Thomas Piketty and the evolution of capital

Having finally read through Capital in the 21^st Century (and spent considerable time with the supplied data) I find many critiques of Thomas Piketty very odd. Part of this is that Piketty (to my mind, anyway) tries very hard to take care with his words. Thus, I get annoyed when I see Branko Milanovic write

Just as a reminder: as we all know by now, $r>g$ implies that the stock of capital is increasing faster than net income.

We know this? Milanovic of course does not. Indeed, his post outlines a case where it may not be true. What irks me is that Milanovic seems to believe it to be generally understood. Frankly, I am not clear if Milanovic is critiquing Piketty or the public. With that in mind, let us begin to review the basic analytical framework presented in Capital.

Piketty defines “capital” or “wealth” as the market value of productive physical capital and net financial assets owned by households and government. Piketty’s capital includes residential real estate, but not consumer durables. It includes corporate-owned equipment through direct stock holdings of households and indirect holdings through pensions and other savings vehicles. Capital also includes holdings of foreign assets net of liabilities. Piketty divides the total value by national income (net of capital depreciation) and expresses the result, $\beta$, in terms of years.

Now define (net) savings as net investment and net acquisition of foreign financial assets so as to equal the change in real wealth, apart from miscellaneous volume adjustments and inflation-adjusted capital gains. If we presently assume these are zero, then the capital-income ratio must evolve as $$ \beta_t=\frac{\beta_{t-1}+s_{t-1}}{1+g_t} $$ where $s_t$ is net savings as a share of net income in period $t$ and $g_t$ is the real (inflation-adjusted) growth rate of net income from period $t-1$ to $t$. Note that regardless of $\beta_{t-1}$ or $g_t$, if $s_{t-1} < g_t\beta_{t-1}$, then $$ \beta_t<\frac{\beta_{t-1}+g_t\beta_{t-1}}{1+g_t}=\beta_{t-1} $$ Thus, 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.

What then is the significance of $r>g$? Stay tuned...

Mitchell finally notices the obvious

Cato’s Daniel J. Mitchell finally takes note of years of flat government spending. Months ago, I suggested this ought to make him “love President Obama” but he is still keeping any love to himself. Instead, Mitchell merely speculates this is all ”very depressing“ for the President.

On the other hand, he damns Obama with faint praise.

And even though we haven’t had impressive growth during the Obama years, there have been modest increases in both nominal GDP as well as inflation-adjusted (real) GDP.

Yes. There has been growth. Whee. Give it another six years and he may even consider the connection between the government’s withdrawal of demand and the piss-poor growth of the economy.

Waste is waste is waste

The fact that we are able to meet the nutritional needs of a great many more people does not mean we have no need for sewage systems and water treatment plants to deal with the slough of disease-spreading shit running down our streets.

Likewise, increased carbon dioxide concentration in the atmosphere is no cause for celebration.

That is all.

On Congressional snack accounting

K. William Watson wrote an amusing little piece yesterday over at Cato. Riffing off a POLITICO article about Congressional staff bartering gifts of snack food, Watson highlighted the natural impetus to trade. In his own words he writes “[i]n order to be insufferably pedantic.”

On the contrary, Watson is insufficiently so. He writes

I think it’s worth pointing out how crazy it would be to restrict this trade. Should offices worry that they’re running a snack trade deficit? Are some snacks being unfairly traded at too low a price? Are other offices inadequately inspecting their exports for safety?

Obviously, the trade is highly regulated. Only certain gifts may be exchanged, and production of these snacks presumably fall under common regulatory schemes before entering into the staff barter system. Ultimately, though, I would like to focus on the question of the “snack trade deficit.” Offices should worry about the existence of such deficits– not because trade deficits do not matter; rather it is unclear what a legal snack-trade deficit might look like.

A trade deficit results from drawing down on currency or other capital stores to finance an excess of imports over exports. To a first approximation, then a snack trade deficit involves paying more cash for snacks obtained than the cash received for snacks given up. But the Congressional snack trade is regulated so no cash may change hands and everyone winds up with exactly the cash they started with. There can be no snack trade deficit so long as staffers merely barter snacks.

