# Why Does Y Equal Real GDP?

I hesitate to post this while Nick Rowe is on vacation, because he’s always so generous with his replies and explanations. Here’s hoping he gets back to this.

But he does get me thinking. I’ve spent several days re-reading and pondering his Identity Economics post and (his) related others, which post begins [my brackets]:

Here are two macroeconomic identities:

1. Y=C+I+G+NX [the National Income Identity]

2. MV=PY [the Identity of Exchange]

Both are true by definition.

Without (for the moment) burdening you with all my thinking, here’s the question that I’m left with:

By convention and practice — “by definition” — Y in these identities equals real GDP. Y *means* ”real GDP”.

Here’s why that doesn’t make sense to me:

Say that in Year 1, U.S. GDP (IOW, Y) is $14 trillion. That’s the total dollars spent on real-world, newly-produced goods and services. This quantity is necessarily counted in dollars, because that’s how the measurement is done (at least in the expenditure approach) — adding up all the dollar-denominated purchases/sales.

Assume: in Year 2, the economy produces that same quantity of real goods and services, and their aggregate human utility is unchanged. Real output is unchanged.

But in Year 2, the prices are 10% higher (for whatever reasons). Measured, counted GDP (a.k.a. “Y,” total dollar transactions) increases by 10%.

If Y is real GDP, then real GDP just went up by 10%. Even though real output didn’t.

This doesn’t make any sense to me. Shouldn’t Y mean nominal GDP?

Even: *mustn’t it*? Because it’s always counted in nominal dollars.

This would give us:

MV = Y

And:

Y/P = Real GDP

This seems much more tractable and conceptually coherent to me. The “real” definition keeps running me into (what seem like) conceptual/arithmetic contradictions.

I was going to stop here, hoping to keep this discussion focused, but as I’m about to post I find that Saturos has given me a very nice response to my comment over at Nick’s place. The confusion expressed here fully explains the more profound confusion in that earlier comment; I simply assumed therein, based on the fundamental construction of the National Income Identity (and the methods of national accounting), that Y means nominal GDP.

Saturos sez:

Talk to any Keynesian and you’ll find that they’re far more inclined to interpret Y = C + I + G + NX as referring to real (CPI or GDP deflator adjusted) quantities. Of course P is here assumed to equal 1, because “prices don’t matter in the short run”. But I agree that it makes far more sense, and is far more consistent with the Keynesian approach, to talk about nominal spending flows. (Really, you should use lowercase for real variables, and uppercase for nominal – and the identity is true in either case, as it’s just a listing of the different categories that any spending or output must fall into.) Matt Yglesias (http://www.slate.com/blogs/moneybox/2012/05/13/fun_with_accounting_identities.html) has a new post in which he takes Scott Sumner’s version: MV = C + I + G + NX. That might be the best approach of all – it shows you that all the changes in “income accounting” variables that get reported on the news must all be manifestations of fluctuations in the overall volume of spending, MV. If we’re talking about fiscal policy or “exogenous shocks” to NGDP, then this must be a fluctuation in V (base velocity).

My responses:

1. Are you telling me that *economists don’t even agree on what Y means?* Not sure if that’s what you’re saying. If so, doesn’t Nick’s “by definition” start this whole discussion (even: the whole discipline of national-accounting-based and monetary economics) on a bed of quicksand?

2. Is it really standard practice to “use lowercase for real variables, and uppercase for nominal”? Seems like a great convention. Do economists adhere to this convention consistently? They don’t seem to. (If Y equals real GDP, shouldn’t it always be lowercase?) Nick uses uppercase throughout in his post, except here in his #3:

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.

I presume that m = M/P. So Y = VM/P. This says rather explicitly that Y is real GDP. (So it should be lowercase, no?) So, also: is Nick a Keynesian? Sometimes, sort of…

3. Mediocre philosophical minds obviously think alike. Matthew’s (and Scott’s) MV = C + I + G + NX is *exactly* where I went in my first stab at this post, which I’ve discarded or at least put aside for the moment.

4. I’ve been coming to the same conclusions about NGDP and velocity. Cf. the subtitle to this post.

I’m going to add one last thought here, to thoroughly muddle the waters: Somebody could presumably argue that price can’t increase by 10% unless the utility derived from produced goods — real output — also increases by 10%. That would resolve the apparent contradiction inherent in the Y = real GDP definition. (At least as that contradiction seems to arise in my example above re: the National Account Identity.) But I can’t really conceive a very convincing empirical and/or theoretical basis for that assertion.

