Path-dependence of measuring real GDP?

I normally try to avoid index number theory. Don't trust me on this.

It is well understood that real GDP is a very imperfect measure of welfare. We teach that in first year macro. That is not what this post is about. What I'm worried about is whether real GDP is an imperfect measure of itself. Does it have internal validity?

If better technology enabled producers of new goods to ramp up production more quickly to meet initial demand, would that cause the measured growth rate to fall?

Initially an economy produces 100 kg of apples at $1 per kg. Then bananas get invented. After a long slow adjustment, the economy eventually reaches a new long run equilibrium and produces 50 kg of apples at $1 per kg and 50 kg of bananas at $1 per kg.

We know that nominal GDP stays the same at $100. But what happens to real GDP? The answer depends on what happened during the long slow process of adjustment. Was it supply, or demand, or both, that caused the slow adjustment?

To keep it simple, assume the price of apples is always $1 per kg (the central bank targets the price of apples). And assume the quantity of bananas produced and consumed increases slowly and continuously from 0 to 50 kg. And assume the statistical agency that measures GDP has access to continuous time data (so we can ignore the difference between Paasche, Laspeyres, and Fisher price indices). Remember that Real GDP = Nominal GDP/Price Level. What happens to the price of bananas during the long slow adjustment period?

1. Assume that supply and demand adjust equally slowly to the invention of bananas. It takes producers time to switch from producing apples to producing bananas, and it takes consumers time to switch from consuming apples to consuming bananas. So the price of bananas is constant at $1 per kg during the adjustment. The weights in the price index slowly change (with a falling number of apples and increasing number of bananas in the basket), but the price index stays constant, because neither price is changing. Real GDP is the same as it was before bananas were invented.
2. Assume that supply adjusts more slowly than demand. Producers need a high price to give them the incentive to adjust. So the price of bananas starts out above $1, and slowly falls over time to $1. So the price index falls over time. Real GDP ends up higher than it was before bananas were invented.
3. Assume that demand adjusts more slowly than supply. Consumers need a low price to give them the incentive to adjust. So the price of bananas starts out below $1, and slowly rises over time to $1. So the price index rises over time. Real GDP ends up lower than it was before bananas were invented.

Even if you say that my third case is implausible, and that demand always adjusts more quickly than supply, so the price of new goods (relative to existing goods) always starts out high and falls over time, that does not resolve the problem. The initial price of bananas, when they first appear on the market, depends on how many bananas can be grown in that very first season. Anything that enables producers of new goods to increase production for the initial roll-out will permanently reduce the measured level of real GDP.

The underlying problem is that we do not observe the price of bananas before bananas are invented. Unless you allow a discontinuous jump in real GDP the moment the new good hits the market, even if initial production is negligible, I don't think you can avoid this paradox.

Thanks to commenters (especially louis) on my previous post, and to Brent Moulton via Twitter and David Rosnick via email. Errors and opinions are mine alone.

Comments

[Lord]

As this starts as an apple economy, the large effect is to lower apple investment, but apple investment can't fall beneath depreciation (or more accurately, can't fall below depreciation at all) without affecting the price of apples, so there is a limit to how fast banana production can ramp up without causing this. Then there is the matter of relative investment costs, higher investment costs would lower growth while lower investment costs would increase growth and only if these were nearly equal would pricing be significant and even then only as one good among many. I expect pricing would have this effect which would differ depending on whether apples and bananas were treated as substitutes. It would take some time for bananas to even make it into baskets reducing their impact on measurements, missing both any discontinuity or initial change.

[Nick Rowe]

Lord: there's lots of things that could be determining what's happening during the adjustment period. But for the question addressed in this post, the only thing that matters is whether the *relative* price of bananas starts out above one apple or below one apple. Dollar prices won't matter. I made the assumption that the *nominal* price of apples stays at $1 throughout, just to make the discussion simpler.

[Market Fiscalist]

Clarifying question about price indexes.