But snacks are not the only medium circulating in the Congressional snack trade. According to the POLITICO article, snacks have been exchanged for use of a cell phone charger. This is probably no big deal. Use of snacks for rewarding unpaid interns is probably a but more sketchy, but pales in comparison to the fact that there are unpaid interns. Ultimately, the critical question is what else might be exchanged for snacks?

Alabama Republican Rep. Robert Aderholt’s chief of staff, Brian Rell, said in an email that he doesn’t see a lot of trading going on; “it is more like a tailgate where food is readily available.”

To the extent this is true, great. Share and share alike. Perhaps there are offices which are snack-rich and others snack-poor and so there may be net transfers of snacks from rich to poor showing up in the snack accounts even if there are no snack trade deficits. But to the extent that there are any expectations for non-snack compensation offices should very much worry about large snack account imbalances.

Farmer’s Folly: The Sequel

The post below is part of an exchange with Roger Farmer with origins which predate the start of this blog. The ultimate question is should (or even can) the government control asset markets for purposes of managing the rate of inflation. I believe Farmer’s call for such interventions is misguided.

More specifically, Farmer declares that the fall in the stock market in 2008 “caused” the Great Recession. What he seems to mean is that current movements in stock prices can be shown to help predict future movements in unemployment. Unfortunately, there is evidence that the relationship has broken down in recent years. Indeed, Farmer dismisses my concern that his initial model produces poor forecasts by making this very point. Furthermore, it is difficult to distinguish between stock prices as forward-looking, as opposed to forward-causing. Thus, even if the actual association today may be discerned, it is not clear that if, say, the Federal Reserve bought up stocks to keep prices high that such action would actually lead to much reduction in unemployment.

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In a new working paper (PDF) UCLA’s Roger Farmer responds to last year’s investigation into his claim that declines in the stock market caused the Great Recession.(PDF) Farmer apparently failed to grasp the nature of the critique.

In his original paper, Farmer claimed to have found a stable relationship between the movements in S&P 500 and unemployment rates, and that the data “leads me to stress asset market intervention as a potential policy resolution to the problem of high and persistent unemployment.” In other words, the government should deliberately prop up the stock market as a way of boosting the economy. Farmer appealed to the apparent forecasting power of his model to support his policy preference.

In response we countered that his visual evidence of forecasting power was deceptive– playing off the serial correlation in the data to trick the naïve observer. Rather, his model was not in fact powerful, as was demonstrated by the fact that a simpler model that ignored stock prices produced superior forecasts. Our analysis showed that even if Farmer’s model was correct, movements in the stock market fail to explain– let alone cause– the Great Recession. Finally, we pointed out that the intervention necessary to prevent the recession was implausibly large to be considered serious.

Farmer now:

• Asserts as fact certain properties of his data shown to be consistent with, but unsupported by his analysis.
• Argues that the asserted properties require the use of a particular type of model.
• Suggests that despite using the proper kind of model, his model was “seriously mispecified” by failing to account for a structural break.
• Reasserts that despite this structural break the observed relationship is somehow “structurally stable.”
• Abandons the “correct way to model” and employs pre-break data in an effort to support the uncontroversial position that stock market data may help forecast unemployment.

We agree that his model may have failed due to structural breaks. In fact, post-2008 data may be completely different in structure than data prior, and therefore any model based on previous data is liable to produce forecasts only spuriously related to the post-2008 economy. In any case, we believe this undermines both his assertion that stock prices caused the Great Recession and his proposed policy solution.

Latvia: First Prize For What, Exactly?

Cato’s Steve Hanke awarded “1st Prize” to Latvia for its lowest misery index score among former Soviet Republics. He notes that to the extent there is misery in Latvia, the “Major Contributing Factor” is Latvia’s unemployment rate, currently at 10.7 percent according to the IMF.

