Is the US Government Debt Sustainable? Exploring the Mathematical Impossibility

The long-standing discourse on the sustainability of US government debt often suggests that future generations will be saddled with the burden of paying off today's debts. However, is this actually true from a mathematical standpoint? This article delves into the mathematical framework that demonstrates why the current path of US debt is unsustainable.

Key Assumptions and Equations

To understand the sustainability of US government debt, we must first lay down some key assumptions and mathematical formulations. Let's begin with the initial debt and growth rates.

Initial Debt

D0 25 trillion

Debt Growth Rate

rd 8% 0.08

GDP Growth Rate

rGDP 2% 0.02

Exponential Growth Functions

The debt at any time t is given by the function:

Dt D0 x e(rd x t)

The GDP at any time t is given by the function:

GDPt GDP0 x e(rGDP x t)

Where D0 and GDP0 represent the initial debt and GDP respectively.

Debt-to-GDP Ratio

The Debt-to-GDP ratio is defined as:

Debt-to-GDP ratio Dt/GDPt

The Mathematical Proof

When examining the nature of exponential functions, it becomes clear why the current growth rate of US debt relative to GDP will eventually lead to a catastrophic outcome. If policymakers do not address the issue, the resulting debt-to-GDP ratio will rise uncontrollably, demonstrating the unsustainable nature of the current trajectory.

Graphical Representation

Here are the graphical representations:

Debt-to-GDP Ratio First Graph

The blue line shows the Debt-to-GDP ratio increasing rapidly, far exceeding the 100 mark, indicating how unsustainable the debt becomes over 30 years.

Debt vs. GDP Growth Second Graph

The red curve representing debt grows exponentially faster than the green curve for GDP, illustrating the widening gap over time.

These graphs highlight why it becomes mathematically impossible to reduce or pay down national debt if debt grows much faster than GDP.

Mathematical Impossibility of Paying Down Debt

Exponential Growth: Debt grows exponentially while GDP grows at a much slower rate. When debt grows faster than GDP, it becomes increasingly difficult to generate enough GDP to cover interest payments and principal debt reduction.

Interest Compounding: As the debt grows, interest payments compound, adding even more debt. With interest payments already exceeding 1 trillion annually, this burden alone swallows a large portion of GDP.

Stagnant or Low GDP Growth: Even with modest GDP growth, it cannot keep pace with the accelerating debt, leading to a vicious cycle where more debt is needed just to service existing debt.

Conclusion

The math is unequivocal: the United States' current trajectory of debt growth is unsustainable. Policymakers have two paths: acknowledge the problem and take the necessary steps to reduce debt, or continue down the course of postponing the inevitable, leading to economic turmoil for future generations. It is crucial to understand the mathematical implications and address the issue proactively.