Support?

Money multiplier

29–30 June 2012

What happens with the money?

At the end of the previous episode the situation was that Villager B had 90 dollars in his purse, borrowed from the bank. The bank could afford that, because Villager A had previously deposited $100. $10 are kept in the bank’s vaults or tills as a cash reserve.

Now Villager B wonders what to do with that money. Perhaps he doesn’t need to buy anything after all. Why not put it back in a checking account? So it doesn’t get stolen or spent on something silly and unnecessary.

Or maybe Villager B does urgently need to go shopping and spends all the money in the supermarket. Then what will the supermarket manager do with it? Knowing that salaries only need to be paid next week, and the supplier the day after tomorrow? Perhaps the supermarket manager will put that money in the bank, instead of Villager B.

(Do I need to mention that wherever I write ‘he’ I equally mean ‘she’? OK, so I don’t mention it.)

Or the supermarket manager does pay his personnel and supplier, and they deposit the money in a bank. (In the bank, being the only one in our hypothetical village. But that doesn’t really change the situation when we look at the whole of this economy.)

So whatever happens, chances are that a large part of those 90 dollars eventually returns to the bank. The bank manager likes this, because he can use that money for new loans.

But at all times, the minimum cash reserve, as required by the central bank, needs to be maintained. If that percentage is 10%, only 90% percent can be re-used for new loans. That is the same as saying that the amount of cash in the bank must at least be 10% of the total of all the money in demand deposits (checking accounts).

Balance sheet

Step 3

Let’s see what that does to the bank’s balance sheet:

Description Debit (assets) Credit (liabilities)
Cash 100
Checking account of Villager A 100
Checking account of Villager B 90
Loan facility of Villager B 90

Step 4

After re-using the excess reserves for granting a new credit (81 dollars):

Description Debit (assets) Credit (liabilities)
Cash 19
Checking account of Villager A 100
Checking account of Villager B 90
Loan facility of Villager B 90
Loan facility of Villager C 81

It is still true that 19 >= (100 + 90) * 10%. So the limit of lending under the fractional reserve banking system (which this typically is) has again been reached and isn’t violated.

This process can and will be repeated: part or all of the 81 dollars will also return to the bank (or to some other bank, but eventually only the totals matter).

I’ll show some more steps, assuming all of the borrowed money returns:

Step 5

Description Debit (assets) Credit (liabilities)
Cash 100
Checking account of Villager A 100
Checking account of Villager B 90
Checking account of Villager C 81
Loan facility of Villager B 90
Loan facility of Villager C 81

Step 6

New loan of 100 − ((100+90+81) * 10%) = 100 − 27.10 = 72.90.

Description Debit (assets) Credit (liabilities)
Cash 27.10
Checking account of Villager A 100
Checking account of Villager B 90
Checking account of Villager C 81
Loan facility of Villager B 90
Loan facility of Villager C 81
Loan facility of Villager D 72.90

Step 7

Loan returns to checking account:

Description Debit (assets) Credit (liabilities)
Cash 100
Checking account of Villager A 100
Checking account of Villager B 90
Checking account of Villager C 81
Checking account of Villager D 72.90
Loan facility of Villager B 90
Loan facility of Villager C 81
Loan facility of Villager D 72.90

Step 8

New loan of 100 − ((100+90+81+72.90) * 10%) = 100 − 34.39 = 65.61.

Description Debit (assets) Credit (liabilities)
Cash 34.39
Checking account of Villager A 100
Checking account of Villager B 90
Checking account of Villager C 81
Checking account of Villager D 72.90
Loan facility of Villager B 90
Loan facility of Villager C 81
Loan facility of Villager D 72.90
Loan facility of Villager E 65.61

Recursion

This process can be repeated over and over: all the money released into the economy can eventually return into a deposit account. But the amount of the new loans gets ever smaller. Meanwhile the amounts are cumulated in ever more accounts in the bank’s books.

What this results in, can be derived mathematically. You could also just repeat the calculation steps and see what happens. That’s a lot of work, so I used a little C program to have the computer do it for me:

#include 
#define RESERVE 10.0
#define ORIG   100.0

int main (void)
{
  double add = ORIG;
  double cumul = ORIG;

  while (add > 0.001)
  {
    add *= 1 - (RESERVE / 100.0);
    cumul += add;
    printf(
      "Cumul = %7.2f, add = %7.2f, 100 - add = %7.2f\n", 
      cumul, add, ORIG - add);
  }
  return 0;
}

We then see that eventually the situation is as follows:

Cash 100
Checking accounts 1000
Loan facilities 900

To make it simpler, I added up all the individual accounts of the same type. We see that the cash amount of $100 is back in the bank. The bank has granted various credits for a total of $900. And all the villagers together have one thousand dollar in demand deposits, that is in checking accounts.

The original amount was 100, but there are now 1000 dollars in bank accounts. That is the money creation blow-up effect. A factor of 10, in this example. The reserve requirement was 10 percent.

That is no coincidence. If the reserve requirement were 20 percent, the factor would be 5. With 2 percent, you get a maximum of 50 times the original amount.

Cash in pocket

I said a maximum of 50. It’s really a maximum. In practice, the actual multiplication effect is far less. One reason for that is that not all the money returns to the bank. People keep some of it in their purses as cash. If we take that into account, assuming 5% of loans is not eventually deposited, the situation in step 3 above will instead look like this:

Description Debit (assets) Credit (liabilities)
Cash 95.50
Checking account of Villager A 100
Checking account of Villager B 85.50
Loan facility of Villager B 90

The maximum loan the bank may now grant, under the reserve requirement of 10%, is: 95.50 − ((100 + 85.50) * 10%) = 76.95. That’s less than the 81 dollar possible under the assumption that people keep no cash.

After iterating this calculation many times too, I found that the eventual situation under these conditions is this:

Cash 68.97
Checking accounts 689.66
Loan facilities 620.69

The total amount of cash in people’s wallets is 31.03. Together with the 68.97 in the bank that accounts for the total of $100 in cash that we started with. 31.03 is indeed 5% of 620.69, which was what I assumed to start with.

So my calculation is correct. I also checked several other combinations of percentages. For that I used a C program, but no doubt this too can be done by deriving a mathematical formula. It’s just that my maths are not good enough for that. Or I am lazy.

Evaluation

Are these results problematic, worrying, dangerous, evil? I’ll look at that in the next article.


Copyright © 2012 R. Harmsen. All rights reserved.

Colours: Neutral Weird No preference Reload screen