# Ege Erdil comments on What Do GDP Growth Curves Really Mean?

• There is a standard reason why real GDP growth is defined the way it is: it works locally in time and that’s really the best you can ask for from this kind of measure. If you have an agent with utility function defined over goods with no explicit time dependence, you can express the derivative of utility with respect to time as

If you divide both sides by the marginal utility of some good taken as the numeraire, say the first one, then you get

where is the price of good in terms of good . The right hand side is essentially change in real GDP, while the left hand side measures the rate of change of utility over time in “marginal units of ”. If we knew that the marginal utility of the numeraire were somehow constant, then changes in real GDP would be exactly proportional to changes in utility, but in general we can’t know anything like this because from prices we can only really tell the utility function up to a monotonic transformation. This means real GDP is by construction unable to tell us the answer to a question like “how much has life improved since 1960″ without some further assumptions about , since the only information about preferences incorporated into it are prices, so by construction it is incapable of distinguishing utility functions in the same equivalence class under composition by a monotonic transformation.

However, real GDP does tell you the correct thing to look at locally in time: if the time interval is relatively short so that this first order approximation is valid and the marginal utility of the numeraire is roughly constant, it tells you that the changes over that time period have improved welfare as much as some extra amount of the numeraire good would have. If you want to recover global information from that, real GDP satisfies

so what you need for real GDP growth to be a good measure of welfare is for nominal GDP (GDP in units of the numeraire) times the marginal utility of the numeraire to only be a function of , which I think is equivalent to being Cobb-Douglas up to monotonic transformation. The special nature of Cobb-Douglas also came up in another comment, but this is how it comes up here.

I think the discussion in the post is somewhat misleading. There’s really no problem that real GDP ignores goods whose price has been cut by a factor of trillion; in the toy example I gave with Cobb-Douglas utility real GDP is actually a perfect measure of welfare no matter which goods have their prices cut by how much. The problem with real GDP is that it can only work as a measure on the margin because it only uses marginal information (prices), so it’s insensitive to overall transformations of the utility function which don’t affect anything marginal.

• I was hoping somebody would write a comment like this. I didn’t want to put a technical primer in the post (since it’s aimed at a nontechnical audience), but I’m glad it’s here, and I basically agree with the content.

• In addition, I’m confused about how you can agree with both my comment and your post at the same time. You explicitly say, for example, that

Also, “GDP (as it’s actually calculated) measures production growth in the least-revolutionized goods” still seems like basically the right intuitive model over long times and large changes, and the “takeaways” in the post still seem correct.

but this is not what GDP does. In the toy model I gave, real GDP growth perfectly captures increases in utility; and in other models where it fails to do so the problem is not that it puts less weight on goods which are revolutionized more. If a particular good being revolutionized is worth a lot in terms of welfare, then the marginal utility of that good will fall slowly even if its production expands by large factors, so real GDP will keep paying attention to it. If it is worth little, then it’s correct for real GDP to ignore it, since we can come up with arbitrarily many goods (for example, wine manufactured in the year 2058) which have an infinite cost of production until one day the cost suddenly falls from infinity to something very small.

Is it “crazy” that after 2058, people will be able to drink wine manufactured in 2058? I don’t think so, and I assume you don’t either. Presumably this is because this is a relatively useless good if we think about it in terms of the consumer surplus or utility people would derive from it, so the fact that it is “revolutionized” is irrelevant. The obvious way to correct for this is to weigh increases in the consumption of goods by the marginal utility people derive from them, which is why real GDP is a measure that works locally.

How do you reconcile this claim you make in your post with my comment?

• The main takeaways in the post generally do not assume we’re thinking of GDP as a proxy for utility/​consumer value. In particular, I strongly agree with:

The problem with real GDP is that it can only work as a measure [of consumer value] on the margin because it only uses marginal information (prices), so it’s insensitive to overall transformations of the utility function which don’t affect anything marginal.

It remains basically true that goods whose price does not drop end up much more heavily weighted in GDP. Whether or not this weighting is “correct” (for purposes of using GDP as a proxy for consumer value) isn’t especially relevant to how true the claim is, though it may be relevant to how interesting one finds the claim, depending on one’s intended purpose.

