Five simple formulas that capture today’s economic challenges and solutions.

March 21st, 2016 at 8:20 am

Contemporary economics is often criticized for being too mathematical. How can we really capture the complex dynamics of what’s happening in people’s economic lives with a bunch of formulas?

In fact, the point of all the math is to distill people’s complex experiences down to something we can model, understand, measure, and maybe predict. Yes, all models are wrong, in the sense that the distillation loses important nuance. But some models are useful.

With that in mind, here are five useful, simple models, all in the form of inequalities (i.e., “x is greater than y”). Each one tells you something important about the big economic problems we face today or, for the last two formulas, what we should do about them. And when I say “simple,” I mean it. It may have taken a lot of gnarly calculations to get to these formulas, but you don’t need any of that to understand and appreciate their critically important messages.


Think of this one as Piketty’s warning. It simply argues that if the return on wealth, or r, is greater than the economy’s growth rate, g, then wealth will continue to become ever more concentrated among a narrow slice of households.

To make it simple, let’s say Sally Capital gets a 3 percent return on her wealth holdings year in and year out, most of which she adds back into to her base wealth of stocks, bonds, Picassos and Stradivariuses. Meanwhile, Joe Paycheck’s income—purely from work—grows 2 percent, and he and his family don’t save (build capital) any of that; they must spend it all to survive. Add in the fact that there are a lot more Joes than Sallies, and you see the problem.

Piketty’s data show some periods when the return to wealth was lower than the growth rate (r<g), but he argues those days are behind us. Others have cast doubt on the dynamics of r>g, or at least pointed out ways in which the returns to wealth could be more diffuse (e.g., Sally bequeaths some wealth to Joe; Joe joins an ESOP), thus questioning Piketty’s warning that wealth inequality will inexorably grow.

But there is little question that wealth concentration is at historically high levels and that mechanisms by which capital ownership could be diffused—like more progressive inheritance taxation or a financial transaction tax—are politically out of favor, particularly in the U.S. Meanwhile, the Joe Paycheck scenario cited above—2 percent growth—is actually better than the experience of too many working households. Thus, we should take Piketty’s warning seriously.


Think of this one as Bernanke’s imbalance, though he calls it the “savings glut” (S is savings, I is investment). Larry Summers’ “secular stagnation” concerns offer a similar, though somewhat more narrow, version. For the record, I think this one is really serious (I mean, they’re all really serious, but relative to r>g, S>I is underappreciated).

If you suffered through econ 101, you might get tripped up by S>I, as we’re taught that what’s saved gets invested, so the correct formula should be S=I. But while that’s true for the world, it’s not true for individual countries, which can and do save more than they produce, thus running a trade surplus. Since S>I in, say, China, and S=I in the aggregate, S must be < I somewhere else, like the U.S., which runs large trade deficits. By “large,” I mean deficits averaging around -2.5 percent of GDP since the late 1970s and -4 percent since 2000, that have created a significant drag on growth. A nation can, and we have, offset this drag with other parts of GDP, but that’s led to deeply damaging bubbles, busts, and intractable recessions.

The really interesting part of the S>I problem is that, in theory, there are key mechanisms in the economy that should automatically kick in and repair the disequilibrium. The currency of the country with a trade deficit should fall relative to the currency of the surplus country, boosting the (now cheaper) exports of the former and suppressing the (now more expensive) exports of the latter. But the surplus country can jam that mechanism by manipulating the exchange rate to keep their currency cheap relative to that of their competitors.

Similarly, interest rates can fall to bring S and I back in line. But Summers and Krugman stress that when rates are already very low—like zero, which is where our central bank rate was for years until recently—the system once again gets jammed.

Central bankers, like Bernanke and Yellen, tend to discuss S>I and the jammed mechanisms just noted as “temporary headwinds” that will eventually dissipate (Summers disagrees). But while it has jumped around the globe—S>I is more a German thing right now than a China thing (Germany’s trade surplus is 8 percent of GDP!)—the S>I problem has lasted too long to warrant a “temporary” label. Ergo, doing something about it, perhaps by going after currency manipulation in trade agreements (or outside of trade agreements as a second-best approach) or otherwise pressuring large surplus countries, should be a serious response.

