## Money Neutrality (or, the Quantity Equation) March 4, 2011

Posted by tomflesher in Macro, Teaching.
Tags: , , , , , , ,

The Macro class I’m TAing has just gotten to money growth and inflation, chapter 12 in Mankiw’s Brief Principles of Macroeconomics. As usual, the quantity equation, MxV = PxY, confuses some of the students a little bit, so I thought I’d see what I can do to clarify it a little.

First, let’s define some terms. M is the size of the (nominal) money supply. V is the velocity of money, or the number of times a given dollar is spent in a year. (It represents how fast people spend money, so that if the money supply is only $100 but GDP, the total of expenditures on all final goods and services in the US in a year, is$500, the velocity of money is 5/year.) P is the nominal price level – that is, just the average price of stuff in the economy. Y is real GDP, so it represents the production level or the amount of stuff produced in the economy.

For reasons of a mathematical nature1 explained at the end of the page, you can think of PxY as all of the expenditures in the economy, and because of that, you can think of it as the product of the average price and the quantity produced. So, the right-hand side of the quantity equation is nominal GDP.

V is determined by many different things. For example, when people feel less confident about the economy, V might drop because people would spend less money. When people feel more confident, they spend more easily, and so V might rise. A lot of things that affect V are difficult to talk about if we only have Principles-level tools, though, so the conventional wisdom is to leave V constant for now.

Then, we basically have the equation:

$M = P \times Y$

What this means is that a change in the money supply has to be matched by the change in price level and the change in production.2 If all we know is that the money supply changed, then it could be due to a change in the price level, a change in real production, or some combination of the two. If on the other hand the price level changes and production doesn’t, then it must be due to a change in the money supply (like if the government started printing too much money).

When there’s too much money in the economy, price levels rise; when businesses produce more, then the money supply will increase; when price levels are changed for some outside reason, then either GDP has to change, the money supply has  to change, or both.

Note:
1 Under the expenditure method, the Gross Domestic Product or GDP is the total market value of all final goods and services produced in the economy in a given period of time. For each good or service, it will either be recorded as a sale if someone buys it or as inventory by the business that produced it if it isn’t sold. So, for every good i in the economy with a given nominal price i, the contribution to GDP of that good is

$p_{i} \times q_{i}$

So, in an economy with n goods, you can add up all of the expenditures on those goods and generate nominal GDP. Thus,

$GDP_{Nominal} =\displaystyle\sum\limits_{i=1}^n p_{i} \times q_{i}$

This is equivalent to multiplying the average price by the total number of goods in the economy.

2 This can be shown using the natural logarithm transformation. Since the interpretation of a change in the natural logarithm is the percentage change in the untransformed variable,

$ln(M \times V) = ln (P \times Y) \implies ln(M) + ln(V) = ln(P) + ln(Y)$

$\Delta V = 0 \implies \Delta ln(V) = 0 \implies \Delta ln(M) = \Delta ln(P) + \Delta ln(Y)$

So, the percentage change in M can be decomposed into two pieces: the percentage change in P and the percentage change in Y.