What is so "real" about "real GDP"

According to the latest report, “Real gross domestic product” grew at a 2.3 percent annual rate in the fourth quarter of 2023. In a very, very vague sense this means that— after accounting for the effects of the seasons— in October through December, we produced about 0.56 percent more than we did in July through September. More accuately, the value of what we produced in those later three months was a bit over half a percent more than the value of what we produced previously.

In the U.S. national accounts, the value of GDP is measured in dollars. Specifically, the market value of all 4th-quarter production included in the measure (again accounting for the season) was a bit over \$7.4 trillion. This is the value assigned by the Bureau of Economic Analysisn (BEA) to that production and called "GDP". Likewise, for the period three months prior, the figure was a bit over \$7.3 trillion. We say "GDP" increased by about \$81 billion at quarterly rates.

So if GDP purports to measure the dollar value of production, why don't we just say the \$81 billion represents the increase in the value of production? Well, we do! It is the (market dollar) value of measured production. The problem, such as it is, is that a dollar value doesn't really connect back to production of real goods and services. To see this, consider what would happen if we used shares of Apple stock in place of the dollar. If we could have purchased all third-quarter production with Apple shares at the July 1 open of \$212.09 per share, we would need to exchange 34.6 billion shares. To purchase all fourth-quarter production with Apple shares on the October 1 open of \$229.52 per share, we would need only 32.4 billion shares. That is, production value fell by a couple billion Apple shares. So did production value go up (dollars) or down (Apple shares)?

The answer is found in neither. The discrepancy is driven by valuing dollars less and Apple shares more. This has nothing to do with production of actual goods and services. To get a “real” sense of production value over time, we need something more consistent. One way might be to try and imagine an exhange between our fourth-quarter selves and out third-quarter selves. What on earth does that mean?

Fun with underspecified games

Thanks a lot to Andrew Gelman for bringing this gem to my attention.

In this particular battle between physicists and economists, I’m taking the economists’ side. https://t.co/fFHGB4owld

— Andrew Gelman (@StatModeling) December 19, 2020

I've been thinking about a particular game of chance. Here is how the game is described in the article to which Gelman links:

Starting with $100, your bankroll increases 50% every time you flip heads. But if the coin lands on tails, you lose 40% of your total. Since you’re just as likely to flip heads as tails, it would appear that you should, on average, come out ahead if you played enough times because your potential payoff each time is greater than your potential loss. In economics jargon, the expected utility is positive, so one might assume that taking the bet is a no-brainer.

Yet in real life, people routinely decline the bet. Paradoxes like these are often used to highlight irrationality or human bias in decision making. But to Peters, it’s simply because people understand it’s a bad deal.

I do not agree. My problem with this “paradox” is that the game (as described) is not entirely clear. What is missing from this description?

• First, it starts the player with a \$100 bankroll. From where did this \$100 appear? Does the player buy in for \$100? Or do they buy in for only \$20?
• Second, the desciption reads “every time” the player flips heads, but how many times must the player flip? Is the player free to cash out at any time, or does the game never end? If the latter, then of course it is a bad deal— the player never has a chance to cash out winnings!
• Third, what happens when the game is over? Can the player buy in to play again?

The answers to these questions matter to whether the game is really a “bad deal” or a rather money-making machine for the player.

Let us investigate some possible rules and evaluate the “deal”

The Dollar Does Matter for Trade

Consider Figure 1, which shows the trade deficit as a percent of total trade. Since the early 1970s, there have been three large swings toward bigger deficits: the mid 1970s, the mid 1980s and the early 2000s. Each of these swings was more sustained and each subsequent movement toward balanced trade was less complete than the last, leaving a trend of greater imbalances. By the second quarter of 2017, the United States spent \$124 in imports for every \$100 worth of goods and services exported. At the end of 1974, however, trade had been balanced.

Figure 1: Trade Balance (Exports-Imports) as a Percent of Total Trade (Exports+Imports)

(Source)

To some extent, this trade imbalance reflects oil imports. However, as Figure 2 shows, removing both petroleum imports and agricultural exports from the trade data does not hugely change the story. If anything, restricting to this core trade amplifies the trend.

Figure 2: Trade Balance (Total and Core Goods)

(Source)

Understandably, increasing trade deficits have coincided with a rising dollar. A product that sells for 100 dollars in the United States would cost 85 euros today compared to only 70 euros a decade ago. (Both dollars and euros have been adjusted for inflation in each area.) This means that U.S. exporters will either have to lower their price to maintain their market, or accept a smaller market, or most likely some combination. In any case, the value of exports will fall.

Similar, but not identical, calculations apply to the foreign producers exporting to the United States. A foreign producer selling an item for 85 euros need only charge 100 dollars today compared to 120 dollars in 2007. They could either keep their profit margin constant and reduce the price of their product to \$100, or they could allow their profit margin to expand by setting the price somewhere between \$100 and \$120. Thus, the foreign producer is usually happy to increase supply to the United States as the dollar rises even if considerable savings were to be passed on to domestic consumers. Typically, real imports should rise with the dollar.

Of course, timing is not exact. It may take time for producers and importers to suppose that a change in currency is going to be relatively long-lived. Retail prices may not move in the short run as importers contract with suppliers over the longer run. By no means would we expect the dollar to be the sole factor in determining trade balances, but taken together we should expect broad co-movement between U.S. trade deficits and the dollar.

Indeed, this is what we observe in Figure 3. Here, we have added a real broad dollar index.

Figure 3: Core Balance and the Real Broad Dollar

(Source)

Certainly since 1980, trade deficits have increased sharply in response to large rises in the dollar and declined similarly in response to rebalancing if with some lag. This is particularly clear in the core trade numbers. Other events are also visible in Figure 3 — the sharp but very temporary rise in the dollar in late 2008 did not result in much movement in the trade deficit; the recession of 1974 saw non-petroleum imports fall nearly 25 percent before recovering — but periods of high dollar are clearly associated with large deficits and lower dollar with much more modest ones.