(Those of you still with me may note an interesting wrinkle at this point. When S>I, r tends to fall such that Piketty’s r>g may be less likely to hold. In fact, the profit share of national income has come down a bit of late relative to the wage share. But we are far from out of the inequality woods, and Piketty’s warning is still very much worth heeding.)


Think of this one as Baker/Bernstein’s slack attack. The first “u” is the unemployment rate and “u*” is a construct called the “natural rate” of unemployment, or the lowest unemployment rate consistent with stable inflation. No one knows what u* is and it’s increasingly hard to identify, but that’s actually not a big problem for this warning, as you’ll see.

What Dean and I have shown in various writings is that for most of the past few decades—about 70 percent of the time, to be precise—u has been > than mainstream estimates of u*, meaning the job market has been slack, characterized by weak labor demand. That, in turn, has led to diminished worker bargaining power, real wage stagnation, and higher inequality.

But if u* is so hard to identify, how do we know that u is greater than it? Because, if anything, the estimates of u* are biased up, not down. Thus, our 70 percent is probably a low-ball estimate.

Back in the 1990s, for example, the experts thought u* was around 6 percent (they’ve since marked it down). So when the actual u fell below that level, they concluded the slack had been squeezed out of the job market (i.e., they thought u<u*). But, in fact, they were overestimating u*—that’s the best explanation for the fact that, when unemployment fell all the way down to 4 percent in the late 1990s, inflation failed to react as predicted, meaning it didn’t speed up. Today’s Federal Reserve has been relearning that same lesson, as they’ve continuously lowered their estimate of the “natural rate” as inflation failed to accelerate when u kept hitting it.

From the 1940s to the late 1970s, u>u* only 30 percent of the time, meaning the job market was mostly at full employment. Since then, as noted, the job market’s been slack 70 percent of the time. That’s terribly persistent slack, and a reason why getting to and staying at full employment has been such an important agenda for us and other economic progressives.

By the way, as Dean argues in this paper, one reason u>u* is that S>I (persistent trade deficits make it a lot tougher to get to full employment without inflating nasty bubbles).

These three formulas—r>g, S>I, and u>u*—are all basically bad news. They’re economic pathologies implying increasing inequality, weak demand, and slack labor markets. But despair not! These next two formulas point to an important way out of the mess implied by the first three.


Think of this one as Kogan’s cushion. In a recent paper, budget analyst Richard Kogan made a critical and underappreciated discovery: for most of the years that our country has existed (he’s got data back to 1792!), the economy’s growth rate (g again) has been greater than the rate the government has to pay to service its debt, which I call t. Kogan calls it r since it’s a rate of return, but it’s not the same r as in Piketty (which is why I’m calling it t). Piketty’s r is the return to wealth holdings; Kogan’s is the interest rate on Treasury bills. They’re related but not the same, and Kogan’s t will typically be lower than Piketty’s r, for good reasons (e.g., being such a safe lender, the U.S. government doesn’t need to pad its t with a “risk premium”).

OK, but why does this make the list? Because it means we should be far less obsessive about our deficits and debts, especially when greater public spending could alleviate some of the pathologies described above. When g>t, the economy is spinning off enough growth, and thus tax revenues, to service the debt burden without going further into debt, or risking the debt spiral you’d get if t<g. As Kogan puts it, “when economic growth rates exceed Treasury interest rates, the burden of existing debt shrinks over time.”

Since 1947, g has been > t about two-thirds of the time, by 1.3 percentage points on average. But, as Kogan stresses, CBO tends to assume that t will exceed g, which a) makes their debt forecasts too pessimistic, and b) may well be too cautious given how low interest rates have been in recent years (partially a function of S>I, btw; lots of idle capital makes for low rates, as does accommodative Fed policy). Those forecasts give rise to austere budget politics that overemphasize the importance of paying down the debt, even at the expense of critical public spending and investments.