*Cross-posted at Asymptosis.*

The problem is that Y, C, I, G, NX, M, V, and P are all measured in terms of money, but money is only worth what it can buy, so the ratio of C and Y is meaningful, but Y and C itself are unscaled quantities. It’s a lot like astronomy which for most of its history dealt with ratios. It wasn’t until the 17th century that astronomers started getting a grasp on the actual distances and locations of the planets, and it wasn’t until the 18th century that they started getting realistic estimates.

If you want a metric basis, you have to choose one, for example work hours, food calorie equivalents, glory units or risk reductrons. You have to remember that the common existing metric of money or inflation adjusted money has its problems as well. For example, consider the divergence between the GDP deflator and the consumer deflator. Choosing your deflator is, of necessity, political as much as anything else.

Astronomers have this problem as well, though the Galilean framework works pretty well, more accurate and more distant predictions and measuremnts require general relativity and sometimes more esoteric cosmology. These astronomers tend to characterize all the elements beyond helium as “metals” and use units in which c and G are set to one. Once again, everything is back to ratios.

I think economists would do better to drop the absolute measurements and look more closely at the relative measurements. Do everything in terms of nominal Y, or, better yet, in terms of nominal Y per capita. We can measure, sort of, nominal Y and the population. Everything else should flow from ratios with things we can measure.

Hey, it worked for the astronomers.

This is especially timely since there is a transit of Venus coming up, and it is by measuring the transit of Venus that we can determine actual astronomical distances.

“Do everything in terms of nominal Y, or, better yet, in terms of nominal Y per capita.”

Generally, yes. To look at some things you have no choice but to work with inflation-adjusted values. But a good rule of thumb is to use nominal values whenever possible to avoid at least one (potentially large) possibility of error. X as a % of GDP, for instance, instead of “real” X.

But what I’m talking about here is more fundamental. I really wonder whether the arithmetic of these identities makes coherent sense if Y means real GDP.

Oh and to add: the “absolute” measure ultimately has to be utility, or unit: utils. But nobody has ever figured out, as far as I can tell, any method for measuring or converting to utils (whatever they are). Inflation adjustment is ultimately trying to do just that conversion, largely without actually saying so. Naked emperor and all that.

Given: Y=C+I+G+NX

If Y is real, then so are C, I, G, and NX.

What is the problem? 🙂

“What is the problem?”

If prices go up by 10%, Y (real gdp) does too. Even though real output didn’t.

Also:

Again, Y (at least in the expenditure approach to counting) is total dollars spent — GDP

Total dollars spent equals MV.

Which equals PY.

So Y = GDP = MV = PY

So Y = PY

So is Y real or nominal?

This unless MV also counts money “spent” on non-real goods — all those financial transactions/asset swaps. In which case MV (and hence PY) is massively larger than GDP.

Ah yes, the utility problem does make it interesting.

There is also the resources problem. Since at least AD1000, Europe has been on a 200 year cycle where there is a divergence of fuel and land prices as against labor and product of labor prices. It has been generally linked to resources and population pressure, both natural in the form of land, water, fish and ore, and artificial in the form of land, charters, free labor and the like. We are seeing the divergence again, so industrialization may have extended the cycle somewhat, but may not have eliminated it.

No, it doesn’t. If prices rise by 10% with real output unchanged, then the measured NGDP=PY rises by 10%, but RGDP=Y=NGDP/P is unchanged.

The problem is you assume C, I, G, and NX are nominal values, even as you use Y for RGDP. If someone uses Y for RGDP, then C, I, G, and NX must be deflated by a price index too.

Sometimes people use Y for nominal, sometimes for real. When it is used to mean nominal, then you’re right that the velocity equation should be MV=Y.

@Andrew:

Okay. But:

GDP = Total dollars spent = MV

Which equals PY.

So Y = GDP = MV = PY

So Y = PY

And how does a price increase of 10% mean that real Y increases by 10%? It could stay the same, for instance.

Replying to myself, real Y, in the sense of not having a money term, is a vector. E. g., Y = (100 bushels of wheat, 37 hogs, 50 massages, 242 chickens, . . .)

And in that case, in MV = PY, P is also a vector.

