The fuss about government debt

In the wake of the coronavirus pandemic, governments around the world have passed very large stimulus packages. Some of the numbers are: \$2 trillion in the US, \$990 billion in Japan,  \$814 billion in Germany. There has been little opposition to the measures, with much of the debate centering around which industries get bailouts, what segment of the population receives checks in the mail etc. This is in sharp contrast to the situation about a decade or so ago, when the financial crisis of 2008 and the so-called Great Recession that followed it prompted major interventions by governments.

When the European debt crisis erupted in late 2009 and early 2010, with potential defaults by southern European countries like Greece and Italy, there was a strong reaction against deficit spending, and government austerity became a dominant theme. Politicians talked about "belt tightening" and "living within your means". They claimed that governments should manage their budgets just like individuals or  households do. This analogy has a simplistic appeal, but is flawed because it ignores critical differences between households and governments as far as budgets are concerned:

• Individuals and households have finite lifetimes, governments exist forever

When people die, their debts have to be paid off or assumed by their heirs. If there is a chance that their assets will be insufficient to pay off their debts, creditors will have to write off debts. Governments, on the other hand, continue to exist beyond any human lifetimes. For a sovereign government, there is no date in the near or distant future when all debts have to be cleared. The US federal government has been in debt continuously since 1790, except for a brief period in 1835.

• Individuals may lose their earning power through sickness or job loss, governments always retain their power to impose and collect taxes
• Sovereign national governments control their currency, and thus never have to default on debts in their own currency.
• Governments own many assets whose value is not accurately known: land, forests, rivers, claims to territorial waters, intangible things like rights to usage of the electromagnetic spectrum, etc.

These characteristics of sovereign governments mean that they can never go bankrupt. They may choose to default on debt in times of economic distress, as several governments do, but they remain going concerns  because of their ability to raise revenues, either through taxation or sale of assets.

Politicians sometimes use the rhetorical technique of stressing how much of the national debt "every man, woman, and child" owes. This is misleading in several ways. As we noted above, there is no date when all of the debt is due. Secondly, it is not as if debt collectors are going to be knocking down your doors. Or, as a friend of mine put it colorfully, sovereign debt will not result in your kneecaps being smashed.

While you might be relieved at the presumptive risk to your kneecaps being low, the debts accumulated by various governments are large and impressive. Without accounting for the various stimulus packages described above, the US debt was \$23.9 trillion,  as of April 6, 2020, a number that is updated daily to heroically futile precision. Numbers from other countries are no less impressive. China's debt is 65.6 trillion Yuan (estimated from IMF data and the USD-Yuan exchange rate), the UK's is £1.8 trillion, Japan's is ¥1116.3 trillion, India's is ₹140 trillion (using IMF data for India and a GDP of ₹204 trillion) . We clearly need a scale to make sensible comparisons between countries as well as over time. The most common measure chosen is the GDP, thus giving rise to the debt-to-GDP ratio. In terms of this ratio, the numbers for these countries are:

Country	Debt to GDP
Japan	238%
US	108%
UK	85%
India	69%
China	61%

Optimal debt ratio

There is no consensus on the optimal value for the debt-to-GDP ratio. In Europe, the treaty that set up the structure of the European Union (the Maastricht Treaty) required member states to avoid excessive deficits and provided a reference value for the debt-to-GDP ratio of 60% (in addition to an annual deficit to be restricted to 3% of GDP). This was no theoretically derived number; it was apparently the median ratio at the time [1]. The limit has been breached quite often by several countries within the EU, with no major consequences.



This issue jumped from academic and policy discussions into public consciousness in the aftermath of the recession that resulted from the financial crisis of 2008. An empirical study by two Harvard economists, Carmen Reinhart and Ken Rogoff, first published in 2010, claimed that high debt-to-GDP ratios, beyond 90%,  were bad for economic growth [2]. This study was quoted by politicians such as Paul Ryan in the US, George Osborne in the UK, and EU economics commissioner Ollie Rehn, in support of their efforts to curb government spending and impose austerity measures.  However, when University of Massachussetts, Amherst, graduate student Thomas Herndon tried to verify these conclusions, he failed. He later wrote a paper with two professors about the results [3]. The abstract of the paper lays it out clearly:

We replicate Reinhart and Rogoff (2010a and 2010b) and find that coding errors, selective exclusion of available data, and unconventional weighting of summary statistics lead to serious errors that inaccurately represent the relationship between public debt and GDP growth among 20 advanced economies in the post-war period. Our finding is that when properly calculated, the average real GDP growth rate for countries carrying a public-debt-to-GDP ratio of over 90 percent is actually 2.2 percent, not −0.1 percent as published in Reinhart and Rogoff. That is, contrary to RR, average GDP growth at public debt/GDP ratios over 90 percent is not dramatically different than when debt/GDP ratios are lower. We also show how the relationship between public debt and GDP growth varies significantly by time period and country. Overall, the evidence we review contradicts Reinhart and Rogoff’s claim to have identified an important stylized fact, that public debt loads greater than 90 percent of GDP consistently reduce GDP growth.

