# Money multiplier

In monetary economics, a money multiplier is one of various closely related ratios of commercial bank money to central bank money (also called the monetary base) under a fractional-reserve banking system.[1] In one version it measures the maximum amount of commercial bank money that can be created, given a certain amount of central bank money and ignoring leakages into currency held by the non-bank public. That is, in a fractional-reserve banking system, the total amount of loans that commercial banks are allowed to extend (the commercial bank money that they can legally create) when there are no leakages is equal to a multiple of the amount of reserves. This multiple is the reciprocal of the reserve ratio, and it is an economic multiplier.[2] The actual ratio of money to central bank money, also called the money multiplier, is lower because some funds are held by the non-bank public as currency and most banks hold excess reserves (reserves above the amount required by the central bank)

Although the money multiplier concept is a traditional portrayal of fractional reserve banking, it has been criticized as being misleading. The Bank of England and the Standard & Poor's rating agency (amongst others) have issued detailed refutations of the concept together with factual descriptions of banking operations.[3][4] Several countries (such as Canada, the UK, Australia and Sweden) set no legal reserve requirements.[5] Even in those countries that do (such as the USA), the reserve requirement is as a ratio to deposits held, not a ratio to loans that can be extended.[5] [6] Basel III does stipulate a liquidity requirement to cover 30 days net cash outflow expected under a modeled stressed scenario (note this is not a ratio to loans that can be extended); however, liquidity coverage does not need to be held as reserves but rather as any high-quality liquid assets [7][8]

In equations, writing M for commercial bank money (loans), R for reserves (central bank money), and RR for the reserve ratio, the reserve ratio requirement is that ${\displaystyle R/M\geq RR;}$ the fraction of reserves must be at least the reserve ratio. Taking the reciprocal, ${\displaystyle M/R\leq 1/RR,}$ which yields ${\displaystyle M\leq R\times (1/RR),}$ meaning that commercial bank money is at most reserves times ${\displaystyle (1/RR),}$ the latter being the multiplier ignoring leakages into currency.

If banks lend out close to the maximum allowed by their reserves and there are no leakages into currency holdings, then the inequality becomes an approximate equality, and commercial bank money is central bank money times the multiplier. If banks instead lend less than the maximum, accumulating excess reserves, then commercial bank money will be less than central bank money times the theoretical multiplier.

## Definition

The money multiplier is defined in various ways.[1] Most simply, it can be defined either as the statistic of "commercial bank money"/"central bank money", based on the actual observed quantities of various empirical measures of money supply,[9] such as M2 (broad money) over M0 (base money), or it can be the theoretical "maximum commercial bank money/central bank money" ratio, defined as the reciprocal of the reserve ratio, ${\displaystyle 1/RR.}$ [2] The multiplier in the first (statistic) sense fluctuates continuously based on changes in commercial bank money and central bank money (though it is at most the theoretical multiplier), while the multiplier in the second (legal) sense depends only on the reserve ratio, and thus does not change unless the law changes.

For purposes of monetary policy, what is of most interest is the predicted impact of changes in central bank money on commercial bank money, and in various models of monetary creation, the associated multiple (the ratio of these two changes) is called the money multiplier (associated to that model).[10] For example, if one assumes that people hold a constant fraction of deposits as cash, one may add a "currency drain" variable (currency–deposit ratio), and obtain a multiplier of ${\displaystyle (1+CD)/(RR+CD).}$

These concepts are not generally distinguished by different names; if one wishes to distinguish them, one may gloss them by names such as empirical (or observed) multiplier, legal (or theoretical) multiplier, or model multiplier, but these are not standard usages.[9]

Similarly, one may distinguish the observed reserve–deposit ratio from the legal (minimum) reserve ratio, and the observed currency–deposit ratio from an assumed model one. Note that in this case the reserve–deposit ratio and currency–deposit ratio are outputs of observations, and fluctuate over time. If one then uses these observed ratios as model parameters (inputs) for the predictions of effects of monetary policy and assumes that they remain constant, computing a constant multiplier, the resulting predictions are valid only if these ratios do not in fact change. Sometimes this holds, and sometimes it does not; for example, increases in central bank money may result in increases in commercial bank money – and will, if these ratios (and thus multiplier) stay constant – or may result in increases in excess reserves but little or no change in commercial bank money, in which case the reserve–deposit ratio will grow and the multiplier will fall.[11]

## Mechanism

There are two suggested mechanisms for how money creation occurs in a fractional-reserve banking system: either reserves are first injected by the central bank, and then lent on by the commercial banks, or loans are first extended by commercial banks, and then backed by reserves borrowed from the central bank. The "reserves first" model is that taught in mainstream economics textbooks,[1][2] while the "loans first" model is advanced by endogenous money theorists.

### Reserves first model

In the "reserves first" model of money creation, a given reserve is lent out by a bank, then deposited at a bank (possibly different), which is then lent out again, the process repeating[2] and the ultimate result being a geometric series.

