Appendix
Derivation of derivatives with binding reserve requirements

Digital money is not included in the definition of M1.

Totally differentiate ( 2 ) to get

.

An increase in vault cash, VC, can either be used to expand deposits, D, digital money balances, EM, or it adds to excess reserves, E:

.

Because




, which implies that

.

By assumption




, which implies that

.

Hence,

.



Digital money is included in the definition of M1.

Totally differentiate ( 2 ) to get

.

Again

.

By assumption




which implies that

.




Notes
1. I would like to thank Winand Emons, Yvan Lengwiler, Heidi Seney, and Kevin Siegel for helpful suggestions and the Swiss National Science Foundation for financial support.
2. For an extended definition and characterization of digital money see BIS (1996a, 1996b), Stuber (1996), or CBO (1996). Literature on digital money has created many other expressions such as e-money, electronic money, network money, digital currency, electronic currency, digital cash, e-cash, etc.

3. The security of digital money has been the center of focus of central bankers. In November 1995, the central bank governors of the Group of Ten (G-10) countries commissioned studies on the security of digital money products. They published their results in BIS (1996a).

4. In the United States, M1 consists of currency held by the public, travelers checks, demand deposits, including NOW (negotiable order of withdrawal) and ATS (automatic transfer service) account balances and ahare draft account balances at credit unions (BoG, 1994).

5. The definitions of these aggregates differ from country to country. In the United States, for example, M2 consists of M1, certain overnight repurchase agreements, Eurodollars, saving deposits (including money market deposit accounts), time deposits in amounts less than $100,000, and balances in money market mutual funds (other than those restricted to institutional investors). M3 consists of M2, time deposits of $100,000 or more, certain term Eurodollars, and balances in money market mutual funds restricted to institutional investors (BoG, 1994).

6. The types of liquid assets that are counted as reserves differ from country to country. For a detailed description of the institutional differences in a number of institutional countries, consider Borio (1997).

7. In many models of the banking firm deposit rates are a function of the deposit level. That is, it is assumed that banks have some market power and their actions affect the price in the deposit market. For a discussion of this assumption and a survey banking firm models, consider Baltensperger (1980).

8. On the market for researves, banks trade reserves to meet reserve requirements and to adjust their settlement balances. The price for these reserves, in the U. S. the Federal Funds rate, is the primary interest rate that central banks influence through their monetary instruments. The substitution of central bank currency increases the total supply of reserves in the economy. Since the demand for these reserves hasn't changed, the price for these reserves must fall.

9. In principal, substitution of central bank currency is equivalent to an expansionary open market operation because both events increase the supply of reserves. Thus, the effect of these two events on the size of deposits must be equivalent. Effectiveness of monetary operations relies on the assumption that a change in reserves results in a multiple change in deposits.

10. The multiplier model we study, therefore, implicitly assumes interest rate rigidity. That is, an increase in the supply of loans or an expansion of deposits does not affact the rate of return on loans or the costs of making deposits. One explanation for this price rigidity would be interest-rate ceilings of the "regulation Q" type seen until quite recently in the U. S. (Spencer, 1986).

11. The derivations of the derivatives are in the appendix.




References
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BIS, 1996b. Implications for Central Banks of the Development of Digital Money. ****l: Bank of International Settlements.

Board of Governors (BoG) of the Federal Reserve System, 1994. The Federal Reserve System: Purpose & Functions. Washington, D. C.: Federal Reserve System.

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The Author
Aleksander Berentsen is a Visiting Scholar at the Department of Economics at the University of California, Berkeley.
http://www-vwi.unibe.ch /staff/berentsen/aleks.htm e-mail: aleksander_berentsen@bigfoot*****