on highly liquid substitutes for currency; since the currency ratio negatively affects the money multiplier, the money multiplier is positively affected by the return on these substitutes. The formula above is derived from the following procedure. Let the monetary base be normalized to unity. Define the legal reserve ratio, , the excess reserves ratio, , the currency drain ratio with respect to deposits, ; suppose the demand for funds is unlimited; then the theoretical superior limit for deposits is defined by the following series: . Analogously, the theoretical superior limit for the money held by public is defined by the following series: and the theoretical superior limit for the total loans lent in the market is defined by the following series: By summing up the two quantities, the theoretical money multiplier is defined as where and The process described above by the geometric series can be represented in the following table, where loans at stage are a function of the deposits at the preceding stage: publicly held money at stage is a function of the deposits at the preceding stage: deposits at stage are the difference between additional loans and publicly held money relative to the same stage: Table This re-lending process (with no currency drain) can be depicted as follows, assuming a 20% reserve ratio and a $100 initial deposit: Note that no matter how many times the smaller and smaller amounts of money are re-lended, the legal reserve requirement is never exceeded - because that would be illegal. Loans first model In the alternative model of money creation, loans are first extended by commercial banks – say, $1,000 of loans (following the example above), which may then require that the bank borrow $100 of reserves either from depositors (or other private sources of financing), or from the central bank. This view is advanced in endogenous money theories, such as the Post-Keynesian school of monetary circuit theory, as advanced by such economists as Basil Moore and Steve Keen. Finn E. Kydland and Edward C. Prescott argue that there is no evidence that either the monetary base or M1 leads the cycle. Jaromir Benes and Michael Kumhof of the IMF Research Department, argue that: the “deposit multiplier“ of the undergraduate economics textbook, where monetary aggregates are created at the initiative of the central bank, through an initial injection of high-powered money into the banking system that gets multiplied through bank