(This post originally appeared on the CEPR blog.)

The Evolution of Capital, Part III

In the previous post, we saw how, under restrictive assumptions, $r < g$ means that capital cannot self-perpetuate. Holders of wealth— in the aggregate— must save more than capital income provides or the wealth-income ratio $\beta$ will fall.

Unfortunately, the assumptions behind this conclusion are surely overly restrictive. In particular, we should at the very least investigate the dynamics when there are long-run capital gains. When there are no miscellaneous volume adjustments, $$ \beta_t=\frac{1+q_t}{1+g_t}\left(1+g^{ws}_t\right)\beta_{t-1} $$ where $q$ is the rate of inflation-adjusted capital gains, and $g^{ws}$ is the pure rate of growth of wealth due to saving (that is, $g^{ws}={S}/{W}$– the savings-wealth ratio. We may rewrite the evolution of $\beta$ as $$ \beta_t=\frac{1+q_t}{1+g_t}\left(\beta_{t-1}+s_{t-1}\right) $$ and therefore $$ \left(\beta_t-\bar{\beta}_t\right)=\frac{1+q_t}{1+g_t}\left(\beta_{t-1}-\bar{\beta}_t\right) $$ where $$ \bar{\beta}_t=\frac{1+q_t}{g_t-q_t}s_{t-1} $$ Thus, so long as $g>q$— the rate of capital gains is less than the growth rate of net income— then $\beta$ tends toward a finite ratio. However, if $q>g$, then $\beta$ grows without bound. The rate at which wealth appreciates may become more critical to the dynamics than the interest and dividends it may provide.

The Evolution of Capital, Part II.

A long while back, I promised to get into the significance of $r>g$ to Piketty’s framework. To review where I left off,

Piketty’s “stock of capital is increasing faster than net income” if and only if there is sufficient net savings irrespective of the rate of return on capital.

This result depended upon the assumptions that there are zero miscellaneous volume adjustments to the capital stock and zero inflation-adjusted capital gains. Under these assumptions, the evolution of the wealth-income ratio $\beta$ follows $$ \beta_t=\frac{1}{1+g_t}\left(\beta_{t-1}+s_{t-1}\right) $$ Equivalently, we may write $$ \left(\beta_t-\bar{\beta}_t\right)=\frac{1}{1+g_t}\left(\beta_{t-1}-\bar{\beta}_t\right) $$ where $\bar{\beta}_t={s_{t-1}}/{g_t}$. That is, $\beta$ is always tending toward ${s}/{g}$ so long as there is real growth in net income ($g>0$). This is Piketty’s Second Law in its simplest form.

Now, $s$ is defined as net savings as a share of net income. If we put savings instead in terms of net capital income, $$ \zeta\equiv\frac{S}{Y^k} $$ then starting with Piketty’s First Law (the identity $\alpha=r\beta$) we find that the economy is tending toward $$ \frac{\alpha}{r}=\beta=\frac{s}{g}=\frac{\alpha\zeta}{g} $$ If we then assume that all net savings come out of net capital income, we find $$ \frac{r}{g}=\frac{1}{\zeta}\geq 1 $$ or $r>g$.

Put another way: if, in the long run, Piketty’s Second Law holds and $r < g$, then $\zeta>1$. That is, under these very restrictive conditions, capital owners must save more than their capital income— in the aggregate, capital cannot self-perpetuate.

Unfortunately, real capital gains are something we do observe in the real world, so the story is surely more complex. We’ll look at that in the next (very mathy) post.

[From May, 2013] Reinhart and Rogoff Trip Over Data While Attacking Krugman

I am re-upping this post, originally at the CEPR blog to amplify a new paper by Michael Ash, Deepankar Basu, and Arindrajit Dube. See also Dube’s contemporaneous piece,“A Note on Debt, Growth, and Causality” for something more sophisticated than mine.

---

Yesterday, Carmen Reinhart—she of the infamous Excel error—wrote an open letter to Paul Krugman taking issue with his “spectacularly uncivil behavior.” That his “characterization of our work is selective and shallow.” In particular, Reinhart cites Krugman’s views on Italy. She writes:

However, [falling interest rates in “high-debt Italy”] is meant to re-enforce your strongly held view that high debt is not a problem (even for Italy) and that causality runs exclusively from slow growth to debt. You do not mention that in this miracle economy, GDP fell by more than 2 percent in 2012 and is expected to fall by a similar amount this year. Elsewhere you have stated that you are sure that Italy’s long-term secular growth/debt problems, which date back to the 1990s, are purely a case of slow growth causing high debt. This claim is highly debatable.

In fact, Reinhart recently cited Italy as an example of a “more recent public debt overhang episode.” She cites another paper to back up her claim that the evidence shows the direction of causality runs from high debt to slow growth. But even a cursory examination of the data undermines that case.

Figure 1 takes data from Reinhart’s paper in the Journal of Economic Perspectives and shows very clearly that Italy built up its debt after growth slowed significantly— not the other way around. In fact, when growth slowed back in 1974, Italy’s debt-to-GDP was only 41.3 percent. Italy did not reach 90 percent debt-to-GDP until 1988—some 14 years later.

Figure 1: Real GDP Index (Italy Since 1947) (log)

Source: Reinhart, Reinhart, and Rogoff and author’s calculations.
Note: Specified years indicate first year of high-debt episode (see Reinhart, Reinhart, and Rogoff)

Indeed, there is a clear association in Italy’s post-war data between high debt and slow growth, but it clearly tells a story very different than what Reinhart would have us believe.

From 1947-74, real economic growth in Italy averaged 5.8 percent per year. Over the period 1975-88 (when Italy’s debt grew from 41.3 to 90.9 percent of GDP) economic growth averaged only 2.7 percent per year—a fall of 3.2 percentage points. It is clear, based on Reinhart’s data, that high debt could not have caused this slowdown in Italy’s economic growth, even if Italy’s period of low debt is associated with much faster growth.

