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Appendix 15.A The Debt-GDP Ratio

In this appendix we derive Eq. (15.4), which shows how the debt-GDP ratio evolves. If we let Q represent the ratio of government debt to GDP, by definition

where B is the nominal value of government bonds outstanding (government debt), P is the price level, and Y is real GDP (so that PY is nominal GDP).

A useful rule is that the percentage change in any ratio equals the percentage change in the numerator minus the percentage change in the denominator (Appendix A, Section A.7). Applying this rule to Eq. (15.A.1) gives

Simplifying this expression gives

which in words means the change in the ratio of government debt to GDP = deficit/GDP minus (debt/GDP times the growth rate of nominal GDP). Eq. (15.A.3) is identical to Eq. (15.4).

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Source: Abel A.B., Bernanke B., Croushore D.. Macroeconomics. 10th Edition, Global Edition. — Pearson,2021. — 690 pp.. 2021
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