variable. Since we have assumed aodistributed lag reaction in forming the expectation of unborrowed reserves with the banker giving less importance to an observation as it recedes into the past, it is reasonable to assume that his memory of the variability of that observation also diminishes with the passage of time. The variance of an observation about its expectation is given by 02 = (Ru ' 11111333;+ (Ru - Rup)%-1 +~ - ~+ (Ru - RIPE-n (2.6) n and the standard deviation as 30 O = \FRU - RUP)1Z:+ (RU - lh.ll)):--l +. ' °+ (RU - Rup)%-n (2.7) D What is desired is to form an estimate of 0 such that the bank "forgets" a particular variance as it recedes away. This requires that each root- mean-square deviation be weighted by the appropriate mi weight deter- mined by the "best" 8 for its place in the relevant time horizon [-T,t]. This can be written symbolically as (2.8) o . \CRu - REEL». + (A. -yRuP):-1wt-1 +- - -+ (R. - mine».-. n and the coefficient of variation as v = \[Uhr ' 1%th + (Ru ' Ihip)12;.1“'t-1 J" ' °+ (1h: ’ lhxpLi-n“’t-n n Rup In this form, the contribution of each coefficient of variation to the total variation at time (t) diminishes as the terms recede into the past. An assumption to be made here is that the same weighting pattern applies to the coefficient of variation as to the expected value of unborrowed reserves. CHAPTER III THE EMPIRICAL EVIDENCE The empirical evidence reported in this chapter will be divided into three sections. The first section will discuss the reason for the time period chosen. The second section will describe the evidence relating to the demand function developed in the previous chapters. The third section will contrast and compare the theoretical model and empirical results developed and presented here with the results obtained by other researchers. THE TIME PERIOD The period chosen for the empirical test was from January, 1954 through December, 1966. In choosing a time period the goal was to re- strict the data, within reasonable bounds, so as to most accurately represent the current functions and the time path and significance of the arguments. The only decision therefore was the particular begin- ning date. January, 1954 was chosen as the beginning date for three reasons: first, it is after the depression period of the 1930's; sec- ond, it is after the Federal Reserve--Treasury Accord in 1951; and third, it is after the excess profits tax of the early 1950's. During the 1930's member banks, bitter and pessimistic, accumulated large quantities of excess reserves in response to their increase demand for liquidity as the Federal Reserve System, a potential producer of liquidity, lapsed into passivity. These large excess reserves along with the fact that the discount rate from 1934 through 1941 was seldom below the short-term bill rate and never below the 4-6 month prime com- mercial paper rate made the quantity of borrowed reserves negligible. 31 32 The System, during this period,felt that it was pursuing a policy of ”monetary ease” due to a confusion of looking at the absolute rather than the relative position of the discount rate, a factor that we have corrected for by the use of the (rm - rd) and (r5 - rd) terms. It may seem as if the experiences of the 1930's is in no way in conflict with the model developed here. However, such a ViGW’iS likely to be erron- eous since it was during the 1930's that the banks discovered the true meaning of unborrowed reserves. During the disasterous deposit runs of this decade banks found that only (1 - RR), where RR is the average reserve ratio, of unborrowed reserves were actually available to meet deposit runs. In April, 1942, the Federal Open Market Committee announced that it would peg the yield of 90-day maturities at 3/8 of one percent by offer- ing to purchase a sufficient quantity of bills so as to prevent their yield from rising. The bond support program.was carried out for longer maturities as well with pegged yields ranging from 7/8 to 2.5 percent per year. These rates, which.were a carry over from depression standards had the effect of removing the interest rate risk on bonds. In late 1942, the discount rate was lowered to 1/2 of one percent on secured loans upon which Friedman-Schwartz have commented that ". . . if banks held such securities, it was generally cheaper for them to ac- quire any needed reserves by selling bills yielding 3/8 of one percent rather than by using them as collateral to borrow at 1/2 of one percen ."1 The effect of the bond support program.during the war and into the post- war period when yields were attempting to rise, extended the disuse of 1'M. Friedman and A. J. Schwartz, op. cit., p. 563. 