Nor is Italy the sole example. In all four such recent examples of advanced countries with episodes of high debt, the slowdown precedes the increase in debt.

Figure 2: Real GDP Indices Since 1947 (log)

Source: Reinhart, Reinhart, and Rogoff and author’s calculations.
Note: Specified years indicate first year of high-debt episode (see Reinhart, Reinhart, and Rogoff)

Though less obvious for Belgium, most of the jump in debt-to-GDP came in 1980 and was largely the result of a series break in the data. According to the data on Reinhart and Rogoff’s website, Belgium’s gross general government debt-to-GDP was 62.5 percent in 1970 and falling (debt-to-GDP stood at 57.8 percent in 1974— the year real GDP peaked). Nevertheless, from the peak in real GDP in 1948 to peak in 1974, economic growth in Belgium averaged 4.2 percent per year. When the economy bottomed out in 1975, debt was only 54.4 percent of GDP, and did not reach 90 percent until 1983. Yet from 1975-83, growth averaged only 2.2 percent per year.

For the other countries, it is even more obvious that the economies slowed well before reaching high levels of debt. Clearly, Reinhart should look carefully to her own data before lashing out at Krugman.

In the Wild: Identities Deceive

I have argued before that accounting identities can be deceiving. I specifically argued early on here that the GDP identity $$ Y=C+I+G+X-M $$ does not by itself imply that increased imports $(M)$ reduce Gross Domestic Product $(Y)$; the presentation merely invites the reader to form a model in which it is true. But this, from Noah Smith, is just painful:

@imalatkinson @jodiecongirl @Jacob_Atkinson1 Imports neither add to GDP nor subtract from it. Exports add to it. Econ PhD 2012. :D

— Noah Smith (@Noahpinion) January 30, 2017

To be clear, Smith’s argument is that that this is true “mechanically”— distinct from any model. But this just requires us to ask what Smith means by “mechanically.”

Imports-in-GDP is a correction to avoid double-counting when measuring GDP based on final sales. We start with domestic sales of consumption, fixed investment, and government goods and services $(C+I^*+G)$. To this, we add sales of all goods and services to foreign economies— that is, exports $(X)$ are considered final sales in terms of their disposal with respect to the domestic economy. To get domestic production from final sales requires two adjustments. First, we add in net unsold production— that is, changes in inventories $(\Delta inv)$; second, we subtract foreign production sold domestically— imports. Thus, $$ Y=C+I^*+G+X+\Delta inv-M $$ Changes in inventories are grouped with fixed investment into gross investment $(I=I^*+\Delta inv)$ so we get $$ Y=C+I+G+X-M $$ But this does $not$ tell us one way or another whether imports add or subtract from GDP. It merely tells us that if $M$ does change, that something else must also change. If imports increase, then $Y$ must fall or $C+I+G+X$ must rise. We need a model to tell us anything more.

For example, it could be that in the long run, imports today increase GDP by increasing pressure on domestic industries to become more productive. One might argue that \$10 of additional $M$ means an additional \$1 of $Y$ and \$11 of $C$. But this is clearly not what Smith has in mind.

For imports to “neither add to GDP nor subtract from it” the change in $Y$ must be zero from any change in $M$. The identity thus tells us Smith believes that any change in $M$ is balanced by an corresponding movement in $C+I+G+X$. A \$10 increase in $M$ must “mechanically” raise $C+I+G+X$ by \$10.

Let us suppose for a moment that this makes sense. What in $C+I+G+X$ can be so definitively affected? For imported goods, the only reasonable answer is $\Delta inv$. When I import a consumer good, I might hope to sell that good and therefore have it counted later in final domestic sales of consumption. I might even have an order for something specific and so be extremely confident that eventually that the extra $M$ will become $C$. However, the immediate effect is that I have increased my inventories. Thus, every dollar of goods imports adds directly to gross investment and so the net effect on GDP is zero.

Given that, well, the net effect of goods exports on GDP is also zero by similar logic. Between production and exportation, goods pass through inventory. I might increase production (increasing $Y$) to compensate for the loss of inventory, but every goods export dollar is immediately a dollar taken out of inventories. In this sense, exports do not add to GDP as Smith argues.

At the very least, we cannot argue that exports definitively add to GDP in any immediate sense. Contra Smith, There is no mechanism by which an increase in exports requires an increase in production. And that is what makes him so painful to read.

How Should We Measure Real Savings?

Suppose that at 12:01AM on 1 January I had \$114, and at 11:59PM on 31 December I have \$180. Obviously, I spent less than my income, saving \$66.

On the other hand, at the end of 2015, \$430.89 could be exchanged for one bitcoin; a year later, one bitcoin ran \$966.30. Thus, on 1 January I had Ƀ0.2646 and on 31 December only Ƀ0.1863. Obviously, I overspent my income by Ƀ0.0783.

So which is it? Did I save or dissave over the course of the year? Let us back up a bit.

The saving discussed above we might call “comprehensive.” It is simply my change in wealth over the period. But this wealth is nominal— measured in terms of currency, rather than real goods and services that such wealth might purchase.

Suppose that at 12:01AM on 1 January I had wealth sufficient to purchase 100 pounds of apples, and at 11:59PM on 31 December I had wealth sufficient to purchase 150 pounds of apples. Obviously, I had saved an amount equivalent to 50 pounds of apples. Neither does it matter what the price of apples was on 1 January, nor does it matter what the price of apples was on 31 December. My savings of 50 pounds of apples was “real”— literally comparing pounds of apples to pounds of apples.

At 12:01AM on 1 January, 50 pounds of apples runs \$57. At 11:59PM on 31 December, 50 pounds of apples runs \$60. Clearly, my \$66 saved does not correspond in a direct way to my 50 pounds of apples saved.

The problem, of course, is that (due to inflation) \$114 on 1 January is not real in the same way that \$114 on 31 December is real. It makes no sense to take my 31 December nominal wealth of \$180 and subtract my 1 January nominal wealth of \$114 to get \$66 in real savings. Rather, if we wish to report real savings in terms of dollars, we must choose a consistent price for apples.

