Money multiplier
Encyclopedia
In monetary economics, a money multiplier is one of various closely related ratios of commercial bank money to central bank money under a fractional-reserve banking
system. Most often, it measures the maximum amount of commercial bank money that can be created by a given unit of central bank money. 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) is a multiple of reserves; this multiple is the reciprocal
of the reserve ratio, and it is an economic multiplier
.
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
the fraction of reserves must be at least the reserve ratio. Taking the reciprocal, 
which yields 
meaning that commercial bank money is at most reserves times 
the latter being the multiplier.
If banks lend out close to the maximum allowed by their reserves, 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.
In the United States since 1959, banks lent out close to the maximum allowed for the 49-year period from 1959 until August 2008, maintaining a low level of excess reserves, then accumulated significant excess reserves over the period September 2008 through the present (November 2009). Thus, in the first period, commercial bank money was almost exactly central bank money times the multiplier, but this relationship broke down from September 2008.
As a formula and legal quantity, the money multiplier is not controversial – it is simply the maximum that commercial banks are allowed to lend out. However, there are various heterodox
theories concerning the mechanism of money creation
in a fractional-reserve banking system, and the implication for monetary policy
.
of "commercial bank money"/"central bank money", based on the actual observed quantities of various empirical measures of money supply, 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,
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). 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
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.
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.
textbooks, while the "loans first" model is advanced by endogenous money
theorists.
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 and the ultimate result being a geometric series.
This formula stems from the fact that the sum of the "amount loaned out" column above can be expressed mathematically as a geometric series with a common ratio of
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:
can be used, where Currency Drain Ratio is the percentage of money that people want to hold as cash and the Desired Reserve Ratio is the sum of the Required Reserve Ratio and the Excess Reserve Ratio.
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 totale 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
So then the money multiplier, m, will be calculated as:
This number is multiplied by the initial deposit to show the maximum amount of money it can be expanded to.
Another way to look at the monetary multiplier is derived from the concept of money supply and money base. It is the number of dollars of money supply that can be created for every dollar of monetary base. Money supply, denoted by M, is the stock of money held by public. It is measured by the amount of currency and deposits. Money Base, denoted by B, is the summation of currency and reserves. Currency and Reserves are monetary policy that can be affected by the Federal Reserve. For example, the Federal Reserve can increase currency by printing more money and they can similarly increase reserve by requiring a higher percentage of deposits to be stored in the Federal Reserve.
Mathematically:
M=Money Supply
C=Currency
D=Deposits
B=Money Base
R=Reserve
So that money supply over money base:
Multiply the right side by
. Since this equals to 1, it is mathematically justified to multiply it to only the right side.
Then multiple the right side of the equation by the Money Base
So we get:
is the multiplier. Therefore, if money base is held constant, the ratio of D/R and D/C affects the money supply. When the ratio of deposits to reserves (D/R) reduces, the multiplier reduces. Similarly, if the ratio of deposits to currency (D/C) falls, the multiplier falls as well.
This can be proved mathematically as follows. Letting
and 
for notational conciseness,
the money supply can be written as a function of two variables
defined over the domain
. The partial derivatives of M
with respect to both variables are positive, implying that this function
is marginally increasing (i.e., considered as a function of either
variable keeping the other constant) over its domain in both of its
variables. Note that the domain of definition, which is determined by
economic considerations, is crucial in establishing both the
differentiability and the marginal monotonicity of
.
The multiplier effect is relevant to considering monetary and fiscal policies, as well how the banking system works. For example, the deposit, the monetary amount a customer deposits at a bank, is used by the bank to loan out to others, thereby generating the money supply. Most banks are FDIC insured (Federal Deposit Insurance Corporation), so that customers are assured that their savings, up to a certain amount, is insured by the federal government. Banks are required to reserve a certain ratio of the customer's deposits in reserve, either in the form of vault cash or of a deposit maintained by a Federal Reserve Bank.http://www.federalreserve.gov/monetarypolicy/reservereq.htm#table1 . Therefore, if the Federal Reserve Bank (and hence its monetary policy) requires a higher percentage of reserve, then it lowers the bank's financial ability to loan.
theories, such as the Post-Keynesian school of monetary circuit theory
, as advanced by such economists as Basil Moore
and Steve Keen
.