#### Formula if there are no currency leakages

The money multiplier, m, is the inverse of the reserve requirement, RR:[2]

${\displaystyle m={\frac {1}{RR}}}$

This formula stems from the fact that the sum of the "amount loaned out" column above can be expressed mathematically as a geometric series[12] with a common ratio of ${\displaystyle 1-RR.}$

#### General formula

To correct for currency drain (a lessening of the impact of monetary policy due to peoples' desire to hold some currency in the form of cash) and for banks' desire to hold reserves in excess of the required amount, the formula:

${\displaystyle m={\frac {(1+CurrencyDrainRatio)}{(CurrencyDrainRatio+DesiredReserveRatio)}}}$

can be used, where "Currency Drain Ratio" is the ratio of cash to deposits, i.e. C/D, and the Desired Reserve Ratio is the sum of the Required Reserve Ratio and the Excess Reserve Ratio.[10]

The desired reserve ratio is the amount of its assets that a bank chooses to hold as excess and required reserves; it is a decreasing function of the amount by which the market rate for loans to the non-bank public from banks exceeds the interest rate on excess reserves and of the amount by which the federal funds rate exceeds the interest rate on excess reserves. Since the money multiplier in turn depends negatively on the desired reserve ratio, the money multiplier depends positively on these two opportunity costs. Moreover, the public’s choice of the currency drain ratio depends negatively on market rates of return 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, ${\displaystyle \alpha \in \left(0,1\right)\;}$ , the excess reserves ratio, ${\displaystyle \beta \in \left(0,1\right)\;}$ , the currency drain ratio with respect to deposits, ${\displaystyle \gamma \in \left(0,1\right)\;}$ ; suppose the demand for funds is unlimited; then the theoretical superior limit for deposits is defined by the following series:

${\displaystyle Deposits=\sum _{n=0}^{\infty }\left[\left(1-\alpha -\beta -\gamma \right)\right]^{n}={\frac {1}{\alpha +\beta +\gamma }}}$

.

Analogously, the theoretical superior limit for the money held by public is defined by the following series:

${\displaystyle PubliclyHeldCurrency=\gamma \cdot Deposits={\frac {\gamma }{\alpha +\beta +\gamma }}}$

and the theoretical superior limit for the total loans lent in the market is defined by the following series:

${\displaystyle Loans=\left(1-\alpha -\beta \right)\cdot Deposits={\frac {1-\alpha -\beta }{\alpha +\beta +\gamma }}}$

By summing up the two quantities, the theoretical money multiplier is defined as

${\displaystyle m={\frac {MoneyStock}{MonetaryBase}}={\frac {Deposits+PubliclyHeldCurrency}{MonetaryBase}}={\frac {1+\gamma }{\alpha +\beta +\gamma }}}$

where ${\displaystyle \alpha +\beta =DesiredReserveRatio}$  and ${\displaystyle \gamma =CurrencyDrainRatio}$

The process described above by the geometric series can be represented in the following table, where

• loans at stage ${\displaystyle k\;}$  are a function of the deposits at the preceding stage: ${\displaystyle L_{k}=\left(1-\alpha -\beta \right)\cdot D_{k-1}}$
• publicly held money at stage ${\displaystyle k\;}$  is a function of the deposits at the preceding stage: ${\displaystyle PHM_{k}=\gamma \cdot D_{k-1}}$
• deposits at stage ${\displaystyle k\;}$  are the difference between additional loans and publicly held money relative to the same stage: ${\displaystyle D_{k}=L_{k}-PHM_{k}\;}$
Process of money multiplication
${\displaystyle n\;}$  Deposits Loans Publicly Held Money
${\displaystyle n=0\;}$  ${\displaystyle D_{0}=1\;}$  - -
${\displaystyle n=1\;}$  ${\displaystyle D_{1}=\left(1-\alpha -\beta -\gamma \right)}$  ${\displaystyle L_{1}=\left(1-\alpha -\beta \right)}$  ${\displaystyle PHM_{1}=\gamma \;}$
${\displaystyle n=2\;}$  ${\displaystyle D_{2}=\left(1-\alpha -\beta -\gamma \right)^{2}}$  ${\displaystyle L_{2}=\left(1-\alpha -\beta \right)\left(1-\alpha -\beta -\gamma \right)}$  ${\displaystyle PHM_{2}=\gamma \left(1-\alpha -\beta -\gamma \right)}$
${\displaystyle n=3\;}$  ${\displaystyle D_{3}=\left(1-\alpha -\beta -\gamma \right)^{3}}$  ${\displaystyle L_{3}=\left(1-\alpha -\beta \right)\left(1-\alpha -\beta -\gamma \right)^{2}}$  ${\displaystyle PHM_{3}=\gamma \left(1-\alpha -\beta -\gamma \right)^{2}}$
${\displaystyle n=k\;}$  ${\displaystyle D_{k}=\left(1-\alpha -\beta -\gamma \right)^{k}}$  ${\displaystyle L_{k}=\left(1-\alpha -\beta \right)\left(1-\alpha -\beta -\gamma \right)^{k-1}}$  ${\displaystyle PHM_{k}=\gamma \left(1-\alpha -\beta -\gamma \right)^{k-1}}$
${\displaystyle n\rightarrow \infty }$  ${\displaystyle D_{\infty }=0}$  ${\displaystyle L_{\infty }=0}$  ${\displaystyle PHM_{\infty }=0}$

Total Deposits: Total Loans: Total Publicly Held Money:

${\displaystyle D={\frac {1}{\alpha +\beta +\gamma }}}$  ${\displaystyle L={\frac {1-\alpha -\beta }{\alpha +\beta +\gamma }}}$  ${\displaystyle PHM={\frac {\gamma }{\alpha +\beta +\gamma }}}$