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Highly contrived example with 3% reserve ratio:

Lets say I deposit $100 in a bank. 
Bank loans $97 PersonA, keeps $3 as reserve
PersonA buys something from PersonB for $97
PersonB deposits it in Bank, bank loans 97%of $97 to PersonC, keeps 3% in reserves.
PersonC buys something from PersonD for $94
This cycle goes on until until 100/0.03=$3333 is in deposits,
$3233 in loans and $100 in reserves.

Is it accurate to say, if a bank starts with $100 as reserves, they can loan out $3333 assuming reserve ratio is 0.03%?

In the above example, if everyone repays the loan ($3233) can bank lend it again? That doesn't sound correct. Where am I going wrong here?

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Yes with reserve requirement of $0.03\%$ the maximum amount of money that can be created will be given by $\frac{100}{0.03} = {\\\$}3333.33$, with ${\\\$} 100$ reserves and ${\\\$}3233.33$ new money.

Yes if everyone repays the ${\\\$}3233.33$ and there is still demand for loans bank can lend it again. To be clear the ${\\\$}3233.33$ will not become new reserve, so the reserve ratio would still have to be maintained. If people start paying the loans back with no demand for new loans the money supply will start contracting.

However, with presently reserve requirements being set to zero by Fed the only constraint is the demand for loans. Furthermore, while in past banks tended to hold almost no excess reserves (see this graph from Fred), after 2008 they started holding excess reserves and in such situation the multiplier idea becomes less useful abstraction as banks can always borrow extra reserves from other banks that have extra reserves if they need to. For example, if system holds excess reserves when the ${\\\$}3233.33$ would be repaid (assuming all this is done through lending and borrowing with single bank) bank could just get extra reserves from other banks holding excess reserves, even when banking system as whole would not be allowed to create new reserve (without central bank agreeing to that). McLeay, Radia, & Thomas(2014), money creation in the modern economy is good source for explaining how fractional reserve system works in such situation.

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  • $\begingroup$ Assuming this lending cycle goes on forever, wouldn't that increase money supply indefinitely? $\endgroup$ – Jean Oct 27 '20 at 16:18
  • $\begingroup$ @Jean no in either case. If there is some reserve requirement then multiplier tells you how much new money can be created in economy with existing reserves. So if there are 100 reserves and reserve requirement is 0.03 maximum money banking sector can generate is 3333.33. If there is no reserve requirement the amount of money still won’t be infinite because to create new money banks need to lend money. So creating infinite amount of new money would require people applying for infinite amount of loans - which at any normal interest rate they will most likely not. In addition bank lending is $\endgroup$ – 1muflon1♦ Oct 27 '20 at 16:25
  • $\begingroup$ further constrained by regulations. Banks for better or worse nowadays in most countries can’t lend just willy-nilly. For example, some European countries have regulations that specify how much people can borrow for mortgage given their income and banks are not allowed to lend people more then specified by law (for example I during the last summer heard at a conference that Slovakia recently implemented such restriction and I know it’s not the first European country to do this). In US lending after 2008 crisis is also more tightly regulated $\endgroup$ – 1muflon1♦ Oct 27 '20 at 16:29
  • $\begingroup$ Meaning, bank can reserve 0.03% of $3222.33 = $97 and lend out $3136.33 and start the cycle again and again? Each time increasing reserve by a bit $\endgroup$ – Jean Oct 27 '20 at 17:59
  • $\begingroup$ @Jean no that will not happen within the context of the multiplier problem. That was what I was trying to say in second part of paragraph 2. The 3233.33 dollars are still based on the original 100 dollars. In order to create more money central bank would have to issue more base money/create more reserves $\endgroup$ – 1muflon1♦ Oct 27 '20 at 18:11

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