, and the distinction between the multiplier being the maximum amount of commercial bank money created by a given unit of central bank money and approximately equal to the amount created has important implications in monetary policy.
If banks maintain low levels of excess reserves, as they did in the US from 1959 to August 2008, then central banks can finely control broad (commercial bank) money supply by controlling central bank money creation, as the multiplier gives a direct and fixed connection between these.
If, on the other hand, banks accumulate excess reserves, as occurs in some financial crises such as the Great Depression
and the Financial crisis of 2007–2010, then this relationship breaks down and central banks can force the broad money supply to shrink, but not force it to grow:
Restated, increases in central bank money may not result in commercial bank money because the money is not required to be lent out – it may instead result in a growth of unlent reserves (excess reserves
). This situation is referred to as "pushing on a string": withdrawal of central bank money compels commercial banks to curtail lending (one can pull money via this mechanism), but input of central bank money does not compel commercial banks to lend (one cannot push via this mechanism).
This described growth in excess reserves has indeed occurred in the Financial crisis of 2007–2010, US bank excess reserves growing over 500-fold, from under $2 billion in August 2008 to over $1,000 billion in November 2009.
Fractional-reserve banking
Fractional-reserve banking is a form of banking where banks maintain reserves that are only a fraction of the customer's deposits. Funds deposited into a bank are mostly lent out, and a bank keeps only a fraction of the quantity of deposits as reserves...
system. Most often, it measures the maximum amount of commercial bank money that can be created by a given unit of central bank money. 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) is a multiple of reserves; this multiple is the reciprocal
Multiplicative inverse
In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction a/b is b/a. For the multiplicative inverse of a real number, divide 1 by the...
of the reserve ratio, and it is an economic multiplier
Multiplier (economics)
In economics, the fiscal multiplier is the ratio of a change in national income to the change in government spending that causes it. More generally, the exogenous spending multiplier is the ratio of a change in national income to any autonomous change in spending In economics, the fiscal...
.
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
the fraction of reserves must be at least the reserve ratio. Taking the reciprocal, which yields meaning that commercial bank money is at most reserves times the latter being the multiplier.If banks lend out close to the maximum allowed by their reserves, 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
Excess reserves
In banking, excess reserves are bank reserves in excess of the reserve requirement set by a central bank. They are reserves of cash more than the required amounts. Holding excess reserves is generally considered costly and uneconomical as no interest is earned on the excess amount...
, then commercial bank money will be less than central bank money times the theoretical multiplier.
In the United States since 1959, banks lent out close to the maximum allowed for the 49-year period from 1959 until August 2008, maintaining a low level of excess reserves, then accumulated significant excess reserves over the period September 2008 through the present (November 2009). Thus, in the first period, commercial bank money was almost exactly central bank money times the multiplier, but this relationship broke down from September 2008.
As a formula and legal quantity, the money multiplier is not controversial – it is simply the maximum that commercial banks are allowed to lend out. However, there are various heterodox
Heterodox economics
"Heterodox economics" refers to approaches or to schools of economic thought that are considered outside of "mainstream economics". Mainstream economists sometimes assert that it has little or no influence on the vast majority of academic economists in the English speaking world. "Mainstream...
theories concerning the mechanism of money creation
Money creation
In economics, money creation is the process by which the money supply of a country or a monetary region is increased due to some reason. There are two principal stages of money creation. First, the central bank introduces new money into the economy by purchasing financial assets or lending money...
in a fractional-reserve banking system, and the implication for monetary policy
Monetary policy
Monetary policy is the process by which the monetary authority of a country controls the supply of money, often targeting a rate of interest for the purpose of promoting economic growth and stability. The official goals usually include relatively stable prices and low unemployment...
.
