The Phase Diagram of Debt-to-GDP Limit and Debt-Financing of Interest

This paper proposes an alternative way of assessing government debt sustainability – instead of focusing on the absolute level of debt-to-GDP ratio, compute the time it takes before the government is compelled, if ever, to finance debt interest with additional debt, a condition I refer to as “Defit” (Debt-Financing of Interest), to distinguish it from Default. Defit occurs when the government’s annual revenue falls short of annual debt interest obligation, which is equivalent to the debt-to-GDP ratio reaching the threshold x/r, where x is the effective tax rate of the government revenue and r the average interest rate on outstanding debt. The paper derives the sufficient and necessary conditions of Defitting, and the formula for time to Defit in terms of observable economic and fiscal inputs. The present analysis departs from the prior literature as it models the joint dynamics of government cash and debt balances, establishing a link between the debt-to-GDP ratio and the debt-to-cash ratio. This approach yields a more precise characterization of debt-to-GDP divergence, particularly in relation to the differential g-r, where g is the nominal GDP growth rate. The result replaces the absolute level of primary surplus with the differential s-r, where s is the ratio of primary surplus to primary outlays.