This issue received an unusual amount of coverage in the media. Among others, the BBC provided a narrative, while the New Yorker had a decent explanation of the substantive issues. Reinhart and Rogoff called the issue an "academic kerfuffle", and the surrounding controversy doesn't seem to have changed their views on the matter. A Google search reveals that they are still writing op ed pieces in March 2020 warning about the dangers of high debt.

Apart from the refutation provided by Herndon and colleagues, we can also look at the historical track records of some major economies, as shown below.


The UK has had a debt-to-GDP ratio above 100% for very long periods of time, both in the nineteenth and twentieth centuries, for much of which it was the world's leading imperial and economic power. Japan had a large debt relative to GDP during the second world war and a discontinuity after its defeat and change of regime. In 1996, its debt-to-GDP ratio crossed 100% and is now inching close to 250%.

Debt dynamics

While it's not clear what the limit on the debt-to-GDP ratio should be, the algebra of changes in debt over time is quite straightforward. Let $D_t$ be the debt in year $t$. This debt is the result of the previous year's debt $D_{t-1}$ plus the interest cost, minus the excess of revenues over expenses in the budget.

\[\begin{eqnarray}
 D_t &=& D_{t-1} + r D_{t-1} - \big(Rev_t - Exp_t\big) \nonumber \\
        &=& (1+r) D_{t-1} - PB_t
\end{eqnarray}
\]

where the rate of interest on debt is $r$ and $PB_t$ denotes the so-called primary balance, the excess of revenues over expenses. When this number is positive, it helps reduce the debt and increases it otherwise. Dividing by GDP, we have

\[  \begin{eqnarray}
  \frac{D_t}{GDP_t} &=& (1+r) \frac{GDP_{t-1}}{GDP_t} \frac{D_{t-1}}{GDP_{t-1}} -  \frac{PB_t}{GDP_t} \nonumber \\
d_t &=& \frac{1+r}{1+g}d_{t-1} - pb_t
\end{eqnarray}
\]

where the lower case variable names refer to the debt and primary balance scaled by GDP, and $g$ is the growth rate of nominal GDP. This can be written in a slightly more useful way as
\[
d_t - d_{t-1} = \frac{r-g}{1+g} d_{t-1} - pb_t
\]

This equation shows that the debt-to-GDP ratio can be kept constant if the two terms on the right hand side are equal in magnitude. So, if the growth rate of nominal GDP is greater than the interest rate on debt, the government can have a primary deficit without affecting the debt-to-GDP ratio.

Let's plug in some numbers to get a sense of what is possible. For the US, the current debt-to-GDP ratio is 110%. Taking real GDP growth to be about 2% and inflation to be 2%, the nominal growth rate $g=4\%$. Currently, the US can borrow money for a fairly long term at astonishingly low rates. We can take $r=1.5\%$ [4]. If we keep the debt-to-GDP ratio fixed,

\[
pb_t = \frac{1.5\%-4\%}{104\%} 110\% = -2.6\%
\]

Since US GDP is about \$21.7 trillion, this means that the US can have a primary deficit of \$564 billion without affecting the debt-to-GDP ratio.

What would the effect of the stimulus package be? We will have to make an assumption of the impact of the outbreak on the economy. Let's assume that real GDP falls by 10% (for comparison, it fell by 2.75% in the 2008 recession). Then, assuming inflation of 2%, $g=-8\%$. This implies that GDP will fall to about \$20 trillion. Since debt will increase from \$24 trillion to \$26 trillion, the debt-to-GDP ratio will become 130%. High, but not cause for alarm. 

Other concerns

There are other concerns raised by people who think that high government debt is a bad thing. Among these are:

• The crowding out effect

The crowding out effect refers to the reduction in private investment in the economy in the presence of heavy government borrowing. The existence of this effect was debated by economists even back in the 1970s [5]. The mechanism by which the crowding out is supposed to occur is an increase in interest rates.  We can look at the empirical evidence to see if interest rates have increased with government debt. 

The 10-year constant maturity yield is a benchmark for many rates in the economy and is a good proxy for medium term interest rates for private investment. As the plot shows, the relationship between government debt levels and the 10-year yield seems to be the exact opposite of what the crowding out effect would suggest. 