33 the discount window which had begun in the 1930's. It was not until the Federal Reserve Accord in 1951 that discounting was again to occur in significant quantities. During the post-accord period the enviornment with regard to mem- ber bank borrowing was again upset by the imposition of the excess profits tax in 1952 and 1953. The excess profits tax which was appli- cable between June 30, 1951 and January 1, 1954, allowed up to 75 per-A cent of borrowed capital to be counted as a deduction from the tax base. Borrowed reserves fell into this category which could produce a tax saving equivalent to an after-tax yield of 2.7 percent on borrowed funds.2 The effect of this on borrowings is reflected in the fact that borrowings rose to a peak during 1952 which has not been equaled since. In light of these reasons the year 1954 has been chosen as the earliest date which marks a time span highly similar in relevant consid- erations to those which exist today. The only major change that took place during the 1954-67 period was the change in the role of vault cash in 1960 which we have made an attempt to compensate for by adding vault cash to member bank reserves. THE REGRESSION RESULTS Tables 1 and 2 record the multiple regression results for the de- mand function from 1954-60 and from 1960-67. The time period was split in this particular manner because 1960 was when vault cash was counted as reserves. Because the data before and after this change are not fully comparable, the data was separated at 1960 so as to make each time series consistently defined. 2A. J. Mbigs, pp. cit., p. 73. 34 mm. an. em. omH wHH wNH we «N mm .3930 Ewe—Rpm 05 333 .30an 953qu emcee 333893 93 now mugwfioz 36m .uooflowmmooo some 333 .3an 9883 Begum cm. amNcoc.cV mococ.o cm. fimvooo.oV aauoe.- cN. amaooc.cv amaoo.o cN. nmNNco.cV Aficuo.o ma. amNccc.cV macco.o mu. smacco.oo NmNco.c cN. newooo.oc camco.c cN. Ammaco.oV ~o~oo.- a? - we a? - a: on. Ammwco.cc Hmfiwc.- on. Anamoo.oV ma~mc.- on. Aweaoo.oU caNmo.- mN. AmHmHo.ov HNONO.- mN. memHo.oV ~maoo.- mN. flammao.oc mow~o.- .913 memoc.o mnoao.o ammoc.o menoo.o onaoo.o owaoo.o pgmoou scuooma noncoma senooma oouvmma oeuvmmfi ocuvmmH 820a UHOmN maomm UHomN >chmN > 03H mom mm>mmwmm MzUcoum >moomm >Uocwm >mocwm >Uccmm >moomN >uomom Hmaoeod ”oohzom Humoupodv nephew amou-ummon cm. on. em. om. oH. oH.- om.- om.1 05.- oo.- OH.H- cm.H- I 1 1 I d d I I I d I I u d d d I u C d d “ i u .5 i “d ‘u u“. n i .s i a o S. o. J I e K o o s a m i i o u I o o A com “ “*IK’ ‘ O on u K n i i o n n V I u n u . . v e u a . a . 0:? a u n o. .. . . . 5 a .a u . a . n. o . xi I I . i I O M. i l C35 I C O ‘ ‘ C I u . n . o a u . . .. . OOO . u u . x a m .a‘ . . 1 o 0 1’ + . o o o .. CE: 0 s o i HmNmHHo: do mcoHHHHTO m00§>1< “JEN UHCSOUWH: mumboHu >4H< >n2bzyé 57 with relatively high levels of borrowing. There seems to be a rather consistent pattern, both from Polakoff's original data and the new data, that the reluctance to borrow is (a) strong, and (b) stable. I feel, however, that this conclusion cannot be supported upon a closer examin- ation of the evidence. The separation of the early data from the later observations aids in bringing this point into relief. Netice that the later observations in general lie below their earlier period counter- '7 parts which implies that the demand for borrowed reserves has declined relative to any least cost spread (holding all else constant). If the reluctance to borrow were of equal intensity, the later observations for high least cost spreads should appear below their earlier period counter- parts but as can be seen, they always lie above these observations. The implication from this is that the reluctance to borrow is diminishing in importance and while it has not completely disappeared, is probably of no real importance in determining the reserve positions of banks due to the presence of a widespread and well fUnctioning Federal funds market. The recent article by S. M. Goldfeld and E. J. Kane on the deter- minants of member bank borrowing is the only piece of published research that has as its goal the isolation of the demand function for borrowed reserves and thus is the only piece that bears similarity to this study.20 BGIOW'We‘Will contrast and compare in some detail the Goldfeld-Kane model to the model developed here and undertake an evaluation of their proce- dure. The Goldfeld-Kane model is based on the following set of assumptions: At the beginning of each week, banks are assumed to face a specified need for new reserves (AR), a given rate of interest 205. M. Goldfeld and E. J. Kane, op. cit. 58 on marketable securities (rs), and a given discoUnt rate (rd). Each banker possesses only one decision variable: the total amount of borrowing (B) he will undertake over the upcoming period. Sales of securities (-AS) plus borrowing must add up to (AR), and negative borrowing (at the discount rate) is not allowed. Finally, we assume that borrowing from previous periods expires automatically. Next, we postulate that bank portfolio managers determine each period's borrowing (and consequently adjust their security holdings) so as to strike a balance between the cost of raising whatever reserves they require (C) and the disutility which arises from increased debt to the Federal Reserve. From this set of assumptions they write the utility function to be max- imized as depending on the cost of borrowing and the disutility of in- creasing debt to the Federal Reserve as fl‘. _'.J|‘.' Lam'A . A - U = U(C,B) (3.3) where the relevant signs of the partial derivatives are 3U 3U —< (3.4) 'a'C‘O' 8B 0 Since borrowing is a function of what they call the "reserve need" and takes place under the side constraint of cost minimization, they hypoth- esize a general demand function for borrowed reserves as 211bid., p. 500. 