When a new good is introduced how is it factored int the price index ? For example suppose in period 1 100 apples are produced, and in period 2 50 Apples and 25 Banana. As the price of apples is fixed at $1 and GDP is fixed at $100 we know the price of bananas is $2. But how do we use this information to calculate the change in the price level between period 1 and period 2?

(I can see that once we have sales data and prices for all periods after bananas have appeared the logic that Nick describes kicks in - but I can't see how we deal with the introduction of bananas)

[Brent Moulton]

I think you're right. There's a technical literature on path dependence of Divisia indices (see, for example, a paper from Nick Oulton). The chain indices used for GDP and the CPI are approximations to the continuous-time Divisia indices, so it's clearly relevant. The problem arises whenever new goods are introduced or old ones disappear, and would also be relevant when large shocks (e.g., major wars) take place that result in temporary major changes to consumption patterns. It's a hard problem, and though the literature suggested some potential solutions, they haven't reached the stage of statistical agency implementation.

Regarding your third case, sometimes sellers offer low initial prices for goods or services for which consumption is likely to be persistent -- new social networks, narcotics. So I don't think it's implausible.

[Majromax]

Your price index doesn't have to rise or fall monotonically with time. If we weight the price of each good by its share of nominal spending, then the starting and ending price indexes are the same – the price level is 1.

What happens in the meantime depends on supply and demand effects.

Imagine we're partway through the transition, and our economy sells 50kg of apples at $1 and 25kg of bananas at $2 (the remaining banana trees are waiting to mature). Nominal GDP is $100 as before, but the expense-weighted price index is 50%*($1 + $2) = $1.50, so real GDP is $66. This is lower than the long-run equilibrium of $100, but so is the actual fruit consumption. It would be accurate to call this economy comparatively depressed.

If banana trees mature quickly but can only be planted slowly, then maybe we're selling 90kg of apples at $1 and 10kg of bananas at $2. Nominal GDP is now $110, and the expense-weighted price index is about 1.18 (9/11*1 + 2/11*2), so the economy still looks depressed (real GDP of 93) but not as badly.

In the other case (apple trees are exogenously converted to banana trees but bananas are an acquired taste), then we might sell 90kg of apples at $1 and 10kg of bananas at $0.50. Nominal GDP is $95, and the price index is about 0.973 (9/9.5 + 0.25/9.5), so real GDP is about 97.6. If the new bana

In both cases, a statistician would observe a fall in real GDP due to a preference shift, followed by a rebound to its prior level. The unstated assumption for meaningful RGDP is that the utility of a price-index basket is approximately the same throughout the interval, and that's what's obviously untrue with preference changes.

[Roger Sparks]

Nick: You price the value of bananas at a constant $1 over the long period that it takes to ramp production from zero to 50 Kg.

It seems to me that you need to simultaneously recognize that to establish 'price', you need to have two persons trading and agreeing on a commonly accepted value ('price').

Hence, one option is that the banana producer is otherwise unemployed but now producing his first banana. The GDP should increase. OTOH, the buyer may be substituting goods so his contribution to increased GDP is negative, making the final GDP change zero. Further OTOH, the buyer may be borrowing to fund the purchase which would result in a valid GDP increase.

I think I am pointing out that just considering the producer's contribution to GDP is an incomplete exercise. A new product has several tentacles that impact GDP.

[Nick Rowe]

MF: Brent Moulton (who commented just after you) knows much more than I do about how your question is normally answered by those who actually measure GDP. But I think the short answer is: "with difficulty". I think the point of this post is to say that that the problem posed by new goods does not get very small even if the expenditure share of new goods starts out very small when they are first introduced. The problem gets bigger over time.

Thanks Brent! Glad to hear I'm not totally out-to-lunch on this! Purely speculative, but I'm wondering if this might be part of the productivity growth slowdown puzzle? If new goods don't fall in price nowadays as much as they used to, simply because it's easier to build a big batch of new phones before releasing them (and/or they keep initial prices low because of network effects, as you say).

