# When fractional reserve banking gets pathological

If you own a macroeconomics textbook published before 2009, throw it away. Or rip out the section on fractional reserve banking. Because it’s wrong.

Implicit (or, rarely, explicit) in all the cheer-leading about how banks “create” money by holding only a fraction of deposits in reserve is the assumption that the money multiplier would forever be greater than 1.0.

Here’s the M1 money multiplier as tracked by the St. Louis Fed

The event that turned all your macroeconomics texts into pet-cage liner is pretty obvious right there in the middle of the gray bar that is the Great Recession: the multiplier fell off a cliff, descending from 1.618 in mid-September 2008 to 1.0 in mid-November 2008. It then did the unthinkable and fell below 1.0 where it remains as of this posting (0.913 with the most recent data from 30 August 2017). It dipped below 0.8 four times between 2010 and 2016. The multiplier languished below 0.8 for three years and seven months  in one stretch, falling to a minimum of 0.678 in mid August 2014.

Although FRED doesn’t provide this particular data set that far back, the M1 money multiplier stayed well above 1.0 during the Great Depression. And it took almost 35 years to recover from its minimum in early 1940. So, as they say, get used to it: the multiplier will remain low for a long time. But less than 1.0?

### What were the banks doing all this time?

Those dips after the Great Recession are the result of a quantitative easing monetary policy implemented by the Federal Reserve in three waves between November 2008 and October 2014. Quantitative easing means that the central bank (Fed in this case) continues to push money into the monetary base even though interest rates are so low that banks don’t want to lend it out (thereby “creating” money, as your macro textbook says). Part of the problem was a less noticed but even more controversial Fed policy change at this time – paying interest on money that banks deposited with the Fed. More on this later.

So, back to the textbook story. The money multiplier is defined as

$$M = mB$$

where

$$\begin{array}{rcl}M&=&\textsf{money supply}\\m&=&\textsf{money multiplier}\\B&=&\textsf{monetary base}\end{array}$$

So

$$m = \frac{M}{B}$$

For the simple model, the money supply is simply currency (C) plus demand deposits (D)  – the M1, basically

$$M = C + D$$

The monetary base – the money the Fed puts into the economy (or takes out) – either goes into circulation as currency (C) or held by banks as reserves (R)

$$B = C + R$$

Now we have

$$m = \frac{C + D}{C + R}$$

Dividing top and bottom by D

$$m=\frac{C/D + D/D}{C/D + R/D}$$

Make the substitutions

$$\begin{array}{rccl}pp&=&C/D&\textsf{public preference for currency versus demand deposits}\\bp&=&R/D&\textsf{bank preference for reserves versus demand deposits}\end{array}$$

so finally

$$m=\frac{pp + 1}{pp + bp}$$

Your macro textbook probably refers to bp as the reserve ratio – a central bank (Fed) monetary policy tool. First problem: your book also made the misleading statement that the Fed imposes a minimum reserve ratio and implied that banks would only ever keep that minimum in reserve. In reality, only a few large banks are subject to a minimum reserve ratio requirement and it’s, generally, not very much. Second problem: there is not maximum reserve ratio imposed on banks, so they could, in principle, put all their money in reserves and never make a single loan. Which is why quantitative easing became a necessity.

Let’s look at what happens when the money multiplier is less than 1.0

$$\begin{array}{rcl}\frac{pp + 1}{pp + bp}&<&1\\pp+1&<&pp+bp\\1&<&bp\end{array}$$

That is, it is the banks’ preference to hold more in reserves than they have in deposits. When the interest rate they can receive for making loans is around zero, and the Fed is paying interest on reserve deposits, is it any surprise?

### The calculus of it all

Here’s an interesting explanation using marginal effects. First, note that the marginal effect of changing the monetary base is

$$\begin{array}{rcl}\frac{\partial M}{\partial B}&=&m\end{array}$$

So, when $m > 1$, increasing the monetary base gave the Fed more bang for the buck, but when $m=0.75$, the banks only put 75 cents of each of Fed’s dollar into circulation. The Fed’s dollars are our dollars, by the way.

Now consider the marginal effect of a change in the banks’ preference for reserves versus deposits

$$\begin{array}{rcl}\frac{\partial M}{\partial bp} & = & B\frac{\partial m}{\partial bp}\\& = & B\left[-\frac{pp+1}{\left(pp+bp\right)^{2}}\right]\\& = & -\frac{B}{pp+bp}\left[\frac{pp+1}{pp+bp}\right]\end{array}$$

The term in the square brackets is just the money multiplier, so

$$\frac{\partial M}{\partial bp} = -\frac{B}{pp+bp}m$$

Recall, however, that

$$\begin{array}{rcl}pp+bp & = & \frac{C}{D}+\frac{R}{D}\\& = & \frac{C+R}{D}\end{array}$$

and $$C + R = B$$ so

$$pp + bp = \frac{B}{D}$$

Finally,

$$\frac{\partial M}{\partial bp} = -\frac{B}{B/D}m = -Dm$$

That makes sense: an increase in the banks’ preference for reserves results in a decrease of the money supply proportional to the money multiplier and the amount of demand deposits.

Now consider the marginal effect of a change in the public’s preference for currency versus deposits

$$\begin{array}{rcl}\frac{\partial M}{\partial bp} & = & B\frac{\partial m}{\partial bp}\\& = & B\left[\frac{1}{bp+pp}-\frac{bp+1}{\left(bp+pp\right)^{2}}\right]\\& = & \frac{B}{bp+pp}\left[1-\frac{bp+1}{bp+pp}\right]\\& = & \frac{B}{bp+pp}\left[1-m\right]\end{array}$$

As we saw before, the first term is just D. Reversing the subtraction in brackets gives us

$$\frac{\partial M}{\partial pp} = -D\left(m-1\right)$$

Now, as long as $m > 1$, this says that increased public preference for currency over deposits reduces the money supply proportional to D, but not quite as strong as increased preference for reserves by banks. That, too, makes sense. And, it’s almost certainly what your macro textbook says: increased preference for currency decreases the money multiplier.

But the post-apocalyptic scenario of $m < 1$ probably doesn’t appear in that macro textbook. That is, when the money multiplier is less than one, an increase in the public’s preference for currency actually increases the money supply. Not to anthropomorphize, but it’s like it’s telling us that, when $m < 1$, we can keep the banks from squirreling away all our money by holding on to it (or putting in the freezer, or under the mattress, or wherever).