A Stochastic Model of the Competitive Banking Firm with Lagged and Contemporaneous Reserve Accounting: Note

Abstract

In 1984 the Board of Governors of the Federal Reserve System restored contemporaneous reserve accounting (CRA). Lagged reserve accounting (LRA) had been instituted in September 1968. Briefly, under a contemporaneous accounting system, member banks are required to hold a certain fraction (the required reserve ratio) of current deposits as reserves during each one-week accounting period.' With the lagged accounting system, the current level of required reserves is calculated as a fraction of the deposit liabilities held by the bank two weeks earlier. The literature focuses almost exclusively on the macro policy issue: will the return to CRA enable the Federal Reserve to exercise greater control over the money stock? In this paper the primary question is whether a bank will prefer LRA to CRA, based on the criterion of expected profits. This model is the simplest which can be used to illustrate the fundamental difference between LRA and CRA: changes in deposits affect both total and required reserves with CRA, but only total reserves with LRA. It is assumed that the representative banking firm is a risk-neutral perfect competitor, which maximizes the expected value of profits E(X). The firm earns an interest rate r on loans and is subject to a required reserve ratio p. Failure to meet reserve requirements means the firm must borrow reserves at a penalty rate rp and is subject to a lump sum penalty cost M; end-of-period free reserves are loaned out at a rate rs. Changes in deposits, I\Dt, are a random variable which is uniformly distributed with a mean of zero; the firm selects a reserve ratio p, which maximizes E(7r). This model represents one accounting period for the banking firm. At the beginning of the period (i.e., Thursday morning) the bank has deposit liabilities D, and selects an optimal reserve ratio p. Since the expected intraperiod change in deposits (/\Dt) is zero, pDt is also the expected value of end-of-period reserves. By assumption, the bank's loan portfolio cannot be adjusted during the period, therefore changes in deposits are matched by changes in reserves. The profit-maximizing strategy involves maintaining the optimal level of reserves: a high reserve ratio