What Hanke leaves out is the fact that unemployment is so low because there has been a mass exodus of workers out of the Latvian labor market– and indeed the country. Since 2008, Latvia’s population has fallen 7.4 percent and the labor force has fallen by a whopping 16.7 percent.

(source)

First prize to Latvia indeed!

Some folks just imagine conflicts that do not exist

It may make great theater, but terrible science.

In the latest issue (#67) of the real-world economics review, Egmont Kakarot-Handtke presumes to moderate a debate (PDF) between Paul Krugman and Steve Keen (PDF). Sadly, Kakarot-Handtke’s paper lies at that unfortunate intersection of incomprehension and irrelevance. The paper is irrelevant, in that Kakarot-Handtke imagines a debate between loanable-funds and endogenous-money approaches to macroeconomics that to my eye does not exist (at least in this context.) The paper demonstrates incomprehension in that Kakarot-Handtke attributes to the loanable-funds model a non-existent property.

To the latter, we hardly need look beyond the final words Kakarot-Handtke offers:

The structural axiomatic analysis leads to the prediction that Krugman’s loanable funds model will be clearly refuted. It simply does not happen in the actual monetary economy that saving and dissaving of the households is exactly equal.

Here, Kakarot-Handtke appears to tear down a straw man. The textbook loanable funds model is one in which households lend to businesses– directly or indirectly– for investment purposes. If the loanable funds model predicted that households neither saved nor dissaved on net, then there would also be no investment. What Kakarot-Handtke actually argues is that even in a purely consumption-based economy, that households can save or dissave on net if businesses dissave or save, respectively. Surely nothing Krugman has written can make one wonder if he disputes this point.

Nevertheless, Kakarot-Handtke manages to get there. Specifically, Kakarot-Handtke quotes Krugman (twice!) as saying

If I decide to cut back on my spending and stash the funds in a bank, which lends them out to someone else, this doesn’t have to represent a net increase in demand.

So it is disconcerting to see Kakarot-Handtke then go on to comment

From the quote above it is clear that for Krugman savers and dissavers are not independent. For someone who saves there is someone else who takes the money, courtesy of the intermediation of the banking system, and spends it. Hence there is no effect on the rest of the economy.

(emphasis again added)

Kakarot-Handtke‘s paraphrase is disconcerting for multiple reasons. First, Krugman does not argue that “there is someone else who takes the money.” Rather, he makes a conditional argument– that if someone else takes the money, then this doesn‘t have to represent a net increase in demand. Nothing Krugman wrote implied that he believes saved money must be lent, making Kakarot-Handtke‘s paper a complete non-sequitur.

Further, nothing Krugman wrote implied that he believes that lending necessitated a prior act of household saving. A single offered example need not constitute an exhaustive list of the universe of possibilities. Importantly, this suggests no obvious endorsement of pure loanable funds modeling here, as Kakarot-Handtke insists. Quite to the contrary, Krugman wrote

Keen says that it’s because once you include banks, lending increases the money supply. OK, but why does that matter?

What part of “OK” does Kakarot-Handtke fail to understand? Krugman is not arguing against endogenous money, but rather wondering aloud what the complication adds. I am inclined to wonder as well. Kakarot-Handtke‘s contrived example sheds no light on the subject, arguing only that households may save or dissave.

In fact, Krugman is sympathetic to the idea that debt plays a role in influencing aggregate demand. He writes

In the kind of model Gauti and I use, lending very much can and does increase aggregate demand, so what is the problem?

Krugman goes on to take issue with the notion that lending by definition adds directly to aggregate demand. In truth, much of the problem is that Keen simply makes up his own definition of “aggregate demand” which includes

income plus the change in debt, and that this is expended on both goods and services and purchases of financial claims on existing assets

He then produces the equation \begin{equation} Y\!\left(t\right)+\frac{d}{dt}D\!\left(t\right)=G\!D\!P\!\left(t\right)+N\!AT\!\left(t\right) \end{equation} Yet this construction is catastrophically non-specific. Consider Keen‘s Figure 3, titled “Aggregate demand as income plus change in debt.” The figure shows, however, nominal GDP plus change in debt. Is $Y$ then equal to nominal GDP? In that case, we are left with \begin{equation} \frac{d}{dt}D\!\left(t\right)=N\!AT\!\left(t\right) \end{equation} and find that the “change in debt” is spent entirely on existing assets, and does not add to aggregate demand in the usual sense at all. Clear as mud, that.

