So I will try my best to reach a general theory of the multiplier.
Some initial points:
- Christmas. For simplification (bear with this please) I will assume that only in the 25th of December do Central Banks inject new Reserves in Banks Balance Sheets, taking into account the Central Bank goal for next year.The Banks had received during the year a bunch of loan applications and now they can finally proceed with lending. (I am not assuming banks need reserves to lend money, but if you assume the deposits a credit creates “fly away” and banks are already fully “loaned up”, banks will eventually need that High Powered Money)
- Banks, unlike the traditional money multiplier story (“The textbook story implicitly assumes that each bank is small relative to the whole banking system, and is looking for the Nash equilibrium.”) are not small, they have market share. Therefore, some of the deposits they create do not “fly away” they remain in the same bank.
- Banks create money by creating an asset (credit) alongside with a liability (deposit).
- Deposits are redeemable with Central Bank currency.
- Other assets provided by other financial intermediaries compete with deposits.
- Banks are obliged to have a percentage of its deposits (liability) as an asset (reserves). They may want to keep a little more than required to face uncertainty about flow of funds in the economy.
- Central Banks inject monetary base (reserves) through Open Mark Operations (they just swap assets in a Bank balance sheet, a bond by reserves)
- Banks have a market share of deposits comparing to the banking system and are expected to maintain that share.
I guess everything is settled now and abstracting from the Christmas assumption, I guess all the other points are straight forward and in accordance with Banking Theory.
Now for the model:
Grab a pencil and a paper.
ER is excessive reserves, DRR is the desired reserve ratio (legal requirement plus precautionary), c is the demand for currency by deposit created, a is the demand for other financial assets (outside the banking system – a la Tobin), X is the amount of deposits a Commercial Bank can create given ER, MS is the bank market share (the amount of deposits it has – and expects to have comparing to the system).
So, my goal is to determine what will be the amount of X we will have given ER.
Lets assume the Central Bank injects ER into a bank by an OMO (which does not affect the liability side of the bank).
The bank will create a credit (asset) by the amount of X (and a deposit in the liability side of the same value).
The deposit will transform part of the ER into Desired Reserves (by DRR) – in the asset side.
Next as part of the deposits (liability) flow out of the Bank depending on the bank’s market share: (1-MS)*X, Reserves will flow out in the asset side: DRR*(1-MS)*X will represent the decrease in Desired Reserves, (ER-DRR*X) will represent the loss in Excess Reserves.
We must assume in the end that c*X*MS and a*X*MS is the proportion of the deposits that stayed in the bank that got transformed either in currency or in other financial assets. (deduction in the liability side of the balance sheet). In the asset side you must deduct: MS*DRR*X*c – MS*(1-DRR)*X*c + MS*DRR*X*a – MS*(1-DRR)*X*a.
I hope you have written everything. Now the fun starts. Let’s find how much X can a bank create for each ER it has received. By applying all the information above we have (left hand side: liabilities, right hand: assets):
– (ER – DRR*X – MS*(1-DRR)*X*c – MS*(1-DRR)*X*a) – DRR*(1-MS)*X – MS*DRR*X*c – MS*DRR*X*a = – (1-MS)*X – MS*X*c- MS*X*a «=» (getting rid of parenthesis)
– ER + DRR*X + MS*X*c – MS*DRR*X*c + MS*X*a – MS*DRR*X*a – DRR*X +DRR*MS*X – MS*DRR*X*c – MS*DRR*X*a = – X + MS*X – MS*X*c- MS*X*a «=» (X to one side)
X + MS*X*c – MS*DRR*X*c + MS*X*a – MS*DRR*X*a + DRR*MS*X – MS*DRR*X*c -MS*DRR*X*a – MS*X +MS*X*c + MS*X*a = ER «=» (X multiplied by the rest)
X * ( 1 + MS*c – MS*DRR*c +MS*a – MS*DRR*a + DRR*MS – MS*DRR*c – MS*DRR*a – MS +MS*c + MS*a) = ER «=»
X * ( 1 + MS * (c – DRR*c +a – DRR*a + DRR – DRR*c – DRR*a – 1 + c + a) = ER «=»
X * ( 1 + MS * (2c + 2a -1 + DRR * (1 – 2c -2a) ) ) = ER
X = ER / ( 1 + MS * (2c + 2a -1 + DRR * (1 – 2c -2a) ) )
Wow…this was a long journey.
But let’s take some conclusions:
First of all, if you assume as in the textbook that each is infinitesimal comparing to the system (MS=0), its individual multiplier is 1! (like in the textbook) A bank can’t lend more than its Excess Reserves. X = ER
Let’s assume now that a bank is as big as the system (this is the textbook multiplier), so that MS=1 we have X = ER / ( 2c + 2a + DRR -2DRRc -2DRRa ) wich transforms into (as ER transform into required reserves)
X/ ER = 1/( 2c + 2a + DRR -2DRRc -2DRRa ) which is (kinda) like the textbook multiplier (added with the other assets).
In between (0<MS<1) banks can create more money than they have initially as Excessive reserves, but they have to take into account desired reserves, competition against other banks deposits, demand for currency and demand for other assets.
So, like I promised, in the end of the day:
- Banks create Money. Yes, they do. If they have the market share.
- Bank Reserves multiply into Bank Deposits. Yes they do, as banks seek to expand their credit, they “use up” Bank Reserves supplied by the Central Bank
- Banks are Financial Intermediaries. Yes, just like others (which they have to compete with), although their liabilities are Medium of Exchange (Money), they are constrained by the laws of the market and don’t have widow’s cruse.
This was a long and exhausting post. I hope I protected the fair Money Multiplier and its usefulness to understand Banking and Central Banking operations.
Nick, I gotchyour back!
(Yes, banks can “cheat” and just ask the Good Ol’ Central Bank for more “juice”)