Why Need a Numeraire Good in Dynamic Analysis

Question

Often in macro pure exchange economy models (sequential trading), authors make the initial period good price $P_0=1$ asserting this set-up is to make the consumption at period $0$ the numeraire.

But I don't see any detailed explanation or justification for doing this.

Just to add some background story:
The usual set-up is that agents are endowed with some units of goods, say oranges. So the unit of their consumptions is oranges. $P_t$ often the price, at $t=0$, of an orange consumption promised for delivery in period $t$. Why make $P_0=1$?

Giskard · Accepted Answer

Is there any reason not to make $P_0 = 1$?
In microeconomics (which your problem seems to be about, rather than macro) consumer's demand is homogeneous of degree zero in prices & income. That is if all parameters measured in money change by the same ratio (say they are all multiplied by 10) consumer decisions are unchanged. The root of this is that what usually matters is the price ratio between two goods. But
$$
frac{p_1}{p_2} = frac{r cdot p_1}{r cdot p_2}
$$
for all $r > 0$, so any proportional price change does not change the price ratio between two goods.
As a result of this, we are usually left with a degree of freedom when solving these models. To do away with it and simplify equations it is best to just arbitrarily select a good and appoint it numeraire.
In macroeconomics you have similar issues if you assume the neutrality of money.

Why Need a Numeraire Good in Dynamic Analysis

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