APPENDIX I: Formula Relating Increase in Interest Rates to Amount of Reduction of Bank Reserves
Let ηS = interest elasticity of supply of demand deposits
ODs = demand deposits supplied
ηD= interest elasticity of demand for demand deposits
ODD = demand deposits demanded
i0 = initial level of short-term interest rate
di = desired change in short-term interest rate
A,B,E are defined in connection with Diagram 1 above.
Combining (3) and (3)’, we have
Given the crudity of the available estimates of elasticity, this formula can be accepted as a sufficiently close approximation where small proportional changes in the interest rate are involved. However, because as much as a one-third rise in the interest rate is at issue, the formula can yield appreciable biases. For S1 in Diagram 1 to have the same elasticity as S0, it would have to have a steeper slope than S0. The steeper slope would mean a higher intersection with the unchanged demand curve, D—i.e., a larger rise in the interest rate as the supply curve was shifted from S0 to S1. It follows that the formula exaggerates the amount of shift (and the amount of open market operations) necessary to produce any substantial rise in interest rates. The exaggeration can be shown to be minor, however. A rise in the slope of S1. sufficient to hold its elasticity equal to that of S0. would reduce the necessary percentage reduction in supply of demand deposits (A + B)/OD0 by only the very small fraction,
A more serious bias in the formula is found if the supply and demand curves are concave upward (instead of being the straight lines shown in the diagram). Concavity is justified both under the “normal” assumption of constant elasticity and by statistical observation. But if the higher ranges of the curves are steeper than the lower ranges, S1 and D must intersect at an interest rate higher than i1, and once more a shift smaller than (A + B) in the supply curve will be sufficient to raise the interest rate to i1 In the absence of data on the actual shapes of the curves, it is possible to say only that an appreciable overestimation of the necessary shift in the supply curve, e.g., a one-tenth overestimate, should exist.
Mr. White, economist in the Finance Division, received his undergraduate and graduate training at Harvard University. He has contributed articles to a number of economic journals.
See J. J. Polak, “International Coordination of Economic Policy,” Staff Papers, Vol. IX (1962), pp. 169–72, for a brief discussion of the problems involved in this type of coordination.
See W. R. Gardner, “An Exchange-Market Analysis of the U.S. Balance of Payments,” Staff Papers, Vol. VIII (1960-61), pp. 195-211, for a discussion of the significance of the “exchange-market balance of payments” for economic analysis.
Data from Germany indicating a close statistical relationship between the short-term interest rate and the free reserves of the banks are provided by F. A. Lutz, “Die Liquiditat des Banksystems und die Zinssatze,” Weltwirt-schaftliches Archiv, Vol. 87, No. 2, 1961, pp. 273-305; see especially the series for free reserves (the excess of the banks’ surplus legal reserves over their borrowings from the central bank) and the day-to-day money rate in 1951-60 charted on page 295. As will be described below, such a relationship suggests the existence of a valid interest-supply curve for bank credit. It is not clear, however, that the series do in fact reflect such a supply relationship. Lutz himself is equivocal. At one point he accepts the data as indicating a valid causal relationship (pp. 294, 296) although he had previously observed that borrowing from the central bank was only an emergency measure adopted when other factors were reducing the bank’s reserves and loan-making power (pp. 275-79)— which may imply a weak or nonexistent interest/credit supply relationship. These views need not be inconsistent, however. On many occasions the observed coexistence of declining reserves and rising debt to the central bank may be simply an adjustment to temporary losses of reserves that are not associated with changes in the levels of the interest rate. When interest rates do rise, the banks may be made willing both to lend out excess reserves and to borrow more reserves in order to expand credit further; since expanding credit means losing some reserves through currency drains, borrowing from the central bank may coincide with reserve losses but still be caused by a different factor, i.e., rising interest rates.
For a fairly extensive bibliography on this subject, see Martin J. Bron-fenbrenner and Thomas Mayer, “Liquidity Functions in the American Economy,” Econometrica, Vol. 28 (1960), pp. 810-34.
J. J. Polak and W. H. White, “The Effect of Income Expansion on the Quantity of Money,” Staff Papers, Vol. IV (1954-55), pp. 422-28; A. J. Meigs, Free Reserves and the Money Supply (University of Chicago, 1962).
This exclusion would not be made for countries where these borrowings are large and their amount is in effect determined by the central bank (through varying rediscount quotas, changing the application of penalty rediscount rates, etc.).
The assumption that central bank open market operations must cause a proportional reduction of only the part of reserves that banks did not acquire by borrowing from the central bank is itself dependent on an implicit assumption that the banks will not raise their ratios of central bank borrowing to deposits when deposits fall. Insofar as the central bank does not reduce a bank’s (formal or informal) rediscount quota fully in proportion to reductions in the bank’s deposit liabilities, the bank may in fact show a rising ratio of borrowings to deposits—a falling ratio of free reserves to deposits—when deposits fall, even though interest rates are held constant. This would mean that the reduction in the bank’s non-borrowed reserve holdings brought about by open market operations would have to be somewhat more than in proportion to the reduction (at constant interest rates) in the bank’s deposits. However, the assumption that open market sales are no more than proportional can be retained because the reserves obtained by borrowing from the central bank are always a negligible proportion of total reserves held.
