Money Multiplier (Textbook Reserve Model)

This page needs its warning before its explanation, so here it is. The money multiplier is a textbook artifact. It is not how money is created in a modern economy, and the Bank of England has said so in print.

The Bank's Quarterly Bulletin for 2014 Q1 carried an article titled "Money creation in the modern economy" whose entire purpose was to correct this. Its position is direct: the majority of money in the modern economy is created by commercial banks making loans, and the act of lending itself creates a deposit. Banks do not wait to receive reserves and then multiply them outward. They lend when they find a creditworthy borrower at a profitable rate, and they obtain the reserves afterwards — from the interbank market, or from the central bank, which supplies them on demand at its policy rate because refusing would push the overnight rate away from its target. Causation runs from lending to reserves, not the reverse.

The institutional fact is even simpler: most central banks no longer have a reserve requirement at all. The US Federal Reserve reduced reserve requirement ratios to zero in March 2020 and has not restored them. Canada abolished them in 1992. The United Kingdom, Australia, New Zealand and Sweden have none. In those systems $rr = 0$, and $m = 1/0$ is not a very large number — it is undefined, which is a fair description of how much the concept is doing.

So why is the formula still taught, and why is this page here? Because the arithmetic is worth understanding on its own terms, and because knowing exactly where a model breaks is more useful than never having met it. The textbook story goes: a bank receives $100 of reserves, must hold 10%, lends $90; that $90 is deposited somewhere, of which $81 is lent on; and the infinite geometric series $100 + 90 + 81 + \ldots$ sums to $1000. The multiplier is the sum of that series, $1/rr$, and the algebra is impeccable.

What the algebra assumes is where it fails. It assumes banks always lend out their full excess reserves, which they demonstrably do not — banks have held enormous excess reserves since 2008 without lending them, because lending is constrained by borrower demand and by capital requirements rather than by reserves. It assumes nobody holds cash outside the banking system, which everybody does. It assumes reserves are the binding constraint on lending, which for a central bank targeting an interest rate they cannot be: if reserves were scarce the overnight rate would rise above target, and the central bank supplies them precisely to prevent that. And it assumes the central bank controls the quantity of reserves as a policy lever, whereas modern central banks set a PRICE — the policy rate — and let the quantity follow.

What actually constrains bank lending is worth listing, because it replaces the multiplier rather than supplementing it: capital requirements, which limit assets relative to equity and are the real binding constraint for most banks most of the time; the availability of borrowers who are creditworthy and want to borrow at the offered rate; the profitability of the loan against the bank's own funding cost; liquidity regulation such as the coverage ratios introduced after 2008; and the bank's own risk appetite. Reserves appear on none of those lists.

The quantity theory framing that usually accompanies this formula — central bank expands reserves, multiplier expands money, prices rise — did not survive the decade after 2008. Reserves in the major economies grew by an order of magnitude under quantitative easing, broad money grew modestly, and inflation stayed persistently below target for years. If the multiplier described a mechanism, that could not have happened.

Use this page, then, for what it is: a piece of geometric-series arithmetic, a fair account of what a generation of textbooks taught, and a worked example of a model that is internally consistent and empirically wrong. Those are common, and recognizing one is a skill worth more than the formula.