Mitchell strikes out again

If Cato’s Daniel J. Mitchell is “particularly impressed” by Sweden’s “genuine fiscal restraint” from 1992-2001, he must love President Obama.

Mitchell shows a graph of total government expenditures in Sweden over the period, noting that “spending grew by an average of 1.9 percent per year” over those nine years. Let’s leave aside the fact that this is in nominal currency– a frequent oversight in Cato’s budget reporting. What I find particularly interesting is that equivalent spending in the United States grew only 0.9 percent per year from 2009-2013. Does that impress Mitchell? I’m guessing not, because spending was especially high in 2009 on account of the bursting of the housing bubble, and resulting unemployment, bailouts, and stimulus. But the story was hardly different in 1992 Sweden– suffering the fallout of a housing bust and banking crisis.

Of course, Sweden went as far as nationalizing banks to deal with the banking crisis and was fully recovered by 1995. So maybe the U.S. should take a lesson from Sweden: nationalize a few banks, get to full employment, and then consider some “fiscal restraint” of its own.

German fiscal policy is not so obviously “onerous”

In Germany last year, households and nonprofits directly consumed 57.4 percent of domestic production. Foreigners (on net) bought another 6.3 percent. Germany invested another 16.7 percent. That comes to 80.5 percent of production. So how can Cato’s Daniel J. Mitchell reasonably claim that “government spending consumes about 44 percent of economic output” as he does in today’s blog post?

The answer is, he cannot. The German government claimed only 19.5 percent of economic output, and even then consumed only 7 percent. The remaining 12.5 percent of economic output– though counted as government expenditure– was in fact consumed by the private sector.

Thus, the domestic private sector eventually claimed almost 93 percent of all economic output.¹ So what did Mitchell really mean? The German government received taxes sufficient to purchase 44 percent of GDP, and spent money sufficient to purchase 44 percent of economic output. That’s fine as far as it goes, but it does not tell us how burdensome the government spending. After all, traders spent $108 trillion on stocks in 2008. Does Mitchell believe that they therefore consumed 175 percent of the economic output of the entire world?

Suppose that government directly confiscated every bit of income produced by the economy, but then returned it dollar-for-dollar to each person. Though the government did nothing, Mitchell claims that government consumed 100 percent of economic output. He may not like that governments take money from some and give to others to spend, and he may not like that governments purchase goods and services and give to others to consume, but he inexcusably exaggerates the amount of output actually consumed by the German government.

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1 claims include the amount saved through net sales to foreigners^↩

Is It Really Time for the Fed to Worry About Inflation?

Ylan Mui at The Washington Post’s Wonkblog had a piece Thursday titled “This is why the Fed should start worrying about inflation again.” The main bit of evidence is a graph attributed to Kevin Logan showing a negative relationship between the unemployment rate and increasing rates of inflation. But this graph actually says far less than Mui says.

Indeed there is a relationship between unemployment and inflation. The Federal Reserve is tasked with balancing inflation and unemployment, and when the Fed fears inflation, it raises interest rates with the intent of slowing the economy and creating unemployment. To some extent, then, the relationship is the Fed’s doing.

Let us put that aside, however, and take the observed relationship at face value. First, it is far from obvious that 6.5 percent unemployment represents a threshold below which inflation is as likely to rise as fall– particularly given the small sample size. In Figure 1, I was unable to reproduce exactly Logan’s figure, but according to data available at the Fed, four of the five years with the highest unemployment rates under 6.5 percent are associated with decreasing inflation.