Real savings is the change in inflation-adjusted stocks of wealth $$ S^{\left(p\right)}_t=\frac{W_t}{P_t}-\frac{W_{t-1}}{P_{t-1}}=\frac{{W_t}\times{p}/{P_t}-{W_{t-1}}\times{p}/{P_{t-1}}}{p} $$ where $p$ is the common price chosen. Presented in EOP prices ($p=P_t$) real savings comes to $$ P_tS^{\left(P_t\right)}_t=W_t-W_{t-1}+W_{t-1}-W_{t-1}\frac{P_t}{P_{t-1}}=W_t-W_{t-1}-\pi_tW_{t-1} $$ where $\pi_t$ is inflation over the period.

	EOY Price Level	EOY Nominal Wealth	EOY Real Wealth
			EOY 2015 Prices	EOY 2016 Prices
2015	57	114	114	120
2016	60	180	171	180
Note: change over 2016		66	57	60

We may describe our 50 pounds of apples saved as either 57 “1-January dollars” or 60 “31-December dollars.” We may even describe real savings in terms of bitcoin:

	EOY Price Level	EOY Nominal Wealth	EOY Real Wealth
			EOY 2015 Prices	EOY 2016 Prices
2015	0.1323	0.2646	0.2646	0.1242
2016	0.0621	0.1863	0.3969	0.1863
Note: change over 2016		-0.0783	0.1323	0.0621

Unlike changes in nominal wealth, changes in real wealth make intuitive sense and are consistent between choices of denomination. At the end of 2016, Ƀ0.0621 could be exchanged for \$60; at the end of 2015, Ƀ0.1323 could be exchanged for \$57. This is because we have employed a consistent set of prices for both periods and denominations. We cannot convert nominal savings measured in dollars to nominal savings in bitcoin because we have not employed a consistent rate of exchange.

To answer our original question, then, the observed savings are real despite the fall in nominal bitcoin wealth. Further, note that real savings is not equal to inflation-adjusted nominal savings; if inflation over the period is zero, then real savings over the period— expressed in EOP dollars— is equal to nominal savings over the period.

Finally, if “comprehensive” savings is defined this way, then real “comprehensive” income (given period-average consumption prices $p_t$) follows naturally as real “comprehensive” savings plus real (inflation-adjusted) consumption: $$ pY^{\left(p\right)}_t=pS^{\left(p\right)}_t+\frac{p}{p_t}C_t=\frac{p}{P_t}\left(W_t-W_{t-1}-\pi_tW_{t-1}\right)+\frac{p}{p_t}C_t $$ noting that this is not equal to inflation-adjusted nominal “comprehensive” income $$ pY^{\left(p\right)}_t\neq\frac{p}{p_t}\left(W_t-W_{t-1}+C_t\right)=\frac{p}{p_t}Y_t $$

Another round on the “elephant curve”

A report from the Resolution Foundation has touched off another round of analysis of Branko Milanovic‘s work on global income growth. Much of what they discuss can be found elsewhere. Including here. But I want to focus for a moment on their claim that

we find that the weak figures for the mature economies as a whole are driven by Japan (reflecting in part its two ‘lost decades’ of growth post-bubble, but primarily due to likely flawed data) and by Eastern European states (with large falls in incomes following the collapse of the Soviet Union after 1988).

Not to suggest that these things did not occur; rather, I find weak the argument that they drive the results. In the figure below, I have estimated the “quasi-nonanonymous” GIC. We see there the growth in real per-capita income among country-deciles sorted by their 1988 income percentiles— with and without China. I have not “reshuffled” incomes when including China, so it is as if Chinese growth had been comparable to its 1988 income peers.

In other words, excess growth in China accounts for nearly all of the fast growth in the 20-70th percentiles. The growth which remains to that half of the world distribution is quite modest— about 1.7 percent per capita annually. True, the more developed 70-99th percentiles seem to have grown somewhat more slowly (1.3 percent per capita annually); even including Japan and Eastern Europe as done here, the severe stagnation which the “elephant” misleadingly suggested does not really exist.

Of course, this latter point was already made clear in Milanovic’s work, as he reminds us.

When we do quasi non-anonymous chart with 1988 country/deciles fixed, we keep 1988 decile populations fixed. Fixed! pic.twitter.com/pdPHcxg6JQ

— Branko Milanovic (@BrankoMilan) September 14, 2016

Ultimately, there are two points:

1. The years 1988-2008 had seen considerable change in what it means to be part of the global middle class.
2. Outside China, the global 10-99th enjoyed only modest growth. To the extent those years saw stagnation among more developed economies, it extended to most of the world. Except China.

Thank Goodness I Did Not Attend GMU

So much Tyler Cowen to criticize here, but I want to focus on one point. The most relevant portion:

At time period zero, a boss hires one hundred workers, who at the time are perceived as being of roughly equal quality and thus are offered the same wage. After a few years on the job, however, some are “keepers,” while others are being paid more than their marginal products.

Because of firing aversion, they are not fired. Because of sticky nominal wages, they also do not take a pay cut. If the economy is imperfectly competitive, and times are good, this nonetheless can be a stable equilibrium.

Now let’s say a negative shock comes along: demand, supply, maybe a bit of both, as is usually the case. At some margin these workers can no longer be carried and the firing aversion of the boss is overcome and they lose their jobs. Then, a few points:

1. They’re not getting those jobs back.
2. They’re not worth a comparable wage elsewhere in many cases.
3. Per hour productivity likely will rise, even adjusting for ex ante measures of changes in worker composition.
4. Companies won’t want to pay higher wages to lure these workers out of leisure, rather they are branded as less productive than average and properly so.