To be clear, my “far less obsessive” point above does not mean we can ignore our public debt, and in fact, one prominent theme of my own work is that we will need more tax revenues in the future if we hope to meet the challenges and obligations faced by the public sector in a fiscally sustainable manner. But too often the prevailing, embedded assumption that g<t is used as a cudgel against government doing what it needs to do. Kogan’s work reveals this assumption to be inconsistent with the historical record.


Think of this one as the DeLong/Summers low-cost lunch. It’s more obscure than the others, but really very intuitive when you break it down. All they’re saying is that when the private economy is weak, government spending can be a very low-cost way to lift not just current jobs and incomes, but future growth as well. In tandem with “Kogan’s cushion,” these formulas provide an important recipe for what to do in recessions and slow-growth economies (hmmm…can anyone think of a slow-growth economy?).

The “h” stands for hysteresis, which describes the long-term damage to the economy’s growth potential when policy neglect allows depressed economies to persist over time. When that happens, the stock of capital grows more slowly and the skills of un- and underemployed workers atrophy. It may sound theoretical, but if you look at our current investment record along with our labor force participation of prime-age workers (non-retirees), you see hysteresis in action. It’s tough to get a job when you’ve been out of work for a couple of years.

So, what’s the 0.05 got to do with it? In a 2012 paper, D&S argue that when the economy’s doing poorly and interest rates are low, it is worthwhile for the government to intervene even when h is very small. They write that “…even a small amount of hysteresis—even a small shadow cast on future potential output by the cyclical downturn—means…that expansionary fiscal policy is likely to be self-financing…[or at least] highly likely to pass the sensible benefit-cost test of raising the present value of future potential output.”

Basically, given reasonable assumptions about a bunch of other moving parts, D&S find that as long as an increase in current output by a dollar raises future output by at least a nickel, the extra spending will be easily affordable. But how do we know if h<0.05?

In a follow-up paper for CBPP’s full-employment project, D&S, along with economist Larry Ball, back out a recent number for h that amounts to 0.24, multiples of the 0.05 threshold, and evidence that, at least recently, h>0.05. QED!


One reason to like all these formulas is that their policy implications are self-evident. Piketty’s warning speaks to the importance of progressive taxation and very much against the type of tax ideas proffered by today’s conservatives. Pushing back on S>I means a) going after countries that manage their currencies to maintain this imbalance and b) offsetting low I with public investment, say in physical and human capital improvements. Given Kogan’s historical finding of g>t, in tandem with h well above 0.05, such investment should be a slam dunk. Were we to go there, we’d surely be pushing back on u>u*, which would, along with making Dean and me very happy, deliver more much-needed bargaining power and wage growth to working people.

To the extent that we fail to heed these formulaic warnings and guideposts, it won’t be because the math is wrong. It won’t be because they’re too arcane. It won’t be because they don’t have solid empirical groundings. It will be because politics and power decide to ignore them.

I unfortunately don’t have a good formula for that one. But I will relentlessly fight back until reason, fairness, and equity regain their long-denied perch.


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7 comments in reply to "Five simple formulas that capture today’s economic challenges and solutions."

  1. David says:

    Surely you have the inequality going the wrong way at the end of this sentence:

    “When g>t, the economy is spinning off enough growth, and thus tax revenues, to service the debt burden without going further into debt, or risking the debt spiral you’d get if t<g."

  2. Ed Walker says:

    The low-cost lunch idea, ” government spending can be a very low-cost way to lift not just current jobs and incomes, but future growth as well”, isn’t that the point of Gerald Friedman’s analysis of Bernie Sanders’ economic program?