Definition
The money multiplier is defined in various ways. Most simply, it can be defined either as the statisticStatistic
A statistic is a single measure of some attribute of a sample . It is calculated by applying a function to the values of the items comprising the sample which are known together as a set of data.More formally, statistical theory defines a statistic as a function of a sample where the function...
of "commercial bank money"/"central bank money", based on the actual observed quantities of various empirical measures of money supply, 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,
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). 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
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.
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.
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 economicsMainstream economics
Mainstream economics is a loose term used to refer to widely-accepted economics as taught in prominent universities and in contrast to heterodox economics...
textbooks, while the "loans first" model is advanced by endogenous money
Endogenous money
In economics, endogenous money refers to the theory that money comes into existence driven by the requirements of the real economy and that banking system reserves expand or contract as needed to accommodate loan demand at prevailing interest rates. It forms part of Post-Keynesian economics...
theorists.
Reserves first model
Formula
The money multiplier, m, is the inverse of the reserve requirement, 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 with a common ratio of
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:
can be used, where Currency Drain Ratio is the percentage of money that people want to hold as cash and the Desired Reserve Ratio is the sum of the Required Reserve Ratio and the Excess Reserve Ratio.
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 totale 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 precedent stage: 
- publicly held money at stage
is a function of the deposits at the precedent stage: 
- deposits at stage
are the difference between additional loans and publicly held money relative to the same stage: 
 |
Deposits | Loans | Publicly Held Money |
|---|---|---|---|
 |
 |
- | - |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
| … | … | … | … |
 |
 |
 |
 |
| … | … | … | … |
 |
 |
 |
 |
| |
|
|
|
| |
Total Deposits: | Total Loans: | Total Publicly Held Money: |
| |
 |
 |
 |
Table
This re-lending process (with no currency drain) can be depicted as follows, assuming a 20% reserve ratio and a $100 initial deposit:| Individual Bank | Amount Deposited | Lent Out | Reserves |
|---|---|---|---|
| A | 100.00 | 80.00 | 20.00 |
| B | 80.00 | 64.00 | 16.00 |
| C | 64.00 | 51.20 | 12.80 |
| D | 51.20 | 40.96 | 10.24 |
| E | 40.96 | 32.77 | 8.19 |
| F | 32.77 | 26.21 | 6.55 |
| G | 26.21 | 20.97 | 5.24 |
| H | 20.97 | 16.78 | 4.19 |
| I | 16.78 | 13.42 | 3.36 |
| J | 13.42 | 10.74 | 2.68 |
| K | 10.74 | |
|
| |
|
|
Total Reserves: |
| |
|
|
89.26 |
| |
Total Amount of Deposits: | Total Amount Lent Out: | Total Reserves + Last Amount Deposited: |
| |
457.05 | 357.05 | 100.00 |
Example
For example, with the reserve ratio of 20 percent, this reserve ratio, RR, can also be expressed as a fraction:So then the money multiplier, m, will be calculated as:
This number is multiplied by the initial deposit to show the maximum amount of money it can be expanded to.
Another way to look at the monetary multiplier is derived from the concept of money supply and money base. It is the number of dollars of money supply that can be created for every dollar of monetary base. Money supply, denoted by M, is the stock of money held by public. It is measured by the amount of currency and deposits. Money Base, denoted by B, is the summation of currency and reserves. Currency and Reserves are monetary policy that can be affected by the Federal Reserve. For example, the Federal Reserve can increase currency by printing more money and they can similarly increase reserve by requiring a higher percentage of deposits to be stored in the Federal Reserve.
Mathematically:
M=Money Supply
C=Currency
D=Deposits
B=Money Base
R=Reserve
So that money supply over money base:
Multiply the right side by
. Since this equals to 1, it is mathematically justified to multiply it to only the right side.Then multiple the right side of the equation by the Money Base
So we get:
is the multiplier. Therefore, if money base is held constant, the ratio of D/R and D/C affects the money supply. When the ratio of deposits to reserves (D/R) reduces, the multiplier reduces. Similarly, if the ratio of deposits to currency (D/C) falls, the multiplier falls as well.This can be proved mathematically as follows. Letting
and for notational conciseness,the money supply can be written as a function of two variables
defined over the domain. The partial derivatives of Mwith respect to both variables are positive, implying that this function
is marginally increasing (i.e., considered as a function of either
variable keeping the other constant) over its domain in both of its
variables. Note that the domain of definition, which is determined by
economic considerations, is crucial in establishing both the
differentiability and the marginal monotonicity of
.
