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 Angry Bear.*

## Comments

## 17 responses to “Why Does Y Equal Real GDP?”

It’s not that Y should “mean nominal GDP.” It’s that you computed nominal GDP and labeled it “Y.”

If you define Y to be nominal GDP, then sure, MV=Y. But that’s a notational difference, not one of accounting.

@D R :

GDP is counted in nominal dollars.

GDP = Y

MV is, ineluctably, nominal dollars.

So:

Y = GDP = nominal dollars spent = MV = PY

So:

Y = PY

??

Never really noticed this before, but came up with the following, through Google:

http://www.cato-unbound.org/2009/09/21/jeffrey-rogers-hummel/explanation-vs-prescription/

He writes:

” MV = Py, in which MV (money times its velocity) is equivalent to aggregate demand, and Py represents nominal GDP, the product of the price level and real output.”

i.e.

y = real output (or GDP)

Y = nominal GDP (I guess)

And, although I haven’t read them, the following two sources may put the issue into perspective:

http://mattrognlie.com/2011/05/03/is-mvpy-useful/

http://econlog.econlib.org/archives/2009/09/i_deny_the_sign.html

@Asymptosis

Nominal GDP is counted in nominal (current) dollars. Real GDP is counted in constant dollars.

The equation is money in circulation times turn over (aggregate demand in a period) is equal to average price times transactions (aggregate supply in that period). It’s all nominal unless adjusted for price instability. The problems generally arise from not sticking to what the equation actually says or to make it say something it does not.

If in 2010, I made 100 hammers and they sold for $10 each, my “nominal GDP” is $1000 and my “real GDP” is 1000 (2010) dollars.

If in 2011, I made 100 hammers and they sold for $11 each, my “nominal GDP” is $1100 and my “real GDP” is 1100 (2011) dollars.

This is all correct, and yet my “real GDP” did not increase by 10%, because 1100/1000=1.1 does not use consistent units of measure. In 2010 dollars, my 2011 “real GDP” was 1000. In 2011 dollars, my 2010 “real GDP” was 1100. Either way, my “real GDP” is unchanged.

Does that help make things clear?

Because my 2011 output of 100 hammers times the 2010 price of $10/hammer is $1000.

Because my 2010 output of 100 hammers times the 2011 price of $11/hammer is $1100…

@D R: Thanks, that does help. P (price index) in year 1 (relative to itself) is inevitably 1. So yes, Y= PY. Not so in year 2.

@JKH: Thanks also. cf this from your second link:

Also this comment from Winterspeak in the second link:

“If the “velocity” of money falls it means that people are saving more. So the MV=PY equation, by its own logic, has saving increasing on one side, and argues that (through lower P) this can be managed by lower incomes on the other. It’s obviously nonsense when you put it plainly, and that’s ignoring higher real debt loans.”

It’s obviously nonsense because the premise is false. The velocity of money may fall with no change in aggregate savings whatsoever. It may be that people spend money at a slower rate, but that the fall in income means that in doing so they fail (on the whole) to increase their savings.

@D R

By way of example, on the first of January, March, May, July, September, and November, I trade you $10 for a loaf of bread you made. On the first of February, April, June, August, October, and December, you trade me $10 for a can of tuna I made. Our GDP is $120 (V=12), and we have no savings– we each earn $60 and consume $60.

Now, if we slow down so that I only buy from you on the first of January, May, and November while you buy from me on the first of February, June, and October, V falls to 6, but we still have no savings– we each earn $30 and consume $30. Instead, GDP falls by half.

@Asymptosis

That particular interpretation is as overly simplistic as the equation itself.

@JKH

sorry, # 13 refers to # 10 above

There are 2 GDP calculations:

Adding up the value of all final goods at market prices. This is nominal GDP.

Adding up the value of all final goods at base-year prices. The base-prices are constant as production increases over a period of time. This is real GDP.

Any textbook will give you the equation of exchange MV = PY. But here Y is Y as in Y = F(K,L).

I.e. the output of the production function, which is real products. Indeed my copy of Mankiw’s Macroeconomics explains this clearly when it introduces the equation: “… Y is real GDP, P is the GDP deflator [the ratio of base prices to current prices] and PY nominal GDP” (Macroeconomics, sixth edition, p. 85). The advantage of using uppercase here is that we can later take logs: lnM + lnV = lnP + lnY

and then use lowercase notation to abbreviate: m + v = p + y.

But in other context you will see that Y is used for NGDP whilst y is used for RGDP. Don’t be confused by uppercase Y in the Equation of Exchange – it stands for RGDP. Read Fisher’s “The Purchasing Power of Money” for more. And to see more examples of economics notation in use, I encourage you to take a university course, where you’ll be exposed to all the jargon and notation you could possible ask for.

Again, when you insist that “GDP is counted in nominal dollars” – that’s only true if you’re measuring nominal GDP. When we measure RGDP, we count in “real dollars” – which means that when we add up the value of goods and services, we don’t take their current prices, but rather their base-year prices.

Prices don’t measure utility, they measure the dollar value of marginal benefits (the consumer prefers a second apple to the $1.50 in his pocket, and to the next best use of that dollar-fifty). When you have more dollars your nominal ability to pay rises, and you bid for more goods and services, which bids their prices higher. As the price level rises, the opportunity cost of each dollar falls – the same utility function now leads you to raise your maximum willingness to pay for each good proportionately, because each dollar buys less goods. Equivalently, the supply of dollars decreases their value, so the same value of goods leaves you more willing to spend dollars on each good.

The Equation of Exchange is commonly regarded as an identity, as V = PY/M by definition. But that’s not quite right: V is the number of times each unit of money (base money, for instance) changes hands each year on average. To measure V, we use the Exchange Equation and calculate V = PY/M.

Another, more “stocks-based” approach is to see V as the number of times the total stock of money held by the public must circulate to equal the public’s nominal income. The stock held is kPY, a fraction of nominal income. Since MV = PY, => k = 1/V.

Assume all income is spent. If you hold a quarter of your nominal income in money balances (k = 1/4), then you must spend four money balances a year. Since all money is held by someone, if people are spending four money balances a year, then the money stock must be passing through each account four times a year, so V = 4. The equation is not so much an identity (true by definition) as a synthetic a priori truth – true because of the synthetic facts that all money is held, and one person’s spending is another’s income.

Alternatively, think of MV = PY as a flow equation. Total buying must equal total selling in a given year. MV = total buying (total spending of money). PY = total selling (total quantity of goods sold, multiplied by their price). All output is bought with money. All the output that is bought and sold must correspond to the quantity of money required to purchase it. Since the same dollars can make multiple purchases each year, we multiply M by V. Not exactly an identity, but still true a priori. Then we use the equation to figure out that we can measure V by calculating PY/M, which are easier to observe.

Of course the above isn’t quite right. Total selling is actually PT, where T is the number of transactions. We then take T = vY, and then MV(t)/v = PY, which is written as MV(y) = PY, or just MV = PY. Now don’t go getting confused over the use of lowercase here – v is an entirey new constant! You must always be sensitive to context. Believe it or not, the same is even true in math.