Money and GDP Multipliers

I love geometric series. It describes so many natural phenomena especially as it relates to finance and economics. For example:

GDP mutiplier and Marginal Propensity to Consume (Save)
Marginal Propensity to Consume (MPC) is for every additional dollar of income, how much will people spend. Marginal Propensity to Save (MPS) is the opposite: for every additional dollar of after tax income how much will people save. By definition:

$1 = MPC + MPS

Different cultures will have different MPC and MPS. Americans are notorious for having high MPC (bordering on higher than $1, using financial instruments like credit cards and lines of credit to boost short term liquidity). Japanese are stereotypically savers in contrast.

However, let's assume a culture with MPC of 40%.

1. A spends $100 on B.
2. B receives $100 and spends $40 (40% of $100) on C.
3. C receives $40 and spends $16 on D.
4. D spends $6.40 on E etc. and the process continues.

Look familiar? It should. This pattern can be described as an infinite geometric series (the same formula which is used to described a perpetuity for DCF evaluation).

For a GDP multiplier, the initial amount is $1 by definition. The "discount rate" or rate of decay is related to the MPC. Recall (Using the same math trick for geometric series):

GDP Multiplier = $1 + $1 x MPC + $1 x MPC^2 + ...
MPC x GDP Multiplier = $1 x MPC + $1 x MPC^2 +...
(1 - MPC) GDP Multiplier = $1
GDP Multiplier = $1 / (1 - MPC)

But recall: $1 = MPC + MPS
MPS = $1 - MPC

Therefore:
GDP Multiplier = $1 / MPS

Money Supply Multiplier and Reserve (Lending) Ratios

This is EXACTLY the same case for Money Supply and Bank Reserves. A bank (by policy or regulation) has a reserve ratio (RR). That is, for every additional $1 in deposits, it keeps a given percentage and lends out the rest. Let's also define lending ratio (LR) as the complement and by definition:

$1 = Reserve Ratio + Lending Ratio

Imagine "the bank" (representing all banks in the economy) has a reserve ratio of 20%.

1. "The Bank" receives a $100 deposit and lends out $80.
2. The $80 it lends out to "the Economy" (representing all depositors and borrowers) takes the $80 and "uses" it and it is redeposited into the Bank.
3. With the new $80 deposit, the Bank lends out $64.
4. The borrower uses it and it is redepositied into the bank.
5. The bank receives $64 and lends out $51.20 etc and the process continues.

Again, this is a pattern described by the same concept and the same formulas apply:

Money Supply Multiplier (MSM) = $1 + $1 x LR + $1 x LR^2 + ...
LR x MSM = $1 x LR + $1 x LR^2 +...
(1 - LR) MSM = $1
MSM = $1 / (1 - LR)

But recall: $1 = RR + LS
RR = $1 - LR

Therefore:
MSM = $1 / RR

Notes:

Some key points about this formula, notice that the multiplier effect is always greater than the initial amount ASSUMING that the reserve (saving) amount is less than the total amount (you don't reserve or save all of it). So when a dollar is spend in the economy (or lent out) the effect on the supply is greater than one.

Also note that for odd values of reserves and saving (aka, American's spending more than a $1 by borrowing), you get a negative MPS and therefore a negative mutliplier which is a non-sense result (in a similar manner as my contest question that Chad got right). Whenever you see non-sense numbers (numbers which tell "stories" that don't make sense) it should always act as a red flag to reinvestigate the initial assumptions of the model.