Figure 1: Unemployment and Changes in Inflation

Source: FRED, series JCXFE and UNRATE and author’s calculations

Rather than cherry picking, we may regress changes in inflation against the unemployment rate. As it turns out, the relationship is statistically weak. The expected change in inflation switches between positive and negative somewhere between 2.5 and 7 percent. Likewise, this suggests that the 50/50 point lies closer to 5 percent than 6.5.

Table 1: Regression results

		(1)	(2)	(3)
$\beta_0$	constant	0.52 (0.37)	0.52 (0.43)	0.52 (0.39)
$\beta_1$	unemployment rate	-0.11 (0.06)#	-0.11 (0.07)	-0.11 (0.06)#
variance/covariance estimator		OLS	jackknife	bootstrap
$-\beta_0/\beta_1$		2.6-7.1	2.9-6.8	3.0-6.7

Standard errors in parenthesis
# Significant at 10% level
Source: FRED, series JCXFE and UNRATE and author’s calculations

In Figure 2, we see the probability that inflation will be higher in 2014 than it was in 2013– assuming various year-round average unemployment rates for 2014. At 6.5 percent unemployment, the probability is closer to one in three than one in two.

Figure 2: Probability of Increased Inflation in 2014

Note: The widest (lightest) confidence band covers 95 percent of outcomes and the most narrow (darkest) band covers 50 percent.
Source: FRED, series JCXFE and UNRATE and author’s calculations

More importantly, increasing inflation is the wrong consideration. The Fed has tolerated inflation below 2.0 percent ever since 2007, and in 2013 core inflation ran only 1.2 percent. If the Fed must target some rate of inflation, it should target a higher rate of inflation. Yet, even if the relationship is meaningful then there is less than a 5 percent chance that 2014 inflation will run even as high as 2.0 percent.

Figure 3: Probability of At Least 2% Inflation in 2014

Note: The widest (lightest) confidence band covers 95 percent of outcomes and the most narrow (darkest) band covers 50 percent.
Source: FRED, series JCXFE and UNRATE and author’s calculations

To the extent that the relationship is both meaningful and a result of Fed activity, then, this suggests that meeting a 2% inflation target would require the Fed to be less hawkish than would be normal for the rate of unemployment. It may yet be some time before the Fed raises interest rates.

(This post originally appeared on the CEPR blog.)

House of Cards and entitlements: embarrassing, but to whom?

UPDATED BELOW

I know I’m late to this little fracas, and I’m not yet caught up on the entire second season, but it is terribly embarrassing that characters in Netflix’s House of Cards would propose raising the retirement age from 65. Now, I understand the original novel was set in the UK. But Netflix’s version is set in the U.S. and in the near present. Were all the writers born before 1938? Because everyone born after 1937 has a Social Security retirement age greater than 65. If you are turning 54 this year, or if you are younger than that, then ever since 1983 your retirement age has been 67. Frank Underwood says the Republicans have wanted this “since Johnson” but Underwood first ran for office in 1986– three years after Republicans got their increase in the retirement age for Social Security.

Now, technically speaking, House of Cards (as far as I have watched) talks only of “entitlements” but connects it to an increase in the age for early retirement, which applies to Social Security but not Medicare. In any case, compared to the typical Medicare recipient, those in their mid-60’s are relatively healthy– while many of those especially unhealthy would be on Medicaid anyway. Consequently, raising the retirement age for Medicare doesn’t even reduce deficits by a noticeable amount. According to the Congressional Budget Office, it would save some $6.7 billion in 2023 (PDF source)– an amount less than 0.03 percent of GDP.

I wonder what the writers had in mind. I’m guessing it’s a slip on the part of the writers. Or– in an age of low employment, wrecked private pensions, and thin household savings– is 70 is just too absurd a proposal for the viewers to swallow? Is this not embarrassing to supposed reformers? Then again, with respect to Matthew Yglesias, maybe the fight over entitlements is not about deficits.

