BOX 3-1

The Arithmetic of the Debt

Suppose borrowing were used only to finance the interest portion of spending, that is, revenues were always sufficient to cover noninterest (“primary”) spending. And suppose that the interest rate on the debt equals the growth rate of incomes (or GDP, given that incomes and GDP grow at nearly the same rate). Under those conditions, the debt will grow at the same rate as incomes, and the debt-to-GDP ratio will remain constant. For example, if the interest rate on the debt is 5 percent and the U.S. Treasury borrows just enough to cover it, the debt will grow 5 percent annually. If GDP also were to grow at 5 percent, the debt-to-GDP ratio would remain constant. At 60 percent of GDP, which is the target ratio the committee proposes, debt service at an interest rate of 5 percent would be 3 percent of GDP (0.6 × 5). If all spending totaled about 20 percent of GDP, interest on the debt would be 15 percent of all spending. With revenues at 17 percent of GDP, the budget would be balanced except for interest on the debt. The annual budget deficit—equal to the additional borrowing or growth of debt—would be 3 percent of GDP and the debt would grow by that amount or at a rate of 5 percent (3/60). If GDP grew 4 percent in one year, then debt at the end of the year would be 60.6 percent of GDP.a If budgeters wanted to prevent the debt from growing, they could either raise revenues or reduce primary spending as a percentage of GDP. This “solution” might not be practical for long, however, given that the study’s baseline projects primary spending to grow faster than GDP if no changes in policy are made. In that case, it would become increasingly difficult to either limit or offset spending growth in order to keep debt stable.

The simple relationship between growth rate of the economy and the federal debt is summarized by the following equation (adapted from von Furstenberg, 1991):

where R is revenues, P is noninterest spending, and Dt−.5 is the average publicly held federal debt during the year, all expressed as percents of GDP. The average interest rate on debt is represented by r, n is the growth rate of GDP, and t is the fiscal year. When r − n is zero, revenues can equal primary spending plus the change in the debt resulting from borrowing the amount needed that year for interest on the debt paid that year. If r − n is negative, either revenues can be lowered or spending increased without increasing the debt. Conversely, in a year when r − n is positive, the debt will grow unless revenues are increased or spending reduced as a percentage of GDP. As the debt grows, so does the burden of interest on that debt.

Even if r − n were always zero, if P is on a rising path, as this study’s baseline projects, then to keep the debt from increasing relative to GDP, R must rise by the same amount. Or, to keep R constant as P rises, D must fall instead. However, r − n can be (and often has been) positive, as in the example above. Although n can be (and often has been) greater than r, it is precisely when the opposite is true that the nation can least afford to run primary surpluses, so these circumstances deserve special attention (Bohn, 1995). The most recent downturn serves as an example. In 2008 and 2009, n turned negative while r fell slowly, P rose as a result of additional spending demands and efforts to stimulate recovery, and R fell with the drop in employment, incomes, and profits: the result was a rapid rise in federal debt, D.

The National Academies | 500 Fifth St. N.W. | Washington, D.C. 20001
Copyright © National Academy of Sciences. All rights reserved.
Terms of Use and Privacy Statement