Module 13 · Chapters 18

Fiscal Policy & Debt Dynamics

Deficits, debt, and the (r − g) condition for sustainability.

“When does a country's debt stay manageable, and when does it spiral?”

~30 min· 3 sub-skills·5 exercises0% mastered

Fiscal policy in macro isn't just about year-on-year stabilization — it's also about a stock variable: public debt. The single equation behind every IMF debt-sustainability analysis is the debt-dynamics formula, which makes the (r − g) gap the master variable.
Government budget identity
$B_{t+1} - B_t \;=\; r\,B_t + (G_t - T_t)$
$B_t$
outstanding debt at start of year t
$r$
real interest rate on debt
$G_t - T_t$
primary deficit (excludes interest)
Debt grows by interest accrued plus the primary deficit.
Debt-to-GDP dynamics
$b_{t+1} - b_t \;\approx\; (r - g)\,b_t \;+\; d_t$
$b_t = B_t/Y_t$
debt-to-GDP ratio
$g$
real GDP growth
$d_t$
primary deficit-to-GDP
**The (r − g) gap is decisive.** If r < g, even with a primary deficit, debt-to-GDP can shrink. If r > g, even with a primary surplus, debt grows.
Figure · Debt-to-GDP path under various (r − g, d)
Debt-to-GDP path
(r − g) = 2.0%
b_{t+1} ≈ (r−g)·b_t + d_t. r > g + d > 0 → debt explodes; r < g → stabilises.
Real rate r4.0%
Real growth g2.0%
Primary deficit d1.0% GDP
Initial b60% GDP
Sustainable when r − g < 0 or when sufficient primary surpluses offset r > g.
Definition·Steady-state debt ratio
Setting Δb = 0: $b^= -d/(r - g)$ (when r > g; with d the primary deficit). Stabilising debt with r > g requires d ≤ (g − r)·b^ — typically a small primary surplus suffices when (r − g) is mildly positive.
Common misconception
A country can never run an indefinite primary deficit.
ActuallyIf r < g consistently (post-2008 most advanced economies), a moderate primary deficit is consistent with stable or falling b. Japan illustrates: huge debt ratios, sustained ultra-low r, no crisis.
Exercise · multiple choice · +8 XP
Primary vs total deficit
What's the difference between primary and overall deficit?
Exercise · numerical · +12 XP
Δb in one year
b_t = 100% (b = 1.0). r = 4%, g = 2%, primary deficit = 1% of GDP. Approximate Δb (in pp).
Your answer (pp of GDP)
Exercise · numerical · +14 XP
Stabilising primary balance
b = 100%. r = 4%, g = 1%. What primary balance (as % GDP) stabilises the debt ratio?
Your answer (% GDP (positive = surplus))
Exercise · true false · +10 XP
(r − g) sign and dynamics
If r < g and the primary balance is zero, the debt-to-GDP ratio falls over time.
"If r < g and the primary balance is zero, the debt-to-GDP ratio falls over time."
Exercise · multi step · +18 XP
Italy hypothetical
Italy: b = 140%, r = 3%, g = 0.5%, primary balance = +1.5% (surplus).
Context: Compute the dynamics for one year and the implied stabilising primary balance.
(a)Δb (in pp):
(b)Primary balance needed to stabilise (in %):
(c)Gap to stabilising surplus (pp):

Mastery check

5 questions · pass with 80%

Answer all five to confirm you've internalised the module. A passing run unlocks the next module.

Q1
(r − g) > 0 means:
Q2
"Primary deficit = government deficit excluding interest payments."
Q3
b = 80%, r = 5%, g = 2%, primary balance = 0. Δb (pp)?
Q4
Why has Japan's >250% debt ratio not triggered crisis?
Q5
b = 60%, r = 3%, g = 1%. Stabilising primary balance (% GDP, +=surplus)?

0 / 5 answered

Exam pitfalls

Confusing primary balance with overall balance. The exam usually asks for primary.
Wrong sign on (r − g). r > g is the danger zone.
Forgetting the formula gives Δb in *pp of GDP*, not absolute euros.
Ignoring inflation channel — high π reduces real debt; the formula uses real r.