“Again, Y (at least in the expenditure approach to counting) is total dollars spent — GDP

“Total dollars spent equals MV.

“Which equals PY.”

That is impossible. If total dollars spent equals MV, then V is dimensionless, as M is measured in dollars. You also claim that Y is measured in dollars. That would mean that P is also dimensionless. That is absurd.

“How much does that cost?”

“13.”

“13 what? Dollars, Euros, Yen? Quarters? Pesos? What?”

AFAICT, the variable Y in the two identities has two related, but different, meanings, and that causes confusion. In 1) Y = C + I + G + NX , Y stands for GDP expressed in terms of money. In 2) MV = PY , Y stands for output expressed somehow or other.

The vagueness of Y in 2) arises this way. If Y is the output expressed as a vector, representing the real gross domestic product, then P is also a vector, representing the prices of the components of Y. The product of these vectors equals Y in 1), IIUC. One way of expressing what I have just said is that Y in 1) is the nominal GDP, while Y in 2) is the real GDP.

The “somehow or other” comes from the concept of the price level, which is a scalar, not a vector. If P stands for the price level, then Y in 2) stands for Y in 2) divided by P. What that means depends upon what the price level means.

There are a number of people in the know here. If I have made a mistake in this note, please let me know. 🙂

Typo: I meant, Y in 2) stands for Y in 1) divided by P. 🙂

Now, to address some of what Nick Rowe says, let’s consider changes in Y.

1) ΔY = ΔC + ΔI + ΔG + ΔNX

Here Δ stands for change, not error.

Given this identity, it would be plausible to suppose that increasing G would tend to increase Y. Nick points out that this identity does not express a causal relationship, however. If increasing G decreased C, I, or NX, it might actually decrease Y. (Besides, Y is nominal GDP. It is possible in 1) for Y to be the real GDP vector, but that is complex and even more uncertain.)

2) can be rewritten using logarithms, if P and Y are scalars.

2) ln M + ln V = ln P + ln Y

Then in terms of change we have

2) Δ ln M + Δ ln V = Δ ln P + Δ ln Y

If we wish to increase Y, then we might suppose that increasing M would tend to increase Y. IIUC, that was pretty much the approach of the Fed after the Great Depression and before Volcker. We might also suppose that decreasing P would tend to increase Y. AFAIK, the Fed has never followed a policy of deflation.

Thanks, Min. That’s what I came to in my comment over at Nick’s. In (2) Y is designated in “some undefined unit of ‘goods.'”

Units-of-goods x price-per-unit = dollars spent = MV.

In any case, we have a term, Y. As far as I can figure, either:

it means different things in the two identities

or

Nobody’s quite sure what it means

Or both.

Which means, in either case:

You can’t do MV = C+I+G+NX

But it’s coherent if:

1. Y means nominal GDP

2. MV=Y

3. Y/P=Real GDP

There is no problem with Y=C+I+G+NX

You measured year 1 output in “Year 1” dollars, and year 2 output in “Year 2” dollars. Hence, you did not compute the change in real GDP, but the change in nominal GDP.

To rewrite…

“But in Year 2, the prices are 10% higher (for whatever reasons). Measured, counted

GDP (a.k.a. “nominalY,” totalPdollar transactions) increases by 10%.”nominalThere is no problem with Y=C+I+G+NX

You measured year 1 output in “Year 1” dollars, and year 2 output in “Year 2” dollars. Hence, you did not compute the change in real GDP, but the change in nominal GDP.

To rewrite…

“But in Year 2, the prices are 10% higher (for whatever reasons). Measured, counted

GDP (a.k.a. “nominalY,” totalPdollar transactions) increases by 10%.”nominalThere is no problem with Y=C+I+G+NX

You measured year 1 output in “Year 1” dollars, and year 2 output in “Year 2” dollars. Hence, you did not compute the change in real GDP, but the change in nominal GDP.

To rewrite…

“But in Year 2, the prices are 10% higher (for whatever reasons). Measured, counted

GDP (a.k.a. “nominalY,” totalPdollar transactions) increases by 10%.”nominalExcept how does Y/P = real GDP? Real GDP is a vector, not a scalar. Right?

It’s not simply some “undefined unit of ‘goods'” in the national accounts. Both nominal and real GDP are dollar aggregates. You just have to make sure you are aggregating consistently… that is, in computing changes in real GDP, you must use a consistent set of prices.

