An alternative framework for a textbook analysis of the money multiplier

In textbooks on the theory of money it is standard practice to hold both reserves and deposits fixed to study the relationship between the quantity of currency in the economy and the money multiplier. But doing so leads to a result that is contrary to the notion that if the public withdraws from their deposits in order to increase currency holdings then bank lending will decrease, causing a fall in both the money supply and the money multiplier. Specifically, when reserves are greater than deposits and both are fixed the money multiplier has a positive relationship with both currency holdings and the currency-deposit ratio. I show that these results are artifacts of implicitly assuming that the monetary authority behaves so as to keep deposits and reserves constant in response to a change in currency held by the public. Dropping these assumptions and abstracting from any response from the central bank results in an unconditionally non-positive relationship between currency holdings and the money multiplier.