Doubt on the meaning of real money balances Mankiw defines real money balances, $\frac{M}{P}$, to be the quantity of goods and services a given amount of money can buy. On page 88 of Macroeconomics 7th edition, he illustrates the concept with the following example:


  Real money balances measure the purchasing power of the stock of
  money. For example, consider an economy that produces only bread. If
  the quantity of money is $\$10$, and the price of a loaf is $\$0.50$, then
  real money balances are $20$ loaves of bread. That is, at current
  prices, the stock of money in the economy is able to buy $20$ loaves.


It is very clear to me what real balances represent in such a simple example. However, I suppose in real life $P$ would be expressed as some index, right? If $P=1.52$ is the implicit price deflator for GDP in some year, for instance, and the quantity of money in the economy is $\$1.000$, what exactly does $\frac{M}{P}=\frac{1000}{1.52}\approx657.9$ represent? How do we interpret that number? Thanks in advance.
The level of a price index (like the GDP deflator) is essentially arbitrary. The usual choice is to set the reference date to 100, but it could just as easily be set to 1. So the level of the real money balance is essentially an arbitrary number. If you were using it in an economic model, you would need to adjust the scaling in the functions to account for this.

For an interpretation, pick a base time $t$. You then divide the monetary aggregate $M(t)$ by annualised nominal GDP at time $t$. This converts the money holding into a fraction of annual GDP. Since GDP is a flow, it has units of \$/(year), and the money number has units of \$. The ratio of the division has units of years. The interpretation is that the ratio is the number of years worth of the goods and services that go into the “GDP basket” that the monetary aggregate could buy. (For example, if the monetary aggregate was 0.5 of annual nominal GDP, it can buy 6 months worth of whatever goes into GDP.)

Let us fix the base period $t$. If we rebase the deflator $P(t)$ so that it equals annualised nominal GDP at time $t$, the “real money balance” using that deflator gives us that same ratio discussed above. (Since the level of the deflator is arbitrary, we can multiply the the entire time series by a constant, and nothing much changes. So in this case, set the scaling constant to be equal to the level of nominal GDP at tine $t$ divided by the value of the original deflator at time $t$. The result is that the new deflator is equal to nominal GDP at time $t$.) So we have an interpretation for the real money balance for period $t$.

If we go to the next period ($t+1$), the interpretation gets more complicated. The problem is that the mix of goods and services in GDP changes. So it may be that I am over-simplifying things here. (This does not happen for bread, which is presumed to be uniform over time.)

My interpretation of the real money balance is that it is volume of real GDP (in years) that can be purchased by the money balance $M(t+1)$ using the prices prevailing at time $(t+1)$, but using the time $t$ real GDP basket. So if moves to 0.75, that means that the money balance at time $t+1$ could buy 9 months worth of the goods and services that were produced at time $t$. However, this will probably not correspond to 9 months worth of the annualised production at time $t+1$, since real GDP may itself have grown (or shrunk, if there was a recession).

One can argue that this interpretation is complicated, but it reflects the difficulties with the definition.