<p>I assume that this question is related to this question on money growth - <a href="https://economics.stackexchange.com/questions/17664/why-did-the-rate-of-money-supply-growth-accelerate-from-1993-96-in-the-united-st/17669#17669">link to question</a></p>

<p>The monetary base and velocity completely pin down the wider monetary aggregate. The behaviour of acceleration is slightly trickier, but I will attack it here.</p>

<p>I will firstly use continuous time for simplicity. I will also use my own notation, which is explained.</p>

<p>Let $M$ be whatever wide monetary aggregate you want to study, and $B$ be the monetary base. Velocity $V$ is defined:
$$
V = \frac{M}{B}.
$$
Importantly, this is a definition, and always holds. There is no reason to ascribe changes in velocity to any particular factor, such as preferences of banks.</p>

<p>Then, the rate of change of the money stock is:
$$
\frac{dM}{dt} = \frac{d(BV)}{dt} = B\frac{dV}{dt} + V\frac{dB}{dt}.
$$
Since we are interested in the percentage rates of change, we define growth rates as the time derivative of the variable divided by the variable. That is,
$$
G_M(t) = \frac{\frac{dM}{dt}}{M},
$$
$$
G_B(t) = \frac{\frac{dB}{dt}}{B},
$$
$$
G_V(t) = \frac{\frac{dV}{dt}}{V}.
$$</p>

<p>Then, we see that:
$$ G_m(t) = \frac{B}{M}\frac{dV}{dt} + \frac{V}{M}\frac{dB}{dt}.
$$
Since $\frac{B}{M} = \frac{1}{V}$ and $\frac{V}{M} = \frac{1}{B}$,
$$
G_m(t) = \frac{\frac{dV}{dt}}{V} + \frac{\frac{dM}{dt}}{M} = G_V(t) + G_B(t).
$$
Since you are interested in acceleration, that would be $\frac{dG_M}{dt}$, which is given by:
$$
\frac{dG_M}{dt} = \frac{dG_V}{dt} + \frac{dG_M}{dt}.
$$</p>

<p>In other words, to characterise the acceleration of the money supply, you need to look at the acceleration of the monetary base and the acceleration of the velocity. If we refer back to the question on the money supply in the 1990s, you would have had to calculate the annual percentage change of velocity, and not look at the <em>level</em> of velocity (which is what was displayed).</p>

<p>If we return to discrete time (and observed data are discrete time), the relationships above become approximations. You would have to take the first differences, to get the rate of change, and the difference of the rate of change to get acceleration. For annual data, that means we end up comparing data two years apart, which may break the continuous time approximation.</p>

<p>Since this question asked about other factors influencing the relationship between the base and wider money: there are none. The above equations hold by definition. The question why velocity would change is a more difficult question.</p>

CC BY-SA 3.0
2017-08-04T00:31:16.327Z
Thanks for the follow up. However, I went back and looked at the FRED data on the acceleration of the M2 velocity as you said and, although it explained most of my confusion, between 1993 & 1994 M2 velocity still accelerated despite the MB accelerating and M2 stock decelerating in that time frame. What accounts for that?
user14013
CC BY-SA 3.0
2017-08-04T01:28:42.747Z
When I write acceleration, that is the rate of change of the growth rate. During that period, the growth rate of M2 was rising, so it was accelerating. The simplest thing would be to download M2 and the base, and examine the data in a spreadsheet. You able to calculate the velocity, and then examine growth rates and the changes in growth rates (acceleration).
Brian Romanchuk
CC BY-SA 3.0
2017-08-04T04:02:52.257Z
If I understand your equation correctly, it stipulates that if the M2 money stock were to accelerate and the monetary base were to decelerate, then the velocity of M2 would in turn accelerate. If this is the case, then how is it that - from 1994-96 - stock sped up from 1.2% to 4.8% while the base slowed down from 9.7% to 4.1% and velocity slowed down from 4.9% to 0.8%?
user14013
CC BY-SA 3.0
2017-08-04T10:51:47.787Z
It's a relationship that holds at a point in time for continuous time series. If the time series are very smooth, it would hold for discrete time, but annual changes may not be smooth enough.
Brian Romanchuk
CC BY-SA 3.0
2017-08-03T13:05:07.497Z
2017-08-04T10:58:25.757Z
2017-08-04T10:58:25.757Z
Brian Romanchuk
<p>I've read that it is not a hard-and-fast rule that the overall money stock growth rate speeds up in response to an accelerated expansion of the monetary base and a growing money multiplier. If this is the case, what factors unrelated to central bank policy and the lending propensity of commercial banks would cause this theoretical deceleration of money stock growth?</p>

CC BY-SA 3.0
2017-08-03T12:07:19.307Z
You first specify acceleration, but wrote deceleration in the last sentance. I assume that is a typo?
Brian Romanchuk
<macroeconomics><money-supply>
Other factors that affect the supply of money
user14013
# Other factors that affect the supply of money

(Asked by: username_0 on 2017-08-03)

I've read that it is not a hard-and-fast rule that the overall money stock growth rate speeds up in response to an accelerated expansion of the monetary base and a growing money multiplier. If this is the case, what factors unrelated to central bank policy and the lending propensity of commercial banks would cause this theoretical deceleration of money stock growth?