(all emphasis added)

Do you see the problem here? Cowen treats marginal product as a property of the worker, so the worker is “properly” branded as less productive. Yet if all the original hires— “perceived as being of roughy equal quality”— are in fact entirely identical, then the dynamics are the same— if not the explanation. When the demand shock hits, the marginal productivity of all one hundred hires falls. All one hundred are being paid more than “their” marginal productivity. As soon as a few are fired, however, marginal productivity rises so not all one hundred are fired. Presumably, workers are fired in sufficient numbers to bring marginal productivity in line with the sticky wage. The same wage the fired workers are presumably holding out for!

The fired workers— though identical to those retained except in their new (un)employment status— are branded as less worthy just for their bad luck. The fired workers are worth just as much in that they could step in perfectly for any of the retained workers.

By passing off marginal productivity as a property of the worker, Cowen manages to blame workers for their unemployment.

A Little Confusion About Inflation

It may help to clear up a little confusion about inflation. Inflation is basically dimensionless— price over price— and does not have dimension of 1/time as suggested by Nick Rowe. Rather, inflation is an exchange rate. Instead of converting between currencies of different countries (say) at a single moment, inflation (and interest rates!) convert between different times the same nominal currency. That is, one may convert today’s dollars in today’s euros; likewise, one may convert yesterday’s dollars into today’s dollars. An inflation rate of 2 percent over the past year means that what cost \$100 last year today costs \$102. Thus \$100 last year is equivalent to \$102 today.

Where folks get a bit mixed up is, hey, that’s 2 percent per year. Doesn‘t that put time in the denominator? No. All we are saying is that we add 2 percent (compounded) in each period— in this case one year. For example, 10 percent inflation over 5 years is not 10/5=2 percent per year, because if we added to prices 2 percent per year for 10 years, we would wind up with a price level about 10.4 percent higher. ”Per year" describes the frequency of compounding at the specified rate.

But we still divide by time to compute the rate, right? We say $$ 1+\pi=\exp{\frac{\ln{\!\left({P_f}/{P_i}\right)}}{t_f-t_i}} $$ do we not? No, we do not. This is obvious shorthand. (Obvious because the dimensions do not work out.) More carefully, we may write $$ 1+\pi=\exp{\!\left[\Delta\frac{\ln{\!\left({P_f}/{P_i}\right)}}{t_f-t_i}\right]} $$ where $\Delta$ is the period of time between compoundings. Equivalently, $$ 1+\pi=\exp{\frac{\ln{\!\left({P_f}/{P_i}\right)}}{n}} $$ where $n$ is the (dimensionless) number of compoundings. That is, “year” specifies $\Delta$— with units of time.

More on the “elephant curve”

Based on feedback from my previous post, I have started a FAQ.

Moving on, I would like to make a couple points. First, the results are not terribly fragile. Previously, I used the data and code of Lakner and Milanovic to produce GICs with and without China. One interesting quirk of their approach is that quantile populations are not necessarily very even. In 1988, the 20th-25th percentile represents less than 150 million people, while the 40th-45th represents more than 275 million. While this might seem like a big problem, it is not. In Figure 1, I use a different method for binning the data. In particular, I allow country-decile groups to split across quantiles. This helps make much more uniform the population size represented by each percentile.

Figure 1: Growth Incidence With Decile Splitting

Source: Lakner and Milanovic and author’s calculations

Clearly, the story is very much the same. But these “anonymous” GICs are easy to misinterpret. With China included in the data, the incomes associated with the 75th-80th percentiles hardly budged; this does not mean that the incomes of people in the those percentiles failed to rise. Lakner and Milanovic also produce “quasi-nonanonymous” GICs which show the average income growth for the actual country-deciles represented in the 1988 quantiles. (pdf) They estimate that across the board, growth exceeded 20 percent and broke 90 percent in the middle of the income distribution.

That method is laid out in their paper, but I take a more direct approach. First, however, I estimate the 1988 and 2008 population and per-capita income for each country-decile based on the observed growth rates. Then, dividing the country-deciles into global quantiles ranked by 1988 per-capita income, I compute the aggregate per-capita growth from 1988 to 2008 treating the quantile as a single aggregate. $$ G_\mathrm{quantile} = 100\times\frac{\sum_{\in\mathrm{quantile}}{\mathrm{Pop}_{2008}\times\mathrm{Inc}_{2008}}}{\sum_{\in\mathrm{quantile}}{\mathrm{Pop}_{1988}\times\mathrm{Inc}_{1988}}}\times\frac{\sum_{\in\mathrm{quantile}}{\mathrm{Pop}_{1988}}}{\sum_{\in\mathrm{quantile}}{\mathrm{Pop}_{2008}}}-100 $$ Then, keeping the quantile definitions, I re-compute the growth without China. The resulting quasi-nonanonymous GICs are seen in Figure 2. We see that— except for the bottom decile— the results are similar to what Lakner and Milanovic.

Figure 2: Quasi-Nonanonymous Growth Incidence

Source: Lakner and Milanovic and author’s calculations

Without China, growth at the 10th-70th percentiles was quite modest— about 1.7 percent per-capita per year. This agrees with earlier analysis based on very different methods. Excluding the global top percentile, higher-income countries did not grow quite as fast— about 1.3 percent per year. Progress in more developed countries has been uneven, favoring the top incomes there. But we do not see the utter collapse of middle-class incomes a naïve reading of the anonymous GIC would suggest.

The Incredible Story of Developing Country Income Growth: Was it Just China?

To what extent has the age of globalization benefitted developing countries—and what of the poor in those countries? To what extent has such progress been driven by local policy decisions rather than a more global phenomenon? Has such development come alongside stagnation of poor and middle incomes within more developed countries and large benefited the extremely rich?

One way—however incomplete—to begin an investigation would be to look at the global “growth incidence curve” (GIC) of Lakner and Milanovic. They estimate the worldwide distributions of income in both 1988 and 2008, which allows them to answer questions such as “How does median (the 50th percentile) income change between the two years.” The GIC is sometimes referred to as the “elephant curve” for its resemblance to the beast.