59 B = G(Kt,rs,AR) (3.5) where Kt is the least cost spread (rs - rd). .After pointing out that their theoretical model is static and that banks operate in a dynamic environment, they attempt to make the model dynamic by inclusion of the following two conditions. The first condition is that banks do not ad- just their actual to their desired portfolios within.a single period. Rather, the adjustment is only partially accomplished during the speci- fied period and thus the borrowing need for any one period includes some borrowing for the adjustments only partially accomplished from previous periods. From this, they say that current borrowing is some function of borrowing lagged one or more periods. The second dynamic consideration comes from taking into account the permanent nature of a change in unborrowed reserves. In order to make this concept opera- tional, they use the concept of separating any measured unborrowed ob- servation into permanent and transitory components. Permanent unborrowed reserves are important, claim Goldfeld and Kane, since "it represents the variable whereby banks incorporate their borrowing decisions into long-run developments. . . ."22 They employ a distributed lag scheme by which they attempt to determine the bank's estimate of permanent unborrowed reserves. They write their estimate of permanent unborrowed reserves at time t, Upt, as 22Ibid., p. 507. 60 h Upt =izlwt-1AUt-i (306) They point out that this permanent—transitory hypothesis is made operational by positing that permanent quantities are estimated as weighted averages of past observations, with the weights declining as the variables recede into the more distant past. . . and where the horizon [i,h] and the weights “t-i will have to be determined empiri- cally.23 The general function that Goldfeld-Kane fit to the weekly time- series data on member bank borrowing is h Bt = B(Kt,Bt_1,Bt-2, . . °;.let-iAUt-i) + Vt (3.7) 1: where Bt is the volume of borrowing at time t, Kt is the least cost spread of the Treasury bill rate minus the discount rate, wt-i is the distributed lag weight for each change in unborrowed reserves, AUt-i’ in the horizon [i,h] and Vt is the error term. The signs of the rele- vant partial derivatives are 3B > 0 3B > 0 BB > 3B < 0 (3.8) 5K;- ’ aBt-l ’ aBt-z ’ a(wt-1AUt-1) 23Ibid., pp. 506-7. 61 A representative sample of their regression results are given in Table 6. .All variables display proper signed coefficients and all but two achieve statistical significance at the .05 level. Goldfeld and Kane suggest that these results provide "clear support" of the importance and signi- ficance of lagged values of borrowed reserves and unborrowed reserves, and the least cost spread. Despite certain similarities in appearance, the Goldfeld-Kane model is much different in spirit as well as in the econometrics than the model developed and tested here. In particular one can view the model here as a "corrected" version of the Goldfeld-Kane model even though.my model was developed independently of theirs. In what follows I intend to show: (1) that their model contains an error, the nature of which destroys the value of the model as a useful hypothesis; (2) that there is an error in their collection of the data which.makes the data in- ternally inconsistent; and (3) that their empirical procedure violates the general method of the distributed lag scheme that they suggest having borrowed from Milton Friedman and contains misspecifications which makes their empirical evidence suspect. .A major error in the construction of the Goldfeld-Kane model which led to equation (3.7) is the omission of the deposit level as an argu- ment. .As was pointed out in Chapter I, the relation between borrowing and the deposit level derives itself from the profit-maximizing behavior of a bank in choosing an optimal stock of reserves to hold during the period against cash withdrawals and clearing drains in an uncertain world. Since total demands on reserves can be reasonably assumed to vary with the volume of deposits, any change in the deposit level will change the safety allowance of reserves thereby affecting the amount of borrowing. 62 oww. 0mm. mma. NNB. No.OH Hee.mO mO.WH HNN.HO OO.N flNO.mO mm.mN new - mpv mcmHummmH .mm>mmmmm szcmmom mom ZOHHUZDm Dz