[Nick Rowe]

Majro: "Your price index doesn't have to rise or fall monotonically with time. If we weight the price of each good by its share of nominal spending, then the starting and ending price indexes are the same – the price level is 1."

The share of nominal spending before bananas are invented? Or after we get to the new long run equilibrium? If "before", then the price of bananas doesn't matter at all.

Roger: "Nick: You price the value of bananas at a constant $1 over the long period that it takes to ramp production from zero to 50 Kg."

Why? And suppose we have a different example, where the (relative) price of bananas never reaches some constant level, because of ongoing technological improvement in growing bananas.

[Brent Moulton]

Market Fiscalist: The way the CPI is calculated by statistical agencies, they would simply ignore the bananas in calculating the period 1 to period 2 price change. The "market basket" that is selected in period 1 doesn't have any bananas in it, so the price change from period 1 to period 2 is based solely on apples (no price change).

Nick: The standard (Konus) constant-utility cost-of-living index theory assumes that each period's consumption is on a stable demand curve. If, for your second case, you assume that it takes time for producers to switch but that each period consumers are consuming along their demand curve, I think the continuous time price index should come close to the true Konus cost-of-living index--especially if there isn't any discrete jump in consumption when the item is first introduced. Real GDP increases, and that's what we expect given that consumers prefer consuming A+B to A. Your first and third cases involve adjustment costs for consumers and thus don't fit easily in the standard cost-of-living index model.

[Nick Rowe]

Brent: I was wondering if something like that would work. We could maybe imagine extending my second case to have a succession of new goods invented.

[Gerard MacDonell]

Do they teach students in first year macro that when they later become professors they should say "we teach that in first year macro"? I see that expression a lot these days. It sounds like, this is truth. ;)

I remember Kotlikoff once using it when I suggested to him that it might not be clarifying to say that the "true" debt is 119 gazillion dollars. "I have been teaching my students that for 20 years." Oh, well could you then please stop writing that AND stop confusing your young charges?

More seriously, I am curious how important you think this index number issue might be compared with the more basic issue of measuring and aggregating individual utility. What is GDP supposed to correlate with? Back in the day it was about the flow of nominal demand but then seemed to morph into aggregate real goodness.

I love your work. Thanks.

[louis]

At risk of straying too far off topic, let's think of this in a trade context.
Instead of bananas being new to the world, let's say they are just new to a market. Country A can produce only apples, Country B produces both apples and bananas. In country B, one apple is tradeable for one banana.
When country B opens up to trade with country A, suddenly there is a global jump in demand for bananas. The price of bananas in terms of apples rises in country B to induce the people to consume fewer bananas and more apples. A certain number of apples flows to country B in exchange for a smaller flow of bananas back to A.
A's GDP, in terms of apples, shouldn't change -- its apple crop is the same as it was the previous year, although consumers are better off due to trade.
You'd think B's GDP should rise, as the value of the bananas it produces is worth more in terms of apples than it used to be. Or would we just say that's a change in the price level and real GDP is unchanged? Seems like this depends on how you construct the price index.
Gains from trade are real, but it seems they would be tricky to capture in GDP framework

[Nick Rowe]

Gerard: thanks!

The main cases where I use that "we teach that in first year" line is (for example, like in this case) some anti-economist says "GDP is a bad measure of welfare!" thinking they are making some devastating new critique of economics.

How important is this particular issue, empirically? I don't know. And I first wanted to make sure my head was straight on the theory. But speculating wildly, I am wondering if the productivity growth slowdown might be caused by something like this? I have no evidence to back that up, but that is what is at the back of my mind about *possible* empirical relevance. It's less about utility per se (which was at the back of my mind in the last post) than productivity.

louis: only a little off-topic. I'm going to be a bit shaky on the answer. I *think* country B's real GDP stays the same, since it is measured in terms of the basket it produces. But its real income *measured in terms of the basket it consumes* will rise or stay the same, depending on whether we use a Paasch or Laspeyres index (old basket or new basket). More simply, if your Terms of Trade improve, real GDP can stay the same, but increase when NGDP is deflated by the CPI.