Figure 1 shows the worldwide GIC as produced directly by Lakner and Milanovic’s public data and code.

Figure 1: Lakner and Milanovic Growth Incidence Curve

Source: Lakner and Milanovic

As seen in the figure, the average income representing the world’s 50-55th percentiles rose more rapidly than any other group. Entrance into the upper half of the world distribution required in 2008 some 76 percent more income—adjusted for inflation—than it did in 1988. Likewise, the average income defining the top 1% rose only 65 percent over the same period. Between, however, the distribution become much more compressed. The average income of the world’s 75th-80th percentiles in 2008 was \$3831—up only 1.3 percent from \$3782 in 1988.

Milanovic looks at this “global reshuffle of income” and finds “it would be hard to dismiss the period 1988-2008... as being one of failure.” While two decades of 2.9 percent annual growth would be reasonable enough, this appears to be much less global and much more local—driven by China’s very rapid progress. Doubtless, China’s poor represented a large fraction of the world’s poor, and growth there greatly increased their incomes. Still, it is critical to investigate how much of the reshuffle is specific to China. With a simple edit of line 12 of their code¹ we may re-run with China excluded from the data.

More on Noah Smith’s Low, Low, Standards

To review, Noah Smith asserted that poor countries “incredible progress” in the last 30 years. Challenged, he erroneously tried to make examples of Mexico and Brazil– both of which have suffered pretty substantial growth failures in the period. (He presented data which was not adjusted for inflation which made gains look much, much larger than an honest defense would permit.) He then pointed to the faster growth of 2000s. Indeed, the 2000s were in the last 30 years. But if he meant ten years then why say 30? This makes no sense, especially as growth again slowed over the last five years.

As a consequence of this spat, Smith has tried to peg me as a poverty-decline-denier, writing

[O]n Twitter, David Rosnick strongly challenged the very existence of a rapid recent drop in poverty. At first he declared that the poor-country boom was purely a China phenomenon.

Yes, I first argued that the “poor-country” boom of the last 30 years was largely China. All evidence points to just this. I am hardly the only person to point out that China can bias your thinking about worldwide trends. Branko Milanović wrote of the “ambivalent role of China in global income distribution” several months back.

But I never argued that, say, \$1/day poverty rates have not actually fallen rapidly. Smith knows this very well and is simply trolling (successfully, to judge by my reaction.) The only difference between us is that I am far less convinced that the rapid fall is as “incredible” as Smith insists.

How could I possibly think that Smith is hyperbolic when he writes things like

This is incredible— nothing short of a miracle. Nothing like this has ever happened before in recorded history.

In truth, this is a ridiculous statement. Of course it has never happened before. So long as poor countries grow and that poor people in those countries share in that growth, then it would be inevitable that fewer and fewer people would live below any threshold of absolute poverty. Believe me, I am thrilled that the percentage of people living in such poverty has come down rapidly. But would I call it a miracle? Please.

Are we supposed to be impressed that someone with income of \$0.99 per day one year now has an income of \$1.01 per day? Starvation is not determined by a bright line. Be happy that such poor people are starving a little less than before, but let us not pretend that their situations have so very much improved.

But poverty rates are falling more rapidly these days, right? Yes, the numbers do bear this out. But it is a mistake simply to infer that they are falling more rapidly because of incredible progress among poor countries. If we found incredible shared growth in poor countries (such as China) then the economy will rapidly pull large numbers of people out of poverty. But there is another important factor: if an incredible number of people live just below the poverty line, then little growth will also pull large numbers of people out of poverty. Such progress in poverty reduction would be best described as “credible.”

That is, even if China had not grown so rapidly, the fact that so large a proportion of people there lived near the poverty line meant that large reductions in poverty rates were all but a matter of time. The fact that China grew so fast for so long meant that large numbers quickly approached and then passed the threshold. So which is it? Faster growth or more people happening to live just short of the poverty line? Consider two countries with exactly the same (high) poverty rate, but one country (dark blue) happens to have more of its population near the poverty line as in Figure 1.

Figure 1: Distribution Matters for Poverty Reduction

Holding constant each country’s level of inequality (measured by the Gini), if both countries grow at the same rate, then the more equal country will pull about 75 percent more people across the poverty line. Put another way, if the more equal country grows at 2 percent, the less-equal country must grow at 3.5 percent in order to keep its poverty rate from rising above that of its neighbor. The important point here is that it takes a lot of growth to make up for not having a population already near the poverty line.

Given any reasonable distribution of world income, an unprecedented share of people would approach the poverty line. And in fact this is exactly what happened in the late 1980s.

(source: Our World In Data)

Of course, fast-growing China pulled its poor across the line much faster than other countries. No matter what Smith thinks about India’s rate of growth, there is a vast chasm between China’s reduction in its poverty rate and what India managed in the same period.

(source: Our World In Data)

So the “miracle” of accelerated poverty reduction is the result of a combination of fast-growing China and the non-China world’s modal income happening to lie close to the poverty line. If Noah Smith wants to sell it as a miracle, I guess that is fine so long as he understands it took China to turn the coincidence into a miracle. Coincidence might be the kind of stuff he considers heady, but not me.

Noah Smith has very low standards

Noah Smith made a ridiculous claim and insists that he is correct. But he does not stop there. For some reason, he chooses to blow it all up by exaggerating the differences in our positions.

To review, Smith asserted “incredible progress in poor countries” over last 30 years. Doubtless, there has been progress over the last three decades. After all, countries do generally grow. But incredible? Progress was generally more slow in comparison to, say, the 1960s and 70s. To be sure, some countries have done well. India– a Smith favorite— grew at an average annualized rate of 4.6 percent per capita between 1985 and 2015. And sure, there are a lot of poor people in India. But I do not consider this “incredibly” fast growth in comparison to 1960-80. The Penn World Tables show countries growing at least 4.6 percent per-capita per year between 1960 and 1980: Spain, Malawi, Tunisia, Portugal, Malaysia, Brazil, Greece, Thailand, Cyprus, Korea, Hong Kong, Japan, Gabon, Taiwan, Singapore, Malta, Botswana, and Romania.