• The burden on future generations

Advocates for limiting government debt often cast the debt as a moral issue, talking about how we are burdening our children and grand children. This is misleading on a couple of fronts. As mentioned above, there is no particular date when all of the debt (or any significant part of it) becomes due. Because the government can roll over debt or refinance it if interest rates are favorable, there is a fair amount of flexibility in debt management, so the burden can be spread out over a few generations. If the economic growth rate is larger than the interest rate on debt, the government can even reduce the debt-to-GDP ratio without increasing taxes.

Secondly, it is useful to remember that a significant part of the debt is held by the public either directly or through mutual fund or retirement account investments. In any generation, while everyone pays taxes, many people benefit from the interest paid on government debt. So it is never the case that an entire generation will have to bear the burden (however defined) of national debt. 

• Foreign holding of debt

Every once in a while, alarms are raised in the US, about China holding a large amount of US debt. This list of foreign holders of US Treasuries shows that China and Japan trade places as the top two holders of US Treasury securities, to the tune of about \$1.1 trillion. These are large numbers, but they amount to about 4% each of total US debt. The fear expressed in the US is that China will start selling US Treasuries, thus causing bond prices to fall and yields to rise. If China does this at any significant scale it would reduce the value of its remaining holdings, resulting in major losses. Governments have done crazier things, but this does seem quite unlikely.

While the foreign holding of debt is not a major concern for the US or other major developed countries, it is a significant issue for low or middle income emerging countries. These countries often borrow internationally in foreign currencies, and then face major problems when the debt servicing costs rise. So it clearly makes sense for emerging countries to keep their external debt low.

• High interest expense

There are plenty of news stories, articles, and opinion pieces highlighting the fact that the US government pays a lot in interest expense. This number was \$575 billion in 2019, but about a quarter of it goes to other branches of government that hold Treasury securities. Nevertheless, it's a big number, so we scale it by GDP, and find that it amounts to 2%. As this time series graph  shows, the interest expense was around 3% of GDP from 1985 to 1996. Of course, this is a consequence of the fact that even as the debt has risen, interest rates have fallen significantly over the last two decades. Nevertheless, the interest expense is not currently a cause for worry.

Again, it is worth highlighting that this argument does not apply to emerging countries, who face much higher interest rates, in addition to their debt being denominated in foreign currencies. For such economies, it makes a lot of sense to reduce external debt as much as feasible. One other good argument is that interest payments come at the cost of development expenditure for such things as education, health care, and infrastructure.

• Deficit spending on undesirable things

This is an important concern, but it is primarily a political and social debate. What is desirable varies according to individual judgment and political ideology. Conservatives in the US seek to reduce deficits by reducing welfare spending or entitlements, whereas those on the left would prefer to do that by progressive taxation of the wealthy, or by reducing the defense budget. Unfortunately, the algebra of debt dynamics, or the history of debt-to-GDP ratios and interest rates is of no help here.

An upper bound on the debt-to-GDP ratio

Does all this mean that there is no limit on how much governments can borrow? The answer is no. We can derive one potential upper bound from a simple requirement: the interest payment should not exceed the government's total revenue. This means that
\[  \begin{eqnarray}
  rD_{t-1} & \le& Rev_t  \nonumber  \\
 d_{t-1} &\le& \frac{(1+g)}{r}\bigg(\frac{Rev_t}{GDP_t}\bigg)
\end{eqnarray}
\]
The bound depends on the growth rate of nominal GDP and the interest rate on debt, along with the share of GDP going to government revenue.

To get an estimate of the bound in a terrible economic situation, let's take the case of Greece during it's debt crisis at the beginning of 2012. It's GDP had fallen by 8.6% in 2011, inflation was 3.3%, and in February 2012, the yield on its 10-year bond had risen to about 30%. The government's revenue as a fraction of GDP was about 45% in 2012. Plugging these numbers in, we get
\[
d_{t-1} \le \frac{1-5.3\%}{30\%} 45\% \approx 142\%
\]
It is worth emphasizing again that it is the economic disaster that results in this upper bound on the debt-to-GDP ratio, and not the other way round.

References

[1] Chowdhury, A. and Islam, I., "Is there an optimal debt-to-GDP ratio?", VOX CEPR Policy Portal, 2010. Link.
[2] Reinhart, C. M., and Rogoff, K. S., "Growth in a Time of Debt", NBER Working Paper number 15639, 2010. Link.

[3] Herndon, T., Ash, M., Pollin, R., "Does High Public Debt Consistently Stifle Economic Growth? A Critique of Reinhart and Rogoff", Political Economy Research Institute, University of Massachussetts, Amherst, 2013. Link.
[4] This number is the 10-year zero coupon rate inferred from the yields of Treasury securities. Source.
[5] Carlson, K. M. and Spencer, R. W., "Crowding Out and Its Critics", Federal Reserve Bank of St. Louis, 1975. Link.