Update: Yup. In the next episode, Underwood specifically mentions Medicare. That makes the CBO report relevant. At least, as relevant as an actual CBO report can be to a fictional show. More importantly, it is not so embarrassing to single out Medicare when talking entitlements. It’s health care costs which are projected to threaten the federal budget– not Social Security. If the United States had health-care costs in line with the rest of the developed world, we would be looking at surpluses, not deficits. But raising the retirement age is no solution.

A note on German austerity

Over at Heritage, Salim Furth talks structural deficits. Sadly, he gets his argument backwards.

The reason Germany did not shrink its structural deficit is that Germany barely had a structural deficit! In 2009, Germany’s structural deficit was just 1 percent of gross domestic product. Greece’s deficit was 19 percent. In fact, across eurozone countries, the change in structural balance from 2009 to 2012 is largely predicted by the size of 2009 deficits—the bigger the deficit, the harder they fell.

That’s a problem for the Keynesian story. According to Krugman’s Keynesian model, government can stimulate aggregate demand by running large deficits in bad times, softening the recession. If government fails in its duty to borrow, the recession will mire on.

[snip]

What did the Germans do that put them in a position for growth right after the recession? Back in 2001, Germany and Greece had the same structural deficit—just above 3 percent. But Germany shrank its deficit from 2004 to 2008 by cutting spending on welfare, unemployment insurance, and pensions.

(source)

Um. Okay. Furth takes data from the latest IMF World Economic Outlook Database. What does the database say about economic growth in these countries over this period? From 2002 to 2009, German output increased 4.7 percent per capita. (That’s only 0.7 percent per year!) By contrast, the Greek economy grew more than three times as fast, per capita (16.3 percent, or 2.2 percent per year.)

So the big-deficit Greeks enjoyed much faster growth than the austere Germans. Now, perhaps Furth might argue the Greek growth was unsustainable on account of all that borrowing, requiring the Greeks to reverse course at the worst possible time. But that hardly represents anything like “a problem for the Keynesian story.”

Economists blithely write “Economists blithely draw…”

Sometimes, I’m going to have to be critical of specific people. Generally, I prefer to be critical of people who disagree with me on policy. Sometimes, a potential ally will make me wince but I let it go. Then there is Steve Keen.

Sometimes, I just don't know what the man could be thinking, driving me to rise to the defense of someone unlikely. Take, for example,

Economists blithely draw diagrams like Figure 23 below to compare monopoly with perfect competition. As shown above, the basis of the comparison is false: given Marshallian assumptions, an industry with many “perfectly competitive” firms will produce the same amount as a monopoly facing identical demand and cost conditions— and both industry structures will lead to a “deadweight loss”. However, in general, small competitive firms would have different cost conditions to a single firm—not only because of economies of scale spread result in lower per unit fixed costs, but also because of the impact of economies of scale on marginal costs.

(PDF source)

Zing! It seems Keen and co-author Russell Standish have Mankiw dead to rights. It appears that Mankiw have made a terrible mistake and did not think about the fact that the marginal cost curve would be different for the industry as a whole. Or so they would have it.

I find this highly unlikely. Their claim that their paper shows that “given Marshallian assumptions, an industry with many ‘perfectly competitive’ firms will produce the same amount as a monopoly” is a matter for another time. For now, it suffices to note that in presenting this figure Mankiw is not referring to “perfect competition” at all. Mankiw leads his discussion saying

We begin by considering what the monopoly firm would do if it were run by a benevolent social planner. The social planner cares not only about the profit earned by the firm’s owners but also about the benefits received by the firm’s consumers. The planner tries to maximize total surplus… the socially efficient quantity is found where the demand curve and the marginal-cost curve intersect. [bold added to original, italics in original]

(source)

The framework for the discussion is monopoly. The discussion concerns the deadweight loss of a profit-maximizing monopoly in contrast to a socially-planned monopoly. In such a context, the monopoly is the industry, so there is no confusion regarding costs. The “efficient quantity” is “efficient” because no monopoly can produce larger total surplus. Mankiw's figure simply does not “compare monopoly with perfect competition” as suggested. Keen and Standish grossly misrepresent Mankiw.