Maybe every single one of these countries simply caught up following a period of poor growth yet no such claim may be made regarding India. Maybe I just have higher standards for incredulity. I find Botswana growing 8 percent per year a bit more impressive.

Smith did look at papers I suggested— which indicates that since 1980 there has slowdown in many indicators of progress across all income levels. But then he moves the goalposts, writing about per-capita GDP growth

By [CEPR’s] measure, the 2000-2010 decade exceeds or ties the supposed golden age of the 60s and 70s, for all but the top income quintile.
...
[T]he graph clearly shows that Rosnick is wrong, and the recent unprecedented progress of global poor countries is not just a China story. Case closed.

No doubt, growth has resumed for a period. And had Smith originally asserted that poor countries enjoyed a decade of growth close to that of middle-income countries in the 1960s and 1970s, I could hardly argue. But again, we are looking for “incredible” growth over three decades— not one. Twenty years of growth one percentage point below is not made-up for by a decade half a percentage point of growth above. Hardly “unprecedented” is growth in recent decades.

I argued that truly “incredible” growth over the last few decades comes from China, rather than “poor countries” in general. Yes, China has a large share of the world’s poor, but it is still not kosher to attribute to “poor countries” what may be specific to one country. Haiti’s poor can take little comfort in China’s success. Still, between 1985 and 2015, China sustained 8.6 percent per-capita growth and now produce 12 times per person than 30 years ago. Assuming the poor in China benefitted anywhere near as well as per-capita output might suggest, we are talking about tremendous gains to very large numbers of people with who had precious little. I will return to the question of poverty later.

First, however, I would like to address Smith’s complaint that CEPR’s several reports are deficient because they averaged countries rather than people. We use countries because that is the unit of observation, and we do not want to say countries generally improved performance if Chinese policymakers did something right while the IMF pushed terrible policies on the most vulnerable. Even Smith used “countries” and not “people.” Perhaps this was just a miscommunication on his part. Twitter is not always the clearest of communication channels.

But I am willing to humor Smith on this point. What happens if we weight real per-capita GDP growth by population and create a worldwide index?

Figure 1: World Growth in per-capita GDP

Sources: Penn World Tables (1950-2011), IMF World Economic Outlook (2011-) including IMF projections

Indeed! Population-weighted growth has accelerated rather than declined. However, my argument is that this is driven by growth in China, which obviously does weight strongly. Has population-weighted growth outside China accelerated?

Figure 2: World (less China) Growth in per-capita GDP

Sources: Penn World Tables (1950-2011), IMF World Economic Outlook (2011-) including IMF projections

Oops. No evidence for “incredible” growth here.

Figure 3: Without China, A Shortfall in World Growth Relative to Trend

Sources: Penn World Tables (1950-2011), IMF World Economic Outlook (2011-) including IMF projections

Yes, we see some catch-up in the 2010s– catch-up which has not as of yet continued past the period examined in CEPR’s paper on the subject. Of course, this mixes higher and lower-income countries. So maybe this is a golden age of poor-country development. Or not. But it is clear that population weights matter because of China rather than because of poor people. Which is nothing new.

It seems to me that not only is Smith willing to stand by data which in no way supports his case but he is easily impressed as well. More on that in the next post.

Noah Smith Shuts His Ears and Cries “La La La”

Recently, Noah Smith made reference to “the incredible progress in poor countries in the last 30 years.” Clearly, Noah has a low bar for incredulity. Based on a variety of measures, countries generally progressed more slowly in recent decades than they had previously. (pdf) (pdf) I challenged Noah on this point, suggesting that he was unduly influenced by China‘s truly extraordinary growth. For example, average annual per-capita GDP growth in Latin America over 2000-10 was little more than half that of 1960-80– clearly better than the 1980’s, which saw growth one-tenth that of the previous two decades.

Nor did growth accelerate in the last five years— averaging 1.1 percent. Progress, to be sure, but the last 30 years saw 1.2 percent annual average growth in per-capita GDP. Smith’s reference to 30 years of incredible growth certainly does not apply there.

So it was extra provocative when Smith cited Mexico and Brazil as examples of places where “living standards have improved a lot.” Mexico grew 38 percent per-capita in the 30 years between 1985 and 2015— a whopping 1.1 percent per year. Brazil grew even more slowly at 0.8 percent per year. When called on it, Smith wrote

@ViscidKonrad @DeanBaker13 Heyyy, look at this, Mexico is almost twice as rich as in 2000! https://t.co/4RJM6mI9fu pic.twitter.com/GjOgv7NEsv

— Noah Smith (@Noahpinion) May 29, 2016

Doubled! Now, the data here shows 85 percent growth from 1999-2013, which seemingly would have resulted in a 16-year doubling from 1999 to 2015 if the trend in this data continued.

How can we reconcile Mexico‘s slow growth 1985-2015 and much more rapid growth between 1999-2013? Does Smith believe Mexico collapsed by a quarter between 1985 and 1999? Actual growth was 6.6 percent— slow, but not that bad. The answer was pretty obvious to those of us who have looked at the data in any depth. Smith’s numbers are in current dollars. Adjusting for inflation, Mexico grew 28 percent from 2000-15. That is a far, far cry from Smith’s 100 percent.

Worse, Smith cited Mexico in support of progress over 30 years— not 15. Over the previous 15, Mexico grew all of 8 percent— a mere half of one percent per year. Brazil grew even more slowly, averaging only 0.8 percent per year from 1985-2015. Smith simply does not know what he is talking about yet insists he did not err.

@ViscidKonrad I probably make a lot of errors in life, but I haven't in this discussion! :-)

— Noah Smith (@Noahpinion) May 29, 2016

What is it called when one reiterates a wrong position once confronted with the facts? Derp, perhaps?