Another reason for expecting cyclical variations in velocity is that the average cost of investing temporarily idle funds declines as the volume of transactions—and hence the amount of funds temporarily idle—increases. (W. J. Baumol, “The Transactions Demand for Cash: An Inventory Theoretic Approach,” Quarterly Journal of Economics, Vol. LXVI , pp. 545-56.) This means that velocity should be high when business activity is high, even if the interest rate has no influence. It has been pointed out (Polak and White, op. cit., p. 416, fn. 15) that this factor should not operate insofar as the cyclical rise in the volume of transactions reflects increases in the price level or in the number of businesses carrying out the transactions. Statistical evidence on the operation of this factor within the individual firm is said to be in favor of stability in the ratio of cash holdings to the transactions of business companies (cf., abstract of a paper by A. H. Meltzer, Econometrica, Vol. 29 , p. 456). “Economies of scale” were found for cash holdings as the size of company rises within the “small” and “medium” ranges, but not for the economically dominant “large” size group.
Karl Brunner, “Some Major Problems in Monetary Theory,” American Economic Review, Vol. LI, Papers and Proceedings, May 1961, p. 56. The formula was
Milton Friedman, “The Demand for Money: Some Theoretical and Empirical Results,” Journal of Political Economy, Vol. LXVII (1959), pp. 327-51.
Paul F. McGouldrick, “A Sectoral Analysis of Velocity,” Federal Reserve Bulletin, December 1962, pp. 1557-71, found that the share of heavy industry in total money transactions is relatively high in prosperity. This sector uses money much more “efficiently” than, say, consumers, whose share declines in prosperity.
M. J. Bronfenbrenner and T. Mayer, op. cit., p. 829. These investigators may have been led to de-emphasize the influence of the wealth variable because of the currently prevalent argument that desired money holdings may not be directly a function of the cost of holding money (the interest income forgone) but instead a function of the value of total wealth held, with the market value of wealth itself an inverse function of the level of interest rates. If so, the wealth variable would in part duplicate the interest variable. Because the interest rate presumably has little influence on the market value of wealth, and because many wealth holders may be affected more by the cost or face value than by the current market value of their assets, the concern over this duplication seems exaggerated.
Minor refinements include allowances for Treasury and other deposits in the Federal Reserve System. Currency held by the member banks as vault cash will be shown to be of minor significance. (Such currency is now counted in legal reserves. Allowance for it would not greatly affect the size of the elasticity—see footnote 14, below—since any portion that the banks are not certain will remain available to meet reserve requirements is probably, in the aggregate, a small and stable fraction of deposits.) When deposits are expanding to levels not previously attained, the banks may feel obliged to raise new capital, and they must use a small fraction of such capital to purchase Federal Reserve shares. This hidden “reserve requirement” is very small, however, and in any case has only limited relevance where there is a tendency for deposits to fall below previous levels. Nonmember bank deposits at the Federal Reserve are unimportant in themselves but introduce the question of the one sixth of the public’s demand deposits that are held in nonmember banks. Allowance for these is made difficult because of the uncertainties about, and inconsistencies in, the reserve requirements of these banks. (In some cases, requirements are equal to those set by the Federal Reserve but are satisfied by deposits maintained in large-city commercial banks.) These banks tend to be small and remote from commercial or financial centers. It is plausible that they do not adjust their credit supplies to the interest rate as sensitively as do member banks. Furthermore, their depositors likewise may tend to have exceptionally low interest elasticities of demand for money. Because of these considerations, the one sixth of the public’s demand deposits held in nonmember banks is disregarded here.
The time and savings components of the unchanging deposits, OD0, may seem to require smaller free reserve ratios, at any given level of interest rates, than do the demand deposits held by the public, DDs. However, the unchanging deposits include interbank demand deposits, and these may require very high free reserves because they tend to be the deposit-owning bank’s first line of reserves. In the absence of means for distinguishing the free reserve ratios associated with OD0 and with DDs, it therefore seems permissible to assume that they are equal. The interest elasticity would not be greatly affected if the observed free reserves were allocated under the assumption that frDD = 2 fγ0D.
The derivation of ηs is achieved by the following steps: Substituting (2) and (3) into (1):
Differentiation of (3a) yields
Dividing by (3a) and multiplying by i yield the following for the interest elasticity of supply:
From (3a) it is seen that the fraction inside the braces in (3c) equals
Derived from Federal Reserve Bulletin, June 1962, p. 709.
A. J. Meigs, op. cit., p. 75. Meigs employs monthly data, using them to estimate the speed with which banks return to their desired free reserve ratios after a change in their excess reserve holdings has occurred.
