Here are two macroeconomic identities:
Both are true by definition.
1. If you start with Y=C+I+G+NX you can immediately see why fiscal policy works. An increase in G works directly to increase Y. "Look, there's G, right there in the equation!". It's only a little bit less obvious to see how tax policy works, and it's only a little bit less direct. "A cut in T increases C, and look, there's C right there in the equation!".
It's hard to see how monetary policy works. I can't see M anywhere in that Y=C+I+G+NX equation. If M does work, it can only work indirectly, which means whether or not it does work is uncertain and unreliable. "So, what exactly is this monetary policy transmission mechanism supposed to be?"
Of course, if you aren't stupid, you recognise the logical possibility that fiscal policy might not work. "Sure, I suppose an increase in G might cause an equal decrease in C+I+NX, but why exactly would we expect that to happen?" The burden of proof is on those who say that an increase in G would cause an equal decrease in C+I+NX, and that transmission mechanism is not at all obvious. And after all, when we talk about fiscal policy, aren't we assuming "other things equal"?
2. If you start with MV=PY you can immediately see why monetary policy works. An increase in M works directly to increase Y. "Look, there's M, right there in the equation!"
It's hard to see how fiscal policy works. I can't see G or T anywhere in that MV=PY equation. If G does work, it can only work indirectly, which means whether or not it does work is uncertain and unreliable. "So, what exactly is this fiscal policy transmission mechanism supposed to be?" If fiscal policy does work, it must work either by increasing M or increasing V, which means it's really just backdoor monetary policy.
Of course, if you aren't stupid, you recognise the logical possibility that monetary policy might not work. "Sure, I suppose an increase in M might cause an equal decrease in V, but why exactly would we expect that to happen?" The burden of proof is on those who say that an increase in M would cause an equal decrease in V, and that transmission mechanism is not at all obvious. And after all, when we talk about monetary policy, aren't we assuming "other things equal"?
3. So far my argument has been symmetric. But there's an obvious asymmetry between the two identities. P appears in the second, but not in the first. "How do we know that an increase in M won't just cause an equal increase in P, with no change in Y?". That's an obvious question when you see MV=PY, because "Look, there's P, right there in the equation!". If I re-wrote the first identity in nominal terms, as PY=PC+PI+PG+PNX, it might invite the same question. Or if I re-wrote the second identity in real terms, as Y=Vm (where m is the real money stock), I could hide that question.
Scott Sumner has written about the very common fallacy that sees fiscal policy as affecting real aggregate demand while monetary policy only affects nominal aggregate demand.
There are three different types of quantities: quantities demanded; quantities supplied; and quantities actually bought-and-sold. Which one of these three types of quantities does Y, C, I, G, and NX refer to? It's very easy to forget the supply side, if you leave P out.
4. There's a second asymmetry. Y=C+I+G+NX is used in National Income Accounting. MV=PY isn't. It's administratively easier to collect the data using Y=C+I+G+NX, because you can divide the questions up between households, firms, governments, and foreigners. Even if those categories don't match at all exactly, and the same new car counts as I if a firm buys it and C if a household buys it. And so Y=C+I+G+NX has a hold on our thinking that MV=PY doesn't, even if that hold is based merely on administrative convenience of data collection. Plus, you can do neat things with Y=C+I+G+NX, like re-write it as S-I+T-G=NX, which looks different from Y=C+I+G+NX, because it has a whole new letter, S (defined as Y-T-C, so it's not really new at all).
5. I was born and bred (as an economist) using Y=C+I+G+NX. That identity is part of my identity. "You can take the boy out of the Y=C+I+G+NX, but you can't take the Y=C+I+G+NX out of the boy". I learned about MV=PY, of course, but it has always felt foreign to me. A couple of years back I decided that MV=PY is a more useful identity (though it should really be MV=PT), but it still feels, well, like wearing someone else's clothes.
6. I think MV=PT is more useful a way of organising our thoughts for business cycle theory because:
Recessions are always and everywhere a monetary (medium of exchange) phenomena, and Y=C+I+G+NX refers equally to a monetary exchange or barter economy.
Recessions are not about Y; they are about T. Production of newly-produced goods Y for home use and for barter with friends and neighbours seems to do very well in recessions; while monetary transactions T of all kinds, whether for newly-produced goods or not, seem to do badly.
7. For the macroeconomics of long run growth I would use a very different identity: Y=C+I+NX. I have left out G altogether. G is divided into government consumption and government investment, and C includes both private and government consumption, while I includes both private and government investment. What matters for long run growth is the division of output between consumption and investment, not between households, firms, and governments.
8. It is common to see amateur "internet economists" placing too great a reliance on identities. But maybe the only real difference is that they are not as good at hiding it as professional economists. (I did see one very very good professional economist recently resort to the purely gratuitous use of Y=C+I+G+NX in the middle of an otherwise respectable argument for why fiscal policy works. It was on Brad Delong's blog a few months back, but now I can't find it. It wasn't Brad himself.) It is easier to see the disease on others' faces.
9. The parallel with "Identity Politics" is deliberate, but I'm not sure how far I can push that parallel. And I can't quite figure out the relationship between this post and my old "Celestial Emporium of Benevolent Knowledge" post.