Money Multiplier Calculator

Where does money come from?

Most people, asked that, say the government prints it.

Almost all of the money you have ever used was not printed by anybody. It exists as a number in a bank's database, and it came into being when somebody borrowed.

The money multiplier is the traditional way of explaining that. You deposit a hundred, the bank keeps ten and lends ninety, that ninety gets deposited somewhere else, that bank keeps nine and lends eighty one, and so on down the chain until the original hundred has become a thousand.

This calculator works that relationship out from the three numbers it needs. It also, if you push it, shows you why the story is not quite right, which is the more interesting half of this page.

Three inputs, four outputs

1. Checkable Deposit. Money held in accounts you can spend from directly, with a unit selector beside it.
2. Reserve ratio. The percentage of deposits banks hold back rather than lend.
3. Currency in circulation. Physical notes and coins held outside banks, with its own unit selector.

Four results come back: bank reserves, the monetary base, the money supply, and the money multiplier itself.

The multiplier is reported to four decimal places, which looks fussy until you start moving the currency figure and watch it swing. Try it. Drop the currency to zero and the answer changes more than you would expect.

What each of the four numbers is

Four short lines, and each one names a different pile of money.

Bank reserves = deposits × reserve ratio

What the banks are holding back rather than lending.

Monetary base = bank reserves + currency in circulation

This is the narrow measure. Central bank money: reserves plus the physical notes and coins in wallets. Sometimes called high-powered money.

Money supply = currency in circulation + checkable deposits

The broad measure. What people can actually spend, which is mostly bank deposits rather than cash.

Money multiplier = money supply ÷ monetary base

How many units of spendable money exist for each unit of central bank money.

Notice that currency appears in both the base and the supply, while deposits appear in the supply and only their reserve slice appears in the base. That asymmetry is what makes the multiplier bigger than one, and it is also what the currency section below is about.

A thousand in deposits

Deposits 1,000, reserve ratio 10 percent, currency in circulation 200.

Four units of spendable money for every unit of central bank money.

Now open a textbook, and you will find a problem.

It will tell you the money multiplier is one divided by the reserve ratio. At 10 percent, that is 10.

We got 4.

Why it is not the number your textbook promised

The textbook version assumes something it rarely states out loud: that nobody holds cash.

Set the currency box to zero and watch:

Reserves 100, base 100, supply 1,000, multiplier 10.0000.

There it is. One divided by the reserve ratio, exactly. The simple formula is not wrong, it is a special case, and the special case is the one where every unit of money sits in a bank account and none of it is in anybody's pocket.

Which is not a world that has ever existed. Check your own wallet.

Here is the same thousand at different reserve ratios, with 200 of currency alongside:

At a 5 percent reserve ratio the textbook says 20 and the real answer is 4.8. Off by more than four times.

So when a calculator asks you for currency in circulation and a textbook does not, the calculator is the one being honest.

Cash under the mattress breaks the machine

Currency is the thing doing the damage, and it is worth seeing on its own. Deposits fixed at 1,000, reserve ratio fixed at 10 percent, only the cash moving:

From 10 down to 1.8, and nothing changed except how much cash people are holding.

Why does a note in your pocket do that?

Because it counts fully in the monetary base and fully in the money supply. It sits on both sides of the fraction at full weight, and anything appearing on both sides of a fraction at full weight drags the ratio toward one.

A deposit does not do that. It counts fully in the supply and only its ten percent reserve slice counts in the base.

So every unit that moves from a bank account into a wallet shrinks the multiplier. Cash under the mattress is not neutral. It is money withdrawn from the lending machine.

This is not just a curiosity. During a banking panic, people withdraw cash, and every withdrawal reduces the multiplier while the banks are already under strain. The behaviour that feels safest to an individual makes the system worse for everyone. Set the reserve ratio to 100 and you get a multiplier of exactly 1, which is a banking system that does no lending at all.

And then the Bank of England said it works backwards

Everything above is the standard account. It has been in textbooks for the better part of a century, and if you studied any economics it is what you were taught.

In 2014 the Bank of England published an article in its Quarterly Bulletin called Money Creation in the Modern Economy, and it said, in effect, that the standard account has the sequence the wrong way round.

Two claims in particular.

