The Money multiplier trilogy (Part III: Immortality)

After the Death and Resurrection came Immortality.

So I will try my best to reach a general theory of the multiplier.

Some initial points:

  1. 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)
  2. 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.
  3. Banks create money by creating an asset (credit) alongside with a liability (deposit).
  4. Deposits are redeemable with Central Bank currency.
  5. Other assets provided by other financial intermediaries compete with deposits.
  6. 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.
  7. Central Banks inject monetary base (reserves) through Open Mark Operations (they just swap assets in a Bank balance sheet, a bond by reserves)
  8. 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),  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:

  1. Banks create Money. Yes, they do. If they have the market share.
  2. 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
  3. 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”)




The Money multiplier trilogy (Part II: Resurrection)

After being killed, it is worthwhile to wonder if the Money multiplier deserves a second chance.

I think it does.

So gather round the Dragon Balls because the Money Multiplier is going to be resurrected.

First things first. Why did the multiplier died, again?

Banks don’t need Reserves to make Loans! Loans create Deposits!


So, do Central Banks supply whatever the amount of reserves Commercial Banks need (to comply with legal requirements, to settle payments or to face cash withdraws) ?

Yes and no.

Yes, in the short run they do. They normally establish a interest rate target, so if they don’t comply with quantity demanded of reserves by Commercial Banks, it would lead to financial havoc as interest rates diverge from their targets.

No, Central Banks have a goal for inflation (or NGDP or …) so they must adjust their intermediate target for interest rates in a way they can achieve their ultimate goal.

So, it’s true to say both Reserves and Interest Rates are endogenous, a trustworthy central bank will only set its ultimate goal as exogenous.

Even though a Commercial Bank is not (usually) reserve constrained, as it can look for reserves in the inter-bank market, or can go to the Central Bank (either by discount or overdraft), it faces uncertainty towards the future.

Uncertainty about the “loyalness” of its depositors, uncertainty about the demand for cash, uncertainty about the demand for non-bank assets. A bank must face the uncertainty of its “usual business” liability side (Deposits) with both the “unconventional” liability side  (Central Bank lending) and the liquidity of its Asset side of the balance sheet.

Banks face restrictions. They don’t possess a widow’s cruse.

The Central Bank creates some of those restrictions. The competition among financial assets create some of those restrictions. The Economy creates some of those restrictions.

My goal is to combine the three theories of banking in a unified one (credit creation theory, Money multiplier, financial intermediary). I believe they must be all the same.

Those 3 theories can be matched with the 3 restrictions stated above. The 3 theories are all different angles of the same reality.

Banks create Money through credit. Yes.

The Central Bank influences the credit creation by supplying Reserves. Yes.

Banks are in the end of the day, just financial intermediaries. Yes.

My next post will try to create a comprehensive model which puts together the 3 theories, which in my view (with some assumptions) will grant a simplified but general view of how Banking and Central Banking works.

Let´s try to give the multiplier, Immortality.

(and create another multiplier during the process)

The Money Multiplier trilogy (Part I: Death)

The Money Multiplier is one real beauty when you start to learn Monetary Economics (most probably from Mishkin textbook). It makes you feel wise, you DO understand how Central Banks influence the Money Supply and Bankers are only pawns in this magnificent Game of Thrones of the Economy in which the Central Bank reigns supreme.

But “they” are trying to kill it!

Everyone, from Monetarists to Keynesians.

Even though in a way or another, most economists still believe in some version of the Money multiplier it’s uncool to express it that way, and you will soon find yourself ridiculed by someone who really understands banking.


I must confess my sins. I too believe the multiplier is dead.

Just look at this graph:


Bernanke killed the Money multiplier!! Quantitative Easing exposed the “truth”!

What are the problems/wrong assumptions of the Money multiplier?

  1. Normally, central banks just “follow along” the demand for reserves, so instead of a “monetary policy driven injection of reserves” Money multiplier, it really is kind of a “Money divisor” as banks look for reserves after they create loans (and deposits). And Central Banks must accommodate the banking system expansion if it wants to ensure financial stability.
  2. A straw man version of the MM assumes Banks lend reserves. This is not true. (as it may be explained later)

So, the Money multiplier has been disproved by Central Banks operations during the crisis as by some “endogenous demand driven vision of the economy”.

But still, I still believe in its beauty and we should not mourn its death, because I know it will rise again, and stronger than ever.

Nick Rowe is not alone in this fight.

Let me do my best to try to defend (my version) of the Money multiplier.