Lars Christensen wants people to stop talking about the National Income Accounting Identity and talk instead about the Equation of Exchange. What are these? Well first National Income:
Y = C+I+G+X
This says that output (Y) equals household consumption (C) plus investment (I) plus government consumption (G) plus or minus net exports (X) and it's how the government adds up GDP.
Then there's the Equation of Exchange:
PY = MV
This says that the prive level (P) times the output level (Y) equals the money supply (M) times the velocity of money (V).
The thing about both of these is that they're accounting identities, notcausal theories. You sometimes hear "more government spending will boost growth because Y = C+I+G+X so if G is higher than so is Y" and you sometimes hear "more government spending won't boost growth because Y = C+I+G+X so if G is higher then C or I will be crowded out and fall." Either of those theories might be right, but neither is supported by the accounting identity. All the identity tells us is that if G rises, then either it will crowd out other things or else Y will go up.
But I think the best thing to do with these equations is probably to combine them:
MV = C+I+G+X
What's happened here is that I've redefined C+I+G+X so that instead of summing to Y (real output) they sum to PY—nominal output. But that's fine, it simply means that we need to think of C and I and G and X as all denoting nominal quantities. And why shouldn't they denote nominal quanties? Despite the names, the real world is nominal. Your salary or wages are denominated in nominal terms, as is your mortgage and the balance on your credit card. Life is nominal. Chained 2005 dollars are an esoteric statistical construct.
What I think this helps emphasize is that there isn't some magical distinction between expansionary policies that promote real output and policies that are just inflationary. If monetary stimulus increases MV then what you'll get is more spending across a wide variety of categories. Since in today's economy some things are scarce (gasoline, apartments in San Francisco) and other things are not (unskilled labor, mall space near Phoenix) that will mean some increase in real output and some increase in prices. Similarly on the fiscal policy side, there's no such thing as an inflation-adjusted tax cut or appropriation. You're pulling on nominal levers, so if crowding out doesn't occur that has to be because the central bank is tolerating an increase in the price level.
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