Accounting Identities are Useful

Nick Rowe directs us today to an eyebrow-raiser. Economist Ankit Mital writes

In the current system of cross-border bookkeeping, unilateral transfers are recorded in the current account (income from trade and foreign business interests) and not the capital account (financial transactions that end up balancing the current account deficit or surplus) side of the transactions.

This is really interesting to those of us who mind our accounting identities. In its simplest form, the balance of payments identity states that the current (CA) and capital account (KA) balances sum to zero. That is, any transaction which increases the CA balance must also decrease the KA balance. If you see the CA balance changing with no corresponding change in the KA balance then you are failing to correctly record the transaction.

So what is Mital missing? First, the KA balance is defined to be the change in foreign ownership of domestic assets less the change in domestic ownership of foreign assets. So if you— residing in the UK— send me £1, then the UK sees foreign ownership of domestic assets rise. That is, the KA rises by £1– exactly balancing the £1 fall in the CA.

It does not matter exactly how the transfer takes place. If you send me \$1.40 instead, then the UK sees domestic ownership of foreign assets fall by \$1.40, raising the KA by \$1.40 and exactly balancing the \$1.40 fall in the CA. If in my name you hand a £1 coin to a London bank to open a deposit account, then again the UK sees foreign ownership of domestic assets rise. (The relevant asset being the deposit and not the coin which moves only between UK actors.)

And there lies the value in minding accounting identities. They force one to think carefully about what else changes.

Depreciation and Income Shares

I would like now to wade briefly into a debate over the gap between growth in productivity and wages by introducing a bit of modeling fun. It seems clear that— in recent decades— although wage income has grown more slowly than GDP there has been little difference between the growth rate of NDP (GDP net of capital depreciation) and the growth rate of total labor compensation.

On the other hand, inequality of compensation has increased quite a bit— driving a large wedge between pay at the top and pay of the ordinary worker. None of this is news. Somewhat less clear is whether net product is more or less appropriate as a comparison. At first blush, workers still have to produce the whole of output no matter how much investment goes to replacing depreciating capital. It might make sense for labor compensation to rise in step with gross production. But...

Differential confusion and a note on “Standard Neoclassical pedagogy”

In my Comment to Standish and Keen, I asserted that it must be that if $P$ is defined as a function $P\!\left(Q\right)$ then it must be that ${\partial P}/{\partial q_i}$ must be zero because $P$ is not a function of $q_i$, but one of $Q$ alone. Standish and Keen wish to argue that if we hold $Q=\sum_i{q_i}$, then ${\partial P}/{\partial q_i}={dP}/{dQ}$, which is assumed to be negative. It is my contention that this may appear correct at first blush, but this compact notation masks hidden assumptions about the underlying economic model.

I tried to explain this in Section 4.2, but it appears my message was lost. Here, I am going to clarify the notation a bit. When specifying the inputs to a function, I am going to use square brackets. Evaluation of a function will employ parentheses. That is, $f\!\left[x\right]$ should be read “$f$— a function of $x$” while $f\!\left(y\right)$ should be read “$f$— a function of one variable evaluated at $y$.”

For example, though Wilfred Kaplan states in Advanced Calculus (3^rd ed.) that

If $z=f\!\left(x,y\right)$ and $x=g\!\left(u,v\right)$, $y=g\!\left(u,v\right)$, then $$ \frac{\partial z}{\partial u}=\frac{\partial z}{\partial x}\frac{\partial x}{\partial u}+\frac{\partial z}{\partial y}\frac{\partial y}{\partial u} $$

Kaplan also clarifies on the following page that $z\!\left[u,v\right]=f\!\left(g\!\left(u,v\right),h\!\left(u,v\right)\right)$ “is the function whose derivative with respect to $u$ is denoted by ${\partial z}/{\partial u}$.”

That is, when $z$ is evaluated at $\left(g\!\left(u,v\right),h\!\left(u,v\right)\right)$ the result is an entirely new function. Kaplan’s chain rule could be written more clearly (if pedantically)

If $z\!\left[x,y\right]=f\!\left(x,y\right)$ and $x\!\left[u,v\right]=g\!\left(u,v\right)$, $y\!\left[u,v\right]=h\!\left(u,v\right)$, then $$ \frac{\partial z^*\!\left[u,v\right]}{\partial u}=\frac{\partial z\!\left[x,y\right]}{\partial x}\frac{\partial x\!\left[u,v\right]}{\partial u}+\frac{\partial z\!\left[x,y\right]}{\partial y}\frac{\partial y\!\left[u,v\right]}{\partial u} $$ where $z^*\!\left[u,v\right]=z\!\left(g\!\left(u,v\right),h\!\left(u,v\right)\right)=f\!\left(g\!\left(u,v\right),h\!\left(u,v\right)\right)$

“Rationality” in the Theory of the Firm... Conclusion

Previously: Introduction; Part 1; Part 2; Part 3; Part 4; Part 5; Part 6; Part 7; Part 8

As we have seen, the response of Keen and Standish is sorely lacking. They continue to misunderstand or misrepresent textbook models, fail to recognize how competitive profit maximization lowers profits, offer no coherent theory to support their contrary arguments, and continue to insist– despite all evidence– that their simulations support their claims.

Defending this failure, the authors write

Admittedly, the agents specified in our paper are not sophisticated enough to do this, but this was a deliberate choice, since we wanted to show that agents following a simple iterative and non-collusive algorithm would choose output levels that clustered around the true profit-maximizing level of output, and not the Cournot level.

The words “not sophisticated” and “simple” are perhaps telling. Keen and Standish seem to argue that their firms are rational profit-maximizers to but insufficiently intelligent to figure out that someone else might have a better strategy. The most frustrating thing about their work is that the claim they fail to support– the existence of a competitive equilibrium at the collusive level within the infinitely-repeated Cournot-Nash game– is itself textbook. Far from debunking textbook economics, they simply do a terrible job of catching up to it.

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Read my original Comment (including Technical Appendix) at World Economic Review.