Multiple Identities Can Deceive Even More

Consider a closed economy (no international transactions) so that \begin{equation} Y=C+I+G \end{equation} Savings is defined as unconsumed income, so national savings ($S$)-- which counts both private and government consumption-- is given by \begin{equation} S=Y-\left(C+G\right)=I \end{equation} Thus, the mystical “savings-investment identity” is born. In a closed economy, savings must equal investment. You like investment, don’t you? You believe that increasing the capital stock makes us more productive, right? So we should strive to increase national savings, don’t you think? And since we only can consume or save our income, we ought to consume less.

Not necessarily. It depends on the model. Suppose I gotta install microwave ovens. Custom kitchen deliveries!¹ If you unexpectedly fail to buy a new oven then very likely I am stuck with a larger inventory and the immediate effect is to increase $I$ by the same amount as the fall in $C$, leaving $Y$ unchanged. Or maybe I will then fail to buy from the manufacturer who then slows production, lowering both $C$ and $Y$. These are not the only possible results, but the point is it matters because equation (2) says that lowering your consumption increased savings only in the former case.

And even then the increased investment came as an inventory increase-- which is nice because it allows for additional future consumption, but it doesn't actually increase productivity.

The bottom line is that while it is tempting to argue from an accounting identity, it is the story that matters. The identity just helps keep the story straight.

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1 Back in 1984, someone totally could have kept his issues to himself instead of calling a guitarist on your MTV a “f****t”.^↩

Identities Can Be Deceiving

Many an economist may be heard complaining that accounting identities are not models. And about this, many an economist is correct. Unfortunately, this sad refrain bears repeating. Accounting identities are not models. In fact, they can be downright misleading. Take for example, the basic national accounting identity defining GDP: \begin{equation}Y=C+I+G+X-M\end{equation} Clearly, imports $M$ count against GDP. But does an increase in imports lower GDP? It looks like imports reduce GDP, but the equation does not tell us this. To see why this might be so, let us divide expenditures into domestic production and foreign imports. That is, $C=C_d+C_m$, $I=I_d+I_m$, $G=G_d+G_m$, $X=X_d+X_m$, and finally $M=C_m+I_m+G_m+X_m$. Then \begin{equation}Y=C_d+I_d+G_d+X_d\end{equation} It appears that imports do not enter into GDP at all. Yet it is no less correct to write \begin{equation}Y=\left(C-C_m\right)+\left(I-I_m\right)+\left(G-G_m\right)+\left(X-X_m\right)\end{equation} which again suggests that imports reduce GDP one-for-one. Which equation is correct? They all are. They all provide exactly the same information, yet invite the reader to different interpretations. Equation (2) invites the reader to believe that $C_d$ is independent of $C_m$ (which may or may not be true.) Equation (3) invites the reader to believe that $C$ is independent of $C_m$ (which also may or may not be true.)

Rather, we require a model to tell us how the various parts move. For example, we might say $G$ and $X$ are fixed, but an additional dollar of $M$ increases $C$ by \$1.50 and reduces investment by \$1.00. The accounting identity would then tell us GDP falls by 50 cents for every dollar of additional imports. Is it true? The result depends on the model, and the model need not be reasonable. Suppose instead that an additional dollar of imports leads to a million dollars of additional consumption. Garbage in, garbage out. But the identity must hold.

As we will see in a future post, identities get more deceptive when used in combination with other identities.

-1/12 is a large friggin’ number

Suppose $$S_n\equiv\sum^n_{i=1}{i}=1+2+\cdots+\left(n-1\right)+n$$ You may have figured out that you can rearrange the sum as $$S_n=n+\left(n-1\right)+\cdots+2+1$$ This is interesting because the first terms in each arrangement sum to $n+1$, the second terms also sum to $n-1+2=n+1$. In fact, this is true of all $n$ terms so we find that $$ S_n+S_n=n\left(n+1\right) $$ Fine. But what about $S_\infty$? The sum of all natural numbers, it is obviously bigger than any natural number. Can we more precisely describe $S_\infty$? Perhaps.