**Tags:** macroeconomics, money-supply

> **Comments:**
> **username_1**: You first specify acceleration, but wrote deceleration in the last sentance. I assume that is a typo?  

---

## 1 answers

### Answer by username_1 on 2017-08-03. Score: 2 | Accepted answer

I assume that this question is related to this question on money growth - [link to question](https://economics.stackexchange.com/questions/17664/why-did-the-rate-of-money-supply-growth-accelerate-from-1993-96-in-the-united-st/17669#17669)

The monetary base and velocity completely pin down the wider monetary aggregate. The behaviour of acceleration is slightly trickier, but I will attack it here.

I will firstly use continuous time for simplicity. I will also use my own notation, which is explained.

Let $M$ be whatever wide monetary aggregate you want to study, and $B$ be the monetary base. Velocity $V$ is defined:
$$
V = \frac{M}{B}.
$$
Importantly, this is a definition, and always holds. There is no reason to ascribe changes in velocity to any particular factor, such as preferences of banks.

Then, the rate of change of the money stock is:
$$
\frac{dM}{dt} = \frac{d(BV)}{dt} = B\frac{dV}{dt} + V\frac{dB}{dt}.
$$
Since we are interested in the percentage rates of change, we define growth rates as the time derivative of the variable divided by the variable. That is,
$$
G\_M(t) = \frac{\frac{dM}{dt}}{M},
$$
$$
G\_B(t) = \frac{\frac{dB}{dt}}{B},
$$
$$
G\_V(t) = \frac{\frac{dV}{dt}}{V}.
$$

Then, we see that:
$$ G\_m(t) = \frac{B}{M}\frac{dV}{dt} + \frac{V}{M}\frac{dB}{dt}.
$$
Since $\frac{B}{M} = \frac{1}{V}$ and $\frac{V}{M} = \frac{1}{B}$,
$$
G\_m(t) = \frac{\frac{dV}{dt}}{V} + \frac{\frac{dM}{dt}}{M} = G\_V(t) + G\_B(t).
$$
Since you are interested in acceleration, that would be $\frac{dG\_M}{dt}$, which is given by:
$$
\frac{dG\_M}{dt} = \frac{dG\_V}{dt} + \frac{dG\_M}{dt}.
$$

In other words, to characterise the acceleration of the money supply, you need to look at the acceleration of the monetary base and the acceleration of the velocity. If we refer back to the question on the money supply in the 1990s, you would have had to calculate the annual percentage change of velocity, and not look at the *level* of velocity (which is what was displayed).

If we return to discrete time (and observed data are discrete time), the relationships above become approximations. You would have to take the first differences, to get the rate of change, and the difference of the rate of change to get acceleration. For annual data, that means we end up comparing data two years apart, which may break the continuous time approximation.

Since this question asked about other factors influencing the relationship between the base and wider money: there are none. The above equations hold by definition. The question why velocity would change is a more difficult question.

> **Comments:**
> **username_0**: Thanks for the follow up. However, I went back and looked at the FRED data on the acceleration of the M2 velocity as you said and, although it explained most of my confusion, between 1993 & 1994 M2 velocity still accelerated despite the MB accelerating and M2 stock decelerating in that time frame. What accounts for that?  
> **username_1**: When I write acceleration, that is the rate of change of the growth rate. During that period, the growth rate of M2 was rising, so it was accelerating. The simplest thing would be to download M2 and the base, and examine the data in a spreadsheet. You able to calculate the velocity, and then examine growth rates and the changes in growth rates (acceleration).  
> **username_0**: If I understand your equation correctly, it stipulates that if the M2 money stock were to accelerate and the monetary base were to decelerate, then the velocity of M2 would in turn accelerate. If this is the case, then how is it that - from 1994-96 - stock sped up from 1.2% to 4.8% while the base slowed down from 9.7% to 4.1% and velocity slowed down from 4.9% to 0.8%?  
> **username_1**: It's a relationship that holds at a point in time for continuous time series. If the time series are very smooth, it would hold for discrete time, but annual changes may not be smooth enough.  

---