The multiplier effect is relevant to considering monetary and fiscal policies, as well how the banking system works. For example, the deposit, the monetary amount a customer deposits at a bank, is used by the bank to loan out to others, thereby generating the money supply. Most banks are FDIC insured (Federal Deposit Insurance Corporation), so that customers are assured that their savings, up to a certain amount, is insured by the federal government. Banks are required to reserve a certain ratio of the customer's deposits in reserve, either in the form of vault cash or of a deposit maintained by a Federal Reserve Bank.http://www.federalreserve.gov/monetarypolicy/reservereq.htm#table1 . Therefore, if the Federal Reserve Bank (and hence its monetary policy) requires a higher percentage of reserve, then it lowers the bank's financial ability to loan.
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. However it is also possible that the loan is used to pay another bank customer who deposits it back with the bank, thus negating the need for any form of reserve to be borrowed by the bank. This view is advanced in endogenous moneyEndogenous money
In economics, endogenous money refers to the theory that money comes into existence driven by the requirements of the real economy and that banking system reserves expand or contract as needed to accommodate loan demand at prevailing interest rates. It forms part of Post-Keynesian economics...
theories, such as the Post-Keynesian school of monetary circuit theory
Monetary circuit theory
Monetary circuit theory is a heterodox theory of monetary economics, particularly money creation, often associated with the post-Keynesian school....
, as advanced by such economists as Basil Moore
Basil Moore (economist)
Basil Moore is a Canadian Post-Keynesian economist, most known for developing and promoting endogenous money theory, particularly the proposition that the money supply curve is horizontal, rather than upward sloping, a proposition known as horizontalism...
and Steve Keen
Steve Keen
Steve Keen is a Professor in economics and finance at the University of Western Sydney. He classes himself as a post-Keynesian, criticizing both modern neoclassical economics and Marxian economics as inconsistent, unscientific and empirically unsupported...
.
Implications for monetary policy
The multiplier plays a key role in monetary policyMonetary policy
Monetary policy is the process by which the monetary authority of a country controls the supply of money, often targeting a rate of interest for the purpose of promoting economic growth and stability. The official goals usually include relatively stable prices and low unemployment...
, and the distinction between the multiplier being the maximum amount of commercial bank money created by a given unit of central bank money and approximately equal to the amount created has important implications in monetary policy.
If banks maintain low levels of excess reserves, as they did in the US from 1959 to August 2008, then central banks can finely control broad (commercial bank) money supply by controlling central bank money creation, as the multiplier gives a direct and fixed connection between these.
If, on the other hand, banks accumulate excess reserves, as occurs in some financial crises such as the Great Depression
Great Depression
The Great Depression was a severe worldwide economic depression in the decade preceding World War II. The timing of the Great Depression varied across nations, but in most countries it started in about 1929 and lasted until the late 1930s or early 1940s...
and the Financial crisis of 2007–2010, then this relationship breaks down and central banks can force the broad money supply to shrink, but not force it to grow:
Restated, increases in central bank money may not result in commercial bank money because the money is not required to be lent out – it may instead result in a growth of unlent reserves (excess reserves
Excess reserves
In banking, excess reserves are bank reserves in excess of the reserve requirement set by a central bank. They are reserves of cash more than the required amounts. Holding excess reserves is generally considered costly and uneconomical as no interest is earned on the excess amount...
). This situation is referred to as "pushing on a string": withdrawal of central bank money compels commercial banks to curtail lending (one can pull money via this mechanism), but input of central bank money does not compel commercial banks to lend (one cannot push via this mechanism).
This described growth in excess reserves has indeed occurred in the Financial crisis of 2007–2010, US bank excess reserves growing over 500-fold, from under $2 billion in August 2008 to over $1,000 billion in November 2009.