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