[marcel proust]

Didn't Hausman address this issue back in the '90s?

[marcel proust]

"this issue" == valuing newly introduced products.
Briefly (IIRC, and I may well not), the solution he presents for getting the price index right is to estimate a demand curve for bananas (presumably right after their introduction) and set the pre-introduction price to the point where demand is zero. This allows for comparison of the CPI's before and after bananas hit the market.

[Nick Rowe]

marcel: yes. But that was a different question (trying to measure utility gains). And that assumes an extreme version of my case 2, where demand adjusts instantly and supply adjusts slowly.

[Matt Young]

Let's get the same result but use the basket brigade model. Instead of a continuous result,m this result is the same but never leaves probability space.

Chiqita introduces bananas, and initially we love them. So the delivery trucks and consumer baskets are fuller than normal. Each time we move goods our cash cards swipe at market price, so the Savings and Loan machine is building the significant probability distribution of cash swipes in (deposit) or out (loan). The S&L machine enforces amortization on the full baskets and everyone pays a suprising 'rate' on their purchases.

As a result of the bananas, the yield curve is steeper. The S&L machine is forcing us to go to work and build bigger baskets, or develop a more granular distribution that we can make more frequent trips and keep our basket from ov er filling. If we choose the latter, we get more term points on the yield curve. Or, we can go home and work out a substitution between bananas and mangos, making our baskets fill normally again.

The key here is our sense of equilibrium, we know when our baskets ate optimally full. We know that because we hate waiting in line, so our sense of frustration stabilizes the queues. When queues ate sable, mean equals variance in our basket. At that condition, then every single person in the monetary zone can independently calculate the probability of the basket overflowing or underflowing. Supply then equals demand and container algebra works, we can pretend to be continuous.

It is Nicks model, fouled up a bit, but just recast as a sequence of probabilistic purchases.

[Market Fiscalist]

'The way the CPI is calculated by statistical agencies, they would simply ignore the bananas in calculating the period 1 to period 2 price change.'

Thanks Brent.

OK, so if the only things that are taken into account to calculate RGDP are NGDP and CPI then new goods affect RGDP only to the extent that they chnage price after they have been launched. If the rate at which the price of new goods fall in price after launch declines over time then other things equal the rate of growth of RGDP will also fall.

Assuming that the price reductions in new goods represents improved productivity in production of these new goods this seems valid. If RGDP is calculated in this way (NGDP and CPI) then the subjective benefits on new goods are just ignored and only improved productivity of all goods drives RGDP.

So in Nick's example 3 the introduction of bananas has no effect on RGDP and then as the price of bananas increases over time (assuming this reflects productivity declines, perhaps bananas make workers fatter and less efficient?) it seems valid this should put downward pressure on RGDP. If the price change of bananas does not reflect changes in productivity then all bets are off.

[Pete Bias]

Hi Nick,

If I'm reading this correctly, I don't think you have enough information to answer your question. What matters for real gdp is the spending in two comparison periods. By keeping the price level constant it becomes trivial that the percentage increase or decrease in spending will completely dictate the percentage change in real gdp.
Holding the price of apples constant is trickier than it appears. All kinds of ways that could happen in the market while the other good's market is adapting, so we need more info.
But if we have total spending, we have it.
I think spending is your key.

Pete Bias

[Mike Sandifer]

Don't know if this helps, but as someone trained in behavioral psychology, I say increasing variety increases sales, which would presumably mean labor being willing to work more/harder producing apples and bananas to afford to buy more overall output. Hence, I would guess that to the degree apples and bananas differ in relevant characteristics, such as nutritional composition for example, is in some proportion to the degree to which overall output would increase.