Monday, May 29, 2017

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.

Tuesday, April 25, 2017

[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.

Tuesday, January 31, 2017

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: 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.

Friday, January 6, 2017

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 $$ pY^{\left(p\right)}_t\neq\frac{p}{p_t}\left(W_t-W_{t-1}+C_t\right)=\frac{p}{p_t}Y_t $$