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Money multiplier
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== Definition == The money multiplier is normally presented in the context of some simple [[Accounting identity|accounting identities]]:<ref name=Goodhart/><ref name="maninter">{{Harv|Mankiw|2022|loc=Chapter 4, Section 3: How Central Banks Influence the Money Supply, pp. 85β92]}}</ref> Usually, the [[money supply]] (''M'') is defined as consisting of two components: [[cash|(physical) currency]] (''C'') and [[deposit account]]s (''D'') held by the general public. By definition, therefore: <math>M = D+C.</math> Additionally, the [[monetary base]] (''B'') (also known as high-powered money) is normally defined as the sum of currency held by the general public (''C'') and the [[Bank reserves|reserves of the banking sector]] (held either as currency in the vaults of the [[commercial bank]]s or as deposits at the [[central bank]]) (''R''): <math>B = R+C.</math> Rearranging these two definitions result in a third identity: <math>M =\frac{1+C/D}{R/D + C/D}B.</math><ref name=Goodhart/> This relation describes the money supply in terms of the level of base money and two ratios: R/D is the ratio of commercial banks' reserves to deposit accounts, and C/D is the general public's ratio of currency to deposits. As the relation is an identity, it holds true by definition, so that a change in the money supply can always be expressed in terms of these three variables alone. This may be advantageous because it is a simple way of summarising money supply changes, but the use of the identity does not in itself provide a behavioural theory of what determines the money supply.<ref name=Goodhart/> If, however, one additionally assumes that the two ratios C/D and R/D are [[exogenous]]ly determined constants, the equation implies that the central bank can control the money supply by controlling the monetary base via [[open-market operations]]: In this case, when the monetary base increases by, say, $1, the money supply will increase by $(1+C/D)/(R/D + C/D). This is the central contents of the money multiplier theory, and <math>\frac{1+C/D}{R/D + C/D}</math> is the money multiplier,<ref name=Goodhart/><ref name="maninter"/> a [[multiplier (economics)|multiplier]] being a factor that measures how much an endogenous variable (in this case, the money supply) changes in response to a change in some exogenous variable (in this case, the money base). In some textbook applications, the relationship is simplified by assuming that cash does not exist so that the public holds money only in the form of bank deposits. In that case, the currency-deposit ratio C/D equals zero, and the money multipli <math>\frac{1}{R/D}.</math><ref name="krugmacro">{{Harv|Krugman|Wells|2009|loc=Chapter 14: Money, Banking, and the Federal Reserve System: Reserves, Bank Deposits, and the Money Multiplier, [https://books.google.com/books?id=dpTBdNGGrtUC&pg=PA393 pp. 393β396] }}</ref> Empirically, the money multiplier can be found as the ratio of some broad money aggregate like [[M2 (economics)|M2]] over [[M0 (economics)|M0]] (base money).<ref name="krugact">{{Harv|Krugman|Wells|2009|loc=[https://books.google.com/books?id=dpTBdNGGrtUC&pg=PA395&q=actual%20money%20multiplier p. 395]}} calls the empirically observed multiplier the "actual money multiplier".</ref>
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