Final answer:
The U.S. federal debt as a fraction of GDP in 2050 will be approximately 56%, assuming a balanced primary budget from 2019 onwards, with nominal GDP growth at 3% and an interest rate on debt at 2%.
Step-by-step explanation:
To calculate the future debt as a fraction of GDP, we need to consider the initial debt-to-GDP ratio, the growth rates of nominal GDP, and the interest rate on the debt. Given that the U.S. federal government starts with a debt-to-GDP ratio of 104% at the end of 2018 and we assume a balanced primary budget from 2019 onwards, only the interest on existing debt will cause changes to the debt-to-GDP ratio. Since the nominal GDP growth rate (3%) is higher than the interest rate on debt (2%), the ratio will decrease over time.
The following formula represents the debt-to-GDP ratio in a future year: Debt-to-GDP in future year = (Debt-to-GDP in previous year) × (1 + interest rate) / (1 + GDP growth rate). Applying this formula iteratively from the year 2019 to 2050 allows us to calculate the expected debt-to-GDP ratio in 2050, considering that expenses equal revenues and thus, the numerator remains constant apart from accruing interest.
Let's denote the debt-to-GDP ratio by D, the annual interest rate by r, and the GDP growth rate by g. Initially, D = 1.04 (104%), r = 0.02 (2%), and g = 0.03 (3%). For each year from 2019 to 2050, the ratio will be recalculated using the formula: D = D × (1 + r) / (1 + g).
By 2050, after applying the formula for 31 years (from 2019 to 2050), we find that the ratio has decreased significantly. Using a spreadsheet, a calculator, or a specialized program, one can compute: D = 1.04 × (1.02/1.03)^{31}, which is approximately 0.56 or 56%.