    • Smith says:

      Gerald Friedman was savagely attacked (by economist columnist standards) for modest predictions of growth that relied on productivity of 3.17 per year for ten years and LFPR (labor force participation rates) close to 70% by the end of ten years. One economist claimed 3.17 for ten years never seen in our lifetime. True if you are ten years old, since productivity grew from 1996 – 2005 at an annual rate of 2.98 percent. In other words, productivity was on track at that rate until the housing market began to collapse and bring down the economy with it. Was that rate then a result of the bubble. Probably not since the tech bust came in the middle, followed by recession, and the housing bust reflected as much a panic as it did overvaluation. These were actual houses and land, not worthless pieces of paper stock, just not liquid when the market froze, they weren’t website flameouts or tulips. Tyler Cowen, recently cited that 10 year productivity growth rate in a Times piece. LFPR was 67% before a 21st Century declined that accompanied slow job growth and stagnant wages (even while productivity grew). Some Nordic countries despite a substantial safety net (maybe because of in part) have rates at or near 70. Just saying.
      Also, Piketty’s r>g is a lot stronger that assumed, just compare historical average stock returns with wage growth or productivity, with all figures adjusted for inflation. Adjust that for distribution of wealth holdings. Also look at $100 billion < $1.8 trillion. That is the amount the leading Democratic candidate advocates spending for stimulus vs. a fairly conservative estimate of the output gap in potential GDP.

  3. Nurhisham Hussein says:

    Great summary, but I have a bone to pick with comments on S>I.

    Given the increase in trade via global value chains, some trade surpluses might be “structural”. That type of trade is not really sensitive to exchange rate movements as exports and imports are cointegrated, and the currency effect will only be on domestic value added. China for example, by all accounts, has very low value added in manufacturing. Taking that into account means the degree of currency undervaluation/manipulation would be overstated based on the gross trade surplus alone.

    The OECD and WTO are making a joint effort in measuring this type of trade:

  4. Nick Estes says:

    Thanks for the good post explaining with some apt formulas what we need to do to have a prosperous and more equal society. At the end you acknowledge that “politics and power” keep us from doing the right thing, but swear to keep relentlessly fighting the good fight. Well, me too.

    A prosperous and more equal society requires primarily that we run bigger budget deficits to get u down much closer to u*. But universal fear of our huge and rising national debt, underscored by scary charts from the CBO, remain a major force preventing us from doing the right thing. Economists earnestly debate whether we have enough “fiscal space” to take action if we enter another recession.

    This conceptual shackle is completely unnecessary. We should never worry about “fiscal space” but only about how much slack there is in the economy so that higher budget deficits will expand output and reduce unemployment instead of causing inflation. If higher deficits are called for (as they are even now), we do not need to borrow the money by issuing bonds, thereby raising the national debt still higher. Instead Congress and the President should politely ask the Fed to create the money and simply transfer it to the U.S. Treasury. The Treasury can memorialize the transfer with a very nice thank-you note. When we get back to full employment, if the Fed thinks that there is too much money in the system, threatening some inflation, it can sell bonds or raise the reserve requirement and reduce the money supply back down again.

    Note this would also address the problem of unemployment caused by our large trade deficit. In the 1990s we had a huge trade deficit, but low unemployment. Aggregate demand was equal to aggregate supply, so it didn’t matter if other countries wanted to send us more goods than we sent them. If a Carrier factory in Indiana closed and moved to Mexico, the workers would retrain and move if necessary, and there would actually be real new jobs for them–maybe paying more than they made at the old ones.

    I know that stimulus without debt is politically unthinkable now (I hear the howls of “printing money!”), but maybe it wouldn’t be so unthinkable if the economics profession would start widely debating the idea (and not leave all the work to British economist Adair Turner).

    Recessions could be as rare as the plague. And perhaps they would be if we would sever the unnecessary connection between deficits and increasing the public debt.

  5. Kevin Rica says:


    On of your bests posts, provided you can clean up your explanations of the multiple meanings of t. Normally g > t is read as a fiscal surplus.

  6. Bob says:

    I’ve saved this most useful post for handy reference. Excellent summary.

    Easy to see from combining the various formulae that the easiest way to bring about a debt-fueled-armageddon so feared by fiscal conservatives is fiscal austerity. Austerity slows the economy, causes underinvestment and stalls growth, increasing the relative burden of debt service even at low nominal rates of interest. Spending, done wisely, decreases the relative burden of debt service by growing the current economy and by creating capital stock that will increase productivity in future years.