National debt

SUMMARY: The accumulation of federal government budget deficits over time is the national debt, which is best considered relative to gross domestic product (GDP) or other factors.

Mathematician Richard Feynman once said, “There are 10¹¹ stars in the galaxy. That used to be a huge number. But it’s only a hundred billion. It’s less than the national deficit! We used to call them astronomical numbers. Now we should call them economical numbers.” In modern society, entities from individuals through governments need money to function. Most government funds are generated by taxing individuals, businesses, goods, and services.

At the same time, governments must spend money for various purposes. If a government has more income than expenditures in a given fiscal period, usually one year, the excess of income over expenditures is called a “surplus”; if a government has more expenditures than income, the excess of expenditures over income is called a “deficit.” The sum of all of these single-year surpluses and deficits over the entire history of the federal government is called the “national debt.”

Mathematics has long been used to quantify expenditures, deficits, and debts. Taxation and deficiency problems were mentioned in the Chinese mathematical text Jiǔzhāng Suànshù (The Nine Chapters on the Mathematical Art), and Indian mathematician Brahmagupta referred to “debts” to mean what are now called negative numbers. William Playfair created some of the earliest graphical representations of social and economic data around the time of the American Revolution, such as trade balances between England and other countries and the English national debt.

By the twentieth century, mathematical measurement, estimation, and modeling were increasingly used. Standard economic measures such as gross domestic product (GDP) were common, and there were theories and research on principles such as return on capital, interest rates, and exchange rates, many of which cannot be known with certainty. Stochastic modeling, random walks, particle theory, and Brownian motion, named for botanist Robert Brown, have been used extensively in the mathematical modeling of financial processes. After events such as the Wall Street crash of 1929, there was also interest in forecasting models that could warn of debt crises.

Mathematicians continue to research and create models to address both historic and new financial concerns, and many people have created representations such as the national debt clock and deficit calculators to extrapolate trends. Others argue against too much aggregation or extrapolation in mathematical models, citing inherent data collection errors in large-scale indices, such as the consumer price index and gross national product, as well as subjectivity in individual perception and often-complex interactions between variables such as debt, deficit, production of goods and services, and allocation of consumer resources.

Inflation

National debt differs uniquely from individual debt in the fact that governments usually have the power to print more money to pay debts. However, doing so often leads to undesirable economic consequences. More money in circulation can lead to increased demand for goods and services, which in turn may lead to inflation.

Mathematicians and economists study inflation trends and cycles, as well as the reciprocal impacts of inflation on factors such as labor costs. While most economies function reasonably well with some level of inflation, too high a level of inflation leads to a host of problems, including hoarding of goods, increases in interest rates for credit and loans, and trade deficits with other countries. For this reason, most governments borrow rather than print the money to finance debt.

In the United States, borrowing is accomplished primarily by selling government bonds. The purchaser, who may be an individual or another country, pays for a bond at the time of sale, and in return is promised a future amount of money, sometimes with interest payments made before the end of the bond period. The United States pays interest on its national debt bonds, which can be significant. For example, in early 2025, interest on the national debt was estimated at $392 billion, or approximately 16 percent of all federal outlays in fiscal year 2025.

Intentional Debt

Having a large national debt poses many risks to an economy. A large national debt can help contribute to inflation and can lead to tax increases. Economists have also determined that GDP tends to grow more in an economy with a moderate level of national debt than in one with a high level of debt. Many variables affect spending, deficit, and debt. For example, governments often run deficits during economic recessions or depressions, spending money to attempt to stimulate the economy, partly under the notion that future gains will compensate and yield a positive long-term average or expected value.

The national debt in the United States was $2.6 billion in 1900 and experienced overall nonlinear growth approaching the twenty-first century. Mathematical analyses have shown that debt increased sharply during World War I, while in the 1920s national debt decreased due to surpluses. It increased sharply again during the 1930s because of the Great Depression. Another increase occurred with spending for World War II. By 1950, the US national debt had grown to $256.8 billion. After several relatively small increases, the national debt grew quickly beginning in the mid-1970s. Using exponential regression, mathematicians and economists have estimated that the national debt was doubling approximately every six years during this latter period up to nearly the end of the twentieth century. Projective models extrapolate such trends to estimate debt, often based on other estimated values, like the future population.

By early 2025, the US national debt had reached $36.2 trillion, of which $28.9 trillion was held by the public, according to the US Treasury. Notable events in the twenty-first century that caused significant spikes in the debt included the Afghanistan and Iraq Wars started in the early 2000s; the 2008 Great Recession; and the COVID-19 pandemic that broke out in early 2020.

Debt Compared to GDP

In the same way that individuals can afford to spend more money when they receive a raise in salary, it can be misleading to look at the dollar amount of the federal debt without considering the overall size of the economy and the time value of money. For this reason, economists often evaluate the economic health of governments by considering national debt as a percentage of the country’s GDP.

In the United States in 1940, the national debt was 51.6 percent (gross federal debt; 43.6 percent, publicly held debt) of the GDP. This number increased during World War II to 118.9 percent (gross; 106.1 percent, publicly held) in 1946, meaning that the national debt was actually larger than the GDP, but fell below 100 percent again in subsequent years. Mathematicians and economists have created models to forecast this index, with some predicting that factors such as the housing and financial crises would cause the United States to once again pass the 100 percent threshold in the twenty-first century. Indeed, this threshold was surpassed in 2012, when the gross federal debt reached 100.1 percent of GDP; however, publicly held debt only amounted to 70.4 percent of that year's GDP. By 2024, the debt to GDP ratio had grown higher, with the United States' gross federal debt estimated at 123 percent of GDP.

Bibliography

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DeSilver, Drew. "5 Facts about the U.S. National Debt." Pew Research Center, 14 Feb. 2023, www.pewresearch.org/fact-tank/2017/08/17/5-facts-about-the-national-debt-what-you-should-know/. Accessed 6 Mar. 2025.

Feigenbaum, Susan K., and R. W. Hafer. Principles of Macroeconomics: The Way We Live. Worth Publishers, 2012.

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"What Is the National Debt?" FiscalData.Treasury.gov, US Department of the Treasury, 5 Mar. 2025, fiscaldata.treasury.gov/americas-finance-guide/national-debt/. Accessed 6 Mar. 2025.