To the extent that we should stop using GDP as a proxy for consumer value, the question of “should a proxy for consumer value more heavily weight goods whose price does not drop?” just isn’t that relevant. The interesting question is not what a proxy for consumer value should do, but rather what GDP does do, and what that tells us.

• The reason I bring up the weighting of GDP growth is that there are some “revolutions” which are irrelevant and some “revolutions” which are relevant from whatever perspective you’re judging “craziness”. In particular, it’s absurd to think that the year 2058 will be crazy because suddenly people will be able to drink wine manufactured in the year 2058 at a low cost.

Consider this claim from your post:

When we see slow, mostly-steady real GDP growth curves, that mostly tells us about the slow and steady increase in production of things which haven’t been revolutionized. It tells us approximately-nothing about the huge revolutions in e.g. electronics.

The way I interpret it, this claim is incorrect. Real GDP growth does tell you about the huge revolution in electronics, the same way that it tells you about the huge revolution in the production of wine in the year 2058. It can’t do it globally for the reasons I discussed, but it does do it locally at each point in time. The reason it appears to not tell you about it is because it (correctly) weighs each “revolution” by how important they actually were to consumers, rather than weighing them by how much the cost of production of said good fell.

I think the source of the ambiguity is that it’s not clear what you mean by a “revolution”. Do we define “revolutions” by decreases in marginal utility (i.e. prices) or by increases in overall utility (i.e. consumer surplus)? If you mean the former, then the wine example shows that it doesn’t really matter if a good is revolutionized in this sense for our judgment of how “crazy” such a change would be. If you mean the latter, then your claim that “GDP measures growth in goods that are revolutionized least” is false, because GDP is exactly designed to capture the marginal increase in consumer surplus.

• The reason it appears to not tell you about it is because it (correctly) weighs each “revolution” by how important they actually were to consumers...

No it doesn’t. It weighs them by price (i.e. marginal utility = production opportunity cost) at the quantities consumed. That is not a good proxy for how important they actually were to consumers.

I think the source of the ambiguity is that it’s not clear what you mean by a “revolution”. Do we define “revolutions” by decreases in marginal utility (i.e. prices) or by increases in overall utility (i.e. consumer surplus)?

I’m mostly operationalizing “revolution” as a big drop in production cost.

I think the wine example is conflating two different “prices”: the consumer’s marginal utility, and the opportunity cost to produce the wine. The latter is at least extremely large, and plausibly infinite, but the former is not. If we actually somehow obtained a pallet of 2058 wine today, it would be quite a novelty, but it would sell at auction for a decidedly non-infinite price. (And if people realized how quickly its value would depreciate, it might even sell for a relatively low price, assuming there were enough supply to satisfy a few rich novelty-buyers.) The two prices are not currently equal because production has hit its lower bound (i.e. zero).

More generally, there are lots of things which would be expensive to produce today, will likely be cheap to produce in the future, but don’t create all that much value. We just don’t produce any of them, To think properly about how crazy the future would be, we need to think about the consumer’s perspective, not the production cost.

A technological revolution does typically involve a big drop in production cost. Note, however, that this does not necessarily mean a big drop in marginal utility.

Now, I do think there’s still a core point of your argument which survives:

Real GDP growth does tell you about the huge revolution in electronics, the same way that it tells you about the huge revolution in the production of wine in the year 2058.

The thing it tells us is that the huge revolution in electronics produced goods whose marginal utility is low at current consumption levels/​production levels.

When I say “real GDP growth curves mostly tell us about the slow and steady increase in production of things which haven’t been revolutionized”, I mean something orthogonal to that. I mean that the real GDP growth curve looks almost-the-same in world without a big electronics revolution as it does in a world with a big electronics revolution. It “doesn’t tell us about things which were revolutionized” in an information-theoretic sense—i.e. we can’t tell by looking at the GDP curve whether or not there was a technological revolution. That still seems basically correct, at least for “revolutions” for which price falls more than consumption increases.

