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Banks can multiply money by using the cash from the people's deposits to give loans and receive an additional interest for the money they lent. If the minimum reserve is...say...20% the maximum ammount a bank could obtain is m=1/R namely 1 x 5/1 which equals 5. So from 100 $ they could create 500 $ by means of successive loans and deposits. However I can't understand how could this help the economy if the quantity of physical cash remains the same. So M0 remains 100$ but M1 or whatever keeps growing until reaching the limit at 500. https://en.wikipedia.org/wiki/Fractional-reserve_banking

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The reason that this helps the economy is that actual physical cash is not terribly important these days.  As electronic transmission of money and payments becomes more and more prevalent, the importance of actual physical cash wanes.  Fractional reserve banking increases the amount of money that is in the economy without necessarily changing the amount of cash.

In today’s world, we in developed countries use relatively little cash.  We get paid by the electronic transfer of virtual money into our bank accounts.  We buy things with credit cards and then pay the credit card companies with electronic transfers of virtual money out of our bank accounts.  It is very rare for us to pick up physical cash and go out and make any kind of a substantial purchase.

Modern economies do not run on cash.  They run on electronic transfers of virtual money.  For this reason, fractional reserve banking can help our economy without creating physical cash.  Fractional reserve banking does this because it creates more of the virtual money that actually drives our economy.  

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