It’s not simply some “undefined unit of ‘goods'” in the national accounts. Both nominal and real GDP are dollar aggregates. You just have to make sure you are aggregating consistently… that is, in computing changes in real GDP, you must use a consistent set of prices.

It’s not simply some “undefined unit of ‘goods'” in the national accounts. Both nominal and real GDP are dollar aggregates. You just have to make sure you are aggregating consistently… that is, in computing changes in real GDP, you must use a consistent set of prices.

Wrong. Real GDP is not a vector. Real GDP is an aggregate, just as is nominal. It’s just a question of how one aggregates. Essentially it’s a question of using consistent prices over time, or not using consistent prices over time.

(And I’m sorry for the triplicates… I have self-reported this bug. I’m definitely

hitting “Post” more than once.)notWrong. Real GDP is not a vector. Real GDP is an aggregate, just as is nominal. It’s just a question of how one aggregates. Essentially it’s a question of using consistent prices over time, or not using consistent prices over time.

(And I’m sorry for the triplicates… I have self-reported this bug. I’m definitely

hitting “Post” more than once.)notWrong. Real GDP is not a vector. Real GDP is an aggregate, just as is nominal. It’s just a question of how one aggregates. Essentially it’s a question of using consistent prices over time, or not using consistent prices over time.

(And I’m sorry for the triplicates… I have self-reported this bug. I’m definitely

hitting “Post” more than once.)notMin, try “13 times as much as 100 years ago.”

Min, try “13 times as much as 100 years ago.”

Min, try “13 times as much as 100 years ago.”

So you are saying that P is dimensionless?

I agree with Andrew. One needs to apply P to both sides of the equation. If you multiply (or divide) Y by P, you have to do the same to C, G, I and NX. Or, put another way: PY = PC + PI + PG + PNX. So, too: MV = PC + PI + PG + PNX. And (MV / P (or Y)) = C + I + G + NX. You just divide P out to get to the standard form of expressing it.

“ Real GDP is not a vector.”

Then what is real about it?

When you write

“GDP = Total dollars spent = MV

Which equals PY.

So Y = GDP = MV = PY

So Y = PY

GDP = Total dollars spent = MV

Which equals PY.”

GDP can be nominal or real. But whichever you choose, you have to be consistent in P. If Y = GDP, then there is no P. So you can’t do that equation Y = GDP = MV = PY. It has to be expressed as Y = GDP = MV/P = PY/P. Or you could say PY = PGDP = MV = PY.

So Y = GDP = MV = PY

So Y = PY

Pls ignore the last two lines. Put my comment into the middle of the quote.

Yes. If you want MV to be in dollars and Y to be in dollars, then yes, P is a price index without dimension. Alternately, you can take Y to be an output index and P to be in dollars– that’s the “generic good” view (of P as price per unit of output) which confuses many. But it’s exactly the same aggregate. You may notice that the NIPA tables report PY in dollars, P without dimension, and Y in dollars… yet NIPA also reports Y separately as a dimensionless index!

What is “real” about it? Output is real. The market value of output is an abstraction.

“You may notice that the NIPA tables report PY in dollars, P without dimension, and Y in dollars… yet NIPA also reports Y separately as a dimensionless index!”

That’s consistency for you! 😉

Real output is a vector.

Besides the nominal/real dilemma there is another related problem: the ignoring of financial assets which also matters when talking about income. If I buy a pound of apples for a dollar and sell it for two dollars my income is one dollar. When I buy a share of General Motors for a dollar and sell it for two dollars my income is again one dollar, but it doesn’t appear anywhere in GDP, which considers only real goods and services. Is it surprising that economics could not predict the Great Depression, Japan’s lost decade or the Great Recession, each of which was caused by the collapse of financial asset markets followed by the collapse of banks. My book “The General Theory of Money” takes care of all these problems: http://www.amazon.com/dp/B0080WPK2I

Har.

To annoy you even further, it even reports both dollar and index values for multiple base years.

That’s because when you swap dollars for stock, you’ve only changed each party’s asset composition– national income is entirely unchanged.

When you buy low and sell high, that’s extra income for you, but only inasmuch as it is a loss of income to the rest of the economy. It’s a wash overall.

The issue is that when you say GDP, it is ambiguous if you mean real or nominal GDP

Nominal GDP=total dollars spent. = PY

Real GDP = (total dollars spent )/ P = (PY)/P=Y