• I think there’s some kind of miscommunication going on here, because I think what you’re saying is trivially wrong while you seem convinced that it’s correct despite knowing about my point of view.

No it doesn’t. It weighs them by price (i.e. marginal utility = production opportunity cost) at the quantities consumed. That is not a good proxy for how important they actually were to consumers.

Yes it is—on the margin. You can’t hope for it to be globally good because of the argument I gave, but locally of course you can, that’s what marginal utility means! This is modulo the zero lower bound problem you discuss in the subsequent paragraphs, but that problem is not as significant as you might think in practice, since very few revolutions happen in such a short timespan that the zero lower bound would throw things off by much.

I’m mostly operationalizing “revolution” as a big drop in production cost.

I think the wine example is conflating two different “prices”: the consumer’s marginal utility, and the opportunity cost to produce the wine. The latter is at least extremely large, and plausibly infinite, but the former is not. If we actually somehow obtained a pallet of 2058 wine today, it would be quite a novelty, but it would sell at auction for a decidedly non-infinite price. (And if people realized how quickly its value would depreciate, it might even sell for a relatively low price, assuming there were enough supply to satisfy a few rich novelty-buyers.) The two prices are not currently equal because production has hit its lower bound (i.e. zero).

I think a pallet of wine that somehow traveled through time would sell at a very high, though not infinite, price. The fact that the price is merely “very high” instead of “infinite” doesn’t affect my argument in the least. Your claim that the two prices aren’t currently equal because of the zero lower bound problem is certainly correct, but it’s a technical objection that can be fixed by modifying the example a little bit without changing anything about its core message. For instance, you can take the good in question to be “sending a spacecraft to the surface of Mars and maintaining it there”, which currently has a nonzero consumption. It’s conceivable, at least to me, that even if the cost of doing this comes down by a factor of a billion, it won’t produce anything like a commensurate amount of consumer surplus.

My problem is, as I said before, that if “revolution” is operationalized as a big fall in production costs then your claim about “real GDP measuring growth in the production of goods that is revolutionized least” is false, because there are examples which avoid the boundary problems you bring up (so relative marginal utility is always equal to relative marginal cost) and in which a good that is revolutionized would dominate the growth in real GDP because the demand for that good is so elastic, i.e. the curvature of the utility function with respect to that good is so low.

A technological revolution does typically involve a big drop in production cost. Note, however, that this does not necessarily mean a big drop in marginal utility.

How does it not “necessarily” mean a big drop in marginal utility if you get rid of your objection related to the zero lower bound? A model in which this is not true would have to break the property that the ratio of marginal costs is equal to the ratio of marginal utilities, which is only going to happen if the optimization problem of some agent is solved at a boundary point of some choice space rather than an interior point.

Nothing in your post hints at this distinction, so I’m confused why you’re bringing it up now.

When I say “real GDP growth curves mostly tell us about the slow and steady increase in production of things which haven’t been revolutionized”, I mean something orthogonal to that. I mean that the real GDP growth curve looks almost-the-same in world without a big electronics revolution as it does in a world with a big electronics revolution.

Can you demonstrate these claims in the context of the Cobb-Douglas toy model, or if you think your argument hinges on the utility function not having a special form, can you write down a model of your own which demonstrates this “approximate invariance under revolutions” property? In my toy model your claim is obviously false (because real GDP growth is a perfect proxy for increases in utility) so I don’t understand where you’re coming from here.

• I think in this case omitting the discussion about equivalence under monotonic transformations leads people in the direction of macroeconomic alchemy—they try to squeeze information about welfare from relative prices and quantities even though it’s actually impossible to do it.

The correct way to think about this is probably to use von Neumann’s approach to expected utility: pick three times in history, say ; assume that where is the utility of living around time and ask people for a probability such that they would be indifferent between a certainty of living in time versus a probability of living in time and a probability of living in time . You can then conclude that

if an expected utility model is applicable to the situation, so you would be getting actual information about the relative differences in how well off people were at various times in history. Obviously we can’t set up a contingent claims market and compare the prices we would get on some assets to infer some value for , but just imagining having to make this gamble at some odds gives you a better framework to use in thinking about the question “how much have things improved, really?”