I'm familiar with the idea of a numeraire when prices come in the form $\frac{p_x}{p_y}$, but I'm still trying to figure out what the more general notion of a numeraire is. For instance, if all that I'm given is $p_x,p_y,$ and $m$: what is supposed to be the numeraire there? Is it just 1 unit of currency?


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numeraire is by definition some unit of measure by which we measure all other units. In your case, let us say that we would decide to use $x$ as a numeraire. In that case we would divide all prices by $p_x$ so the price for $x$ would become $\frac{p_x}{p_x}=1$ (one $x$ can be worth only itself) and for good $y$ the price would be set as $\frac{p_y}{p_x}$.

For example, let us suppose that 1 apple is worth 2 rocks. In such case if we would decide to make apple numeraire everything would be expressed in apples. Now 1 apple would cost 1 apple and 1 rock would cost 0.5 apples (since 1 apple is worth 2 rocks 1 rock would only cost 0.5). If we would choose to make rocks numeraire then 1 apple would now have price 2 rocks and one rock would have price one rock.

If you want to think of it intuitively, you could consider numeraire to be 'currency' used in your particular problem (in real life money is a numeraire - we value all other goods and services in terms of money, let's say dollars, and one dollar itself is just worth one dollar).


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