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Tuesday, August 18, 2020

Primer: Understanding The Money Multiplier Model

I ran into a question about the Money Multiplier Model, and I realised that explaining it is a lot more complicated than I expected.  The story of the model is how an initial cash deposit at one bank creates a chain of smaller and smaller loans, until the aggregate loan book has grown by a multiple of the initial deposit. The key to understanding the model is that it is based on a defective behavioural assumption. That is, it would apply -- if bankers operated in a way that they do not do in the real world.

It is entirely possible that the reader is taking a course that teaches the model. This can be viewed as the "Devil's Guide" to it.

Questionable (If Not Wrong) Assumptions

My assumption is that the reader has a grasp on how banks work. For such readers, the Money Multiplier Model is an enigma.

In order for it to apply, the following assumptions are made. They are all obviously questionable, but one could imagine work-arounds (as described below).
  • The first issue is that reserve requirements have to exist, which is largely no longer the case in the developed world. (The United States finally abandoned them in 2020, and American textbook authors might finally be forced to catch up to reality.) However, since this is model of reserve requirements, technically not wrong.
  • We assume that a non-bank financial sector ("shadow banking") does not exist. We could try to pretend that shadow banking does not matter, but it raises a lot of questions.
  • We ignore the existence of bank capital and liquidity regulations. This is defensible on the basis that we want to model the effect of reserve requirements; those regulations just impose added constraints.
  • All deposits are subject to reserve requirements. Once again, we can argue this is an approximation.
  • Reserve requirements are applied in real time, not with a lag. (We can pretend that bankers are forward-looking, or that real-time regulations were imposed.)
  • The explanations I have seen have skated over the difference between dollar bills and Fed settlement balances. We will assume that these are immediately fungible as a simplification.
  • Cash withdrawals do not happen. (This could be worked into the story, so not a big deal.)
  • Customers cannot force a loan to be made (loan pre-approvals, lines of credit). This is obviously not descriptive of the real world.

Behavioural Assumptions

The killer assumptions are behavioural assumptions.
  • It is assumed that deposits are sticky - people leave all deposits untouched. However, loan proceeds are spent immediately, with the proceeds transferred to somebody else. (This assumption is defensible as an approximation.)
  • It is assumed that banks process loan applications one at a time, with the loan officer consulting with the bank Treasury with regards to the bank's real time liquidity position. (This is a defective description of bank operations.)
  • It is assumed that banks limit their loan sizes to the amount of excess reserves that they hold. (This assumption is drop-dead wrong.)

The "Model"

We assume that reserve requirements are 10%, and that all banks start the day with no excess reserves. Then, a depositor at Bank A gets a $1000 transfer from the Federal Government. We assume that this initial deposit sticks at Bank A.
  1. Bank A gains a $1000 settlement balance asset, along with a new $1000 deposit liability. The $1000 deposit implies $100 increase in required reserves. The result is that required reserves rise by $100, and excess reserves by $900.
  2. This then allows the loan officer at Bank A to make a $900 loan to somebody, who then spends it. This increases both assets (the loan) and liabilities (deposits) by $900. (Jump to point 3 or 4.) 
  3. If the recipient of the spending is also at Bank A, then Bank A has no external transfers. Since deposits rise by $900, that means that $90 of excess reserves turn into required reserves. This leaves Bank A with $810 of excess reserves. (Go to 5.)
  4. If the recipient of the spending is at Bank B, then Bank A has to make a $900 transfer to Bank B. Bank A loses its $900 in excess reserves, but its deposit liabilities shrink by $900. Meanwhile. Bank B has a settlement balance increase of $900 (an asset), but deposits go up by $900 (liability). The $900 deposit implies $90 in reserve requirements, so Bank B has $810 in excess reserves. (Go to 5.)
  5. Whichever bank was the recipient has $810 in excess reserves, allowing it to make a $810 loan. It then does so, starting the process again, with a smaller loan. This repeats until the loan size goes to zero. Total deposits = $1000 + $900 + $810 + ... = $10,000. ($1000 /(1 - 10%)).
  6. The final deposit size ($10,000) is divided by the initial reserves injection ($1000) is the multiplier (10).

Why This Does not Describe The Real World

The multiplier chain story does not happen because bank loan officers do not limit themselves to loan amounts that are less than the bank's excess reserve position. There are two obvious reasons.
  1. Bank loan officers and Treasury departments are not in constant communications during the day, unless this is a hokey small American community bank. Bank treasurers are not going to have frank discussions of the bank liquidity position with hundreds (thousands) of loan officers, as this would pose severe run risk in any sort of crisis.
  2. A well-run bank starts the day with zero excess reserves. They are not going to have their loan officers sitting on their rear ends drinking coffee until some excess reserves make their way through the front door. 
The way banks actually work is that they make loans within guidelines (which have the bank's capital/liquidity position behind them), and then the Treasury trades reserves or money market instruments to hit the regulatory target for end-of-day balances. That is, they make the loan, then seek out reserves. If there are no excess reserves in the system (not an issue for a long time...), then the central bank has to do open market operations to supply them.

One final point is that this model explains the story that there is a belief that banks lend out reserves, which is obviously crazy. To the extent that this is believed, it is a result of looking at the assumed bank behaviour, and then converting this arbitrary constraint that there is a 1:1 relationship between excess reserves and lending.

Concluding Remarks

The reader should now know enough to be able to pass a mainstream course that teaches the money multiplier, as well as enough to explain why the teacher should not be teaching it.

(c) Brian Romanchuk 2020


  1. A number of questions:

    (I) Is the money multiplier model insinuating that "money multiplication" describes

    (Ia) the exact daily lending practice of banks, or does it purport

    (Ib) to delimit a ceiling, that need not be reached necessarily.

    Is it correct to say?

    (Ia) is wrong, since it is not reserves that drive loan extensions but loan profitability, customer quality, customer demand, and capital requirements. When these criteria are satisfactorily fulfilled, loan volumes drive reserve requirements (which are typically accommodated by the central bank after the event).

    As for (Ib), does the money multiplier theory (or do versions of it) concede to be defining just a ceiling that may not be reached owing to not fulfilling the above criteria?

    1. I am referring to how the model is taught in textbooks, and online courses. (A question about such a video on Stack Exchange triggered this article.) The description relies on the infinite sequence happening as I describe.

      If banks ignore that behavioural rule, Bank A in my example can just lend out $9000 immediately, eliminating the excess reserves in one shot. That is not how the model is taught.

    2. Addendum: the teacher might note that the limit is not reached (which it obviously can’t in the absence of infinitely fast transactions). However, the whole point of this as a teaching tool is to sneak in a math component - calculating the sum of an infinite series. That’s probably why teachers liked this monstrosity.

    3. When you say, “…loan profitability, customer quality, customer demand, and capital requirements. When these criteria are satisfactorily fulfilled, loan volumes drive reserve requirements.” You are almost there. Loan volumes (bank assets) drive bank liability needs. The cheapest liability (normally) for a bank is to acquire deposits, but a healthy bank will have other options if need be. Now to clarify, the need for the offsetting deposits comes after the loan not before… think of a line of credit. The bank only needs to fund the draw on the line of credit at the time it is used, not pre-funding everything at the time of approval which would actually lower the profitability of the overall loan. Bank lending is about using access to liquidity to fund loans and depositor withdrawals in a fashion that makes the bank profit. Going back to the reserve requirement… it’s really an ex-post accounting activity to meet regulatory requirements whose origins are debatable in value.

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