Expectations, Output & Policy

Deficit & Debt Dynamics

coreExam · medium

Debt-to-GDP evolves according to Δ(B/Y) ≈ (r − g)·(B/Y) + primary deficit/Y. If r > g, debt is unstable without primary surpluses. Countries need primary surpluses proportional to (r − g)·(B/Y) to keep debt stable. Sovereign risk rises when this is infeasible — spread widens, r rises further.

Derivation

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The Debt-to-GDP Law

\Delta\left(\frac{B}{Y}\right)_t \approx (r - g) \cdot \frac{B_{t-1}}{Y_{t-1}} + \frac{G_t - T_t}{Y_t}

Two components:

Snowball: $(r - g) \cdot (B/Y)$ — interest compounding offset by growth
Primary deficit: $(G - T)/Y$ — direct new borrowing

Stabilisation Condition

For $(B/Y)$ to stay constant:

\frac{\text{PS}}{Y} = (r - g) \cdot \frac{B}{Y}

where PS = primary surplus.

When $r > g$ , you need positive primary surpluses just to tread water. The Euro-area periphery faced this in 2010–14: bond yields rose, growth collapsed, and the required PS was infeasible.

r vs g Regimes

| Regime | Dynamics | Example | |--------|----------|---------| | $r < g$ | Debt self-stabilising even with deficits | US most of postwar, Japan 2010s | | $r > g$ | Need primary surplus; vulnerable to shocks | Italy 2011, Greece 2010–15 |

Multiple Equilibria

The risk premium $x$ depends on perceived default risk, which depends on debt sustainability, which depends on $r = i + x - \pi$ . This creates a feedback loop:

Low- $x$ equilibrium: debt sustainable, $x$ stays low.
High- $x$ equilibrium: fears of default raise $x$ → $r$ rises → debt unsustainable → actual default risk rises.

The ECB's 2012 OMT programme was designed to eliminate the bad equilibrium by committing to conditional intervention — a "whatever it takes" backstop.

Worked Example

Italy 2012: B/Y = 130%, r = 6%, g = −1% (recession), primary surplus = 2% of GDP.

Required PB for stability: (r − g)·B/Y = (0.06 − (−0.01)) × 1.30 = 0.07 × 1.30 = 9.1% of GDP.
Actual PB = 2%. Shortfall = 7.1 pp. Debt/Y rising rapidly.
Italy needed to close a 7 pp gap via austerity or restored growth.
ECB OMT announcement lowered x and hence r — shifted the debt dynamic back toward sustainability.

Required primary surplus 9.1% of GDP. Actual only 2%. Gap of 7 pp — debt on unsustainable trajectory until r fell via OMT intervention.

Common Mistakes

—Confusing primary balance (excl. interest) with overall balance (incl. interest) — debt dynamics depend on primary.
—Missing the (r − g) term — the snowball driver of debt accumulation.
—Assuming r is exogenous — sovereign spread x makes r rise with debt/GDP.
—Ignoring the multiple-equilibria story — belief-driven sovereign-debt crises.

Exam Cues

→Debt dynamics: Δ(B/Y) = (r − g)·(B/Y) + primary deficit/Y.
→Stabilisation: PS required = (r − g)·B/Y. If r > g, need PS > 0.
→Multiple equilibria: high-x possible, eliminated by central-bank backstop.
→Austerity debate: cutting G raises PS but also ↓g (recession), may not lower debt ratio.