# Debt is relative

One of the memes that is dominating American political discourse is that the US national debt is too high. The debt is currently around 16 trillion dollars which exceeds the current GDP of 15 trillion. This may seem large but keep in mind that the debt-to-GDP ratio in Japan is over two. The federal government will bring in about 2.6 trillion in revenues this year but spend about 3.7 trillion dollars, giving us an annual deficit of a trillion dollars. However, our borrowing costs are also very low. The yield on a 10 year Treasury bond is under 2% and the 30 year yield is under 3%. The general fear of having large debt is that it may cause interests rates to rise because people fear a default. This then leads to higher borrowing costs until you get to a point where you can never repay the debt. This is where Greece is now and where Spain, Portugal, Ireland, and Italy may be heading.

However, an important fact that should be kept in mind when thinking about debt is that the absolute amount is irrelevant. This is because economics, like biology, is about flux and growth.  As long as the nominal GDP growth rate (real GDP growth plus inflation) exceeds the borrowing rate, then the debt ratio will shrink in the future.  In fact the power of exponential growth shows that you can always be in deficit and the debt to GDP ratio can shrink.  We can see this in a simple calculation.  Let $D$ be the debt, $I$ be the annual deficit, and $y$ the borrowing rate.  The debt then grows as$\dot{D}=I+y D$, which has the solution $D(t)=(I+D(0))e^{yt}/y-I/y$.  Now suppose that the nominal GDP $G$ grows with rate $r$ so $G(t) = G(0)e^{rt}$.  So in the short term deficits do matter but as long as $r > y$, the debt-to-GDP ratio will always shrink in the long run.  In fact, this is what happened after World War II.  The debt was never officially retired.  It just faded away into insignificance because of growth and inflation.  Given the low interest rates, there is an arbitrage opportunity to borrow as much as possible and invest the money in infrastructure to promote future growth.