Federal Reserve Bulletin, June 1962, p. 709. The figure for each deposit group was between $92 billion and $93 billion.
As noted above, this formula does not provide for the amount of vault cash which banks must hold for operating purposes. However, as the definition of minimum required reserves has been changed to include vault cash, these holdings may reasonably be ignored; they are now part of the reserve base. Some smaller banks may not treat vault cash as a complete substitute for deposits at the Federal Reserve, because of its erratic behavior. (See “The Vault Cash Provision: Has It Changed the Way Banks Manage Their Reserves?” Federal Reserve Bank of Philadelphia, Business Review, September 1961, pp. 11-15.) Ample allowance for the extra desired “excess” reserve holdings that may now exist because of this consideration could be provided by raising the constant term, a, from .005 to .015 or .02 (Polak and White, op. cit., p. 427, fn. 30, found that the vault cash/deposits ratio had been very stable regardless of the level of the short-term interest rate, so that the constant term is the one to be modified).
With the constant term raised to .02, the elasticity is reduced from .085 to .077. This is within the range of .07-.08 found by Polak and White (op. cit., p. 428) for a period ending five years earlier than the period covered by Meigs.
The rise in the long-term rate would actually be less than the rise in the short-term rate, for economic conditions justifying high long-term rates would be absent when the short-term rate was being raised merely for transitory balance of payments reasons. The relevant total yield on long-term securities—that commensurable with the short-term rate—would be the long-term interest yield plus the value of the prospect of a capital gain to be obtained when long-term rates fall back again. (The same consideration operates to hold even cyclical rises in long-term rates below the accompanying rises in short-term rates, although not so strongly.)
This analysis of the influence of the long-term market on the elasticity of the public’s demand curve for deposits could also be applied to the elasticity of the bank’s supply of credit (of deposits), but the observed supply elasticity is already so low that any resulting changes in it would be insignificant.
A study prepared for the Commission on Money and Credit took some account of the interaction between the short- and long-term markets (by including portions of short- to medium-term and of long-term security holdings, along with varying portions of the stock of money plus time deposits, as determinants of the level of the U.S. Treasury bill rate); this permitted the making of a most probable, rather than merely an upper-limit, estimate of the necessary amount of open market sales. (See Arthur M. Okun, “Monetary Policy, Debt Management and Interest Rates: A Quantitative Appraisal,” Cowles Foundation Discussion Paper No. 125, mimeographed, preliminary, esp. 22, 34, 38-39, 43, 44.)
Two estimates of equal statistical reliability were made on the basis of different definitions of the variable representing the stock of money and time deposits. Although noting that the two estimates were “distressingly far apart,” the investigator did not express his preference in this preliminary version of his report. While the estimated open market sales derived from the definition that seems to have greater economic meaning tends to confirm the findings of the present study, the other definition indicates a need for volumes of open market sales which might be impractical.
Federal Reserve Bulletin, January 1963, p. 31. Until 1961, the large New York banks refused to pay any interest on the time deposits of business corporations. They then made time deposits especially attractive to big companies by creating a marketable interest-bearing security, the “negotiable certificate of (time) deposit.”
This is the situation in the United Kingdom. Moreover, if the Bank rate is raised by the amount of the desired rise in Treasury bill rates, the yield on U.K. time deposits (which is linked to Bank rate) will rise by substantially as much as the yield on Treasury bills, so that time deposits can be expected to claim a good share of the funds diverted from British demand deposits. Other things equal, therefore, the amount of open market sales required to offset the consequences of lower demand deposits will be even smaller in the United Kingdom than in the United States.
Less restriction than in the United States is also indicated for Germany, even though German reserve requirements are (somewhat) lower for time than for demand deposits: First, a major part of the reduction in demand deposits must be reflected in transfers to time deposits because (Lutz, op. cit., p. 298) the domestic nonbank public in Germany has little access to other money market investments (except long-term bonds). Second, the yield on German banks’ time deposits rises less than the Bundesbank discount rate, and the net yield obtainable rises still less because the demand deposits themselves receive a (low) interest rate that is linked to the discount rate. A given rise in the Treasury bill rate therefore has an attenuated effect on the size of desired demand deposits even with the same rise for the discount rate.
The reduction of the effect on demand deposits may be counterbalanced by the appearance of an effect on currency: those financially sophisticated enough to be influenced by the level of interest rates obtainable on idle money may hold some of their idle money in the form of currency when the rate of interest is low, because demand deposits in Germany lack the attraction of being a convenient means of making check payments. And a shift from currency to time deposits strengthens the banks’ reserve position by almost the full amount of the funds shifted.
Other relevant considerations are the influence of the development of other convenient and safe short-term outlets for funds (such as finance company commercial paper) which compete with the banks’ negotiable certificates of time deposit, and the possible effects on the demand for currency of the high rates on savings deposits now obtainable when money is made tight.