Banks do not lend out deposits. The Bank states that rather than banks receiving deposits when households save and then lending them out, bank lending creates deposits. When a bank makes a loan, it simultaneously creates a matching deposit in the borrower's account. The money did not come from a saver. It came into existence at the moment the loan was written.

Central bank money is not multiplied up. The Bank is explicit that in normal times the central bank does not fix the amount of money in circulation, and central bank money is not multiplied up into more loans and deposits. Reserves are a consequence of lending rather than the raw material for it.

Which turns the chain around entirely. The textbook says deposits come first and loans follow. The Bank of England says loans come first and deposits follow.

That is not a small correction. That is the arrow pointing the other way.

So is the number useless?

No, and it is worth being precise about what it is and is not, because the distinction decides how you should read it.

The multiplier is a perfectly good ratio. Money supply divided by monetary base is a real measurement of a real relationship, and watching it move over time tells you something genuine about how much lending is happening relative to central bank money.

What it is not is a mechanism. The ratio does not describe a process where reserves get multiplied into loans, because that is not the order things happen in.

So use it as a descriptive figure rather than a predictive one. A falling multiplier says the ratio between broad and narrow money is compressing, which happens when lending slows or when reserves balloon. A central bank creating enormous reserves through asset purchases will crush the measured multiplier without anything in the lending market having changed, and the Bank of England makes exactly this point about quantitative easing: those reserves cannot be multiplied up into additional loans and deposits.

What actually limits lending, on the Bank's account, is profitability in a competitive market, prudential regulation, and how much households and firms want to borrow. Not a reserve ratio, and not a stock of reserves waiting to be multiplied.

Which is a better answer than the textbook one, and a more complicated one. That is usually how it goes.

This is an educational calculation rather than financial advice.

Questions people ask

How is the money multiplier calculated?

Money supply divided by monetary base. The supply is currency plus deposits, and the base is currency plus bank reserves.

Why is my answer not one divided by the reserve ratio?

Because that formula assumes nobody holds cash. Set currency in circulation to zero and you get exactly one over the reserve ratio. With any cash at all, the multiplier is lower.

Why does currency reduce the multiplier?

Because a note counts at full value in both the monetary base and the money supply, so it sits on both sides of the fraction and pulls the ratio toward one. A deposit only contributes its reserve slice to the base.

What is the monetary base?

Bank reserves plus currency in circulation. It is central bank money, sometimes called high-powered money or narrow money.

What is the money supply here?

Currency in circulation plus checkable deposits, which is broad money in the sense of what people can actually spend.

What happens at a 100 percent reserve ratio?

The multiplier becomes exactly 1. Banks hold every deposit and lend nothing, so no additional money is created.

Is the money multiplier theory wrong?

The Bank of England's 2014 Quarterly Bulletin states that banks do not lend out deposits and that central bank money is not multiplied up into loans. Lending creates deposits rather than the other way around. The ratio remains a valid measurement, but it does not describe the mechanism.

Why did the multiplier fall after quantitative easing?

Because asset purchases create large quantities of reserves, which expands the monetary base without a matching expansion in lending. The Bank is explicit that those reserves cannot be multiplied up into additional loans and deposits.

References

The definitions of the monetary base, the money supply and the money multiplier as a ratio between them follow standard monetary economics and the money stock measures published by the Federal Reserve. The findings that commercial bank lending creates deposits rather than intermediating existing savings, that central bank money is not multiplied up into loans and deposits, that reserves created through quantitative easing cannot be multiplied up into additional lending, and that lending is constrained instead by profitability, prudential regulation and demand for borrowing, come from the Bank of England's Quarterly Bulletin article on money creation.

1. McLeay, M., Radia, A., and Thomas, R. (2014). Money Creation in the Modern Economy. Bank of England Quarterly Bulletin, 54(1), 14 to 27. https://www.bankofengland.co.uk/quarterly-bulletin/2014/q1/money-creation-in-the-modern-economy
2. Board of Governors of the Federal Reserve System, H.6 Money Stock Measures. https://www.federalreserve.gov/releases/h6/
3. Federal Reserve Bank of St. Louis, FRED Economic Data: Monetary Base and M1 Money Stock. https://fred.stlouisfed.org/series/BOGMBASE