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.