The Phillips Curve

Disinflation & the Sacrifice Ratio

coreExam · medium

Disinflation — reducing inflation — requires u > un for a sustained period. With adaptive expectations and PC: Δπ = −α(u − un), the sacrifice ratio is 1/α: cumulative unemployment points above natural per pp of inflation reduction. Credible, rapid disinflations can have lower sacrifice ratios if expectations move (Volcker 1979–82, Chile 1975).

Derivation

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The Core Formula

Under adaptive expectations ( $\theta = 1$ ), the Phillips curve is:

\pi_t - \pi_{t-1} = -\alpha(u_t - u_n)

To cumulatively reduce inflation by $\Delta \Pi$ , the economy must accumulate an unemployment gap:

\sum_t (u_t - u_n) = \frac{\Delta \Pi}{\alpha}

The Sacrifice Ratio

SR = \frac{1}{\alpha}

Cumulative unemployment-points above natural per 1 pp of disinflation. For $\alpha = 0.5$ , $SR = 2$ .

Fast vs Slow Disinflation

| Strategy | Duration | Average $u - u_n$ | |----------|----------|-------------------| | Gradual (4 years) | 4 years | 4 pp | | Cold-turkey (2 years) | 2 years | 8 pp |

Under purely adaptive expectations, the total cost is the same. The difference is the distribution of pain.

Credibility and Rational Expectations

If the disinflation is announced and credible, forward-looking wage setters revise $\pi^e$ downward immediately. The PC shifts down without needing a large unemployment gap — the sacrifice ratio shrinks or disappears.

Empirically, this is rare: Volcker (US) and Thatcher (UK) had sacrifice ratios of 4–5 despite CB independence, because expectations adjusted only after the painful recessions confirmed policy commitment.

Historical Values

| Episode | Approx SR | |---------|-----------| | US Volcker 1979–82 | 5 | | UK Thatcher 1980s | 4 | | Chile 1975 (rapid credible reform) | 1 | | Argentina Convertibility (1991) | ~1 |

Worked Example

α = 0.5, un = 5%. Initial π = 10%, target π = 2%. Find cumulative unemployment-years needed.

ΔΠ = 10 − 2 = 8 pp of disinflation needed.
Cumulative u gap = ΔΠ/α = 8/0.5 = 16 pp-years.
Over 4 years: average u − un = 4 pp → u = 9% for 4 years.
Fast (2 years): u = 13% for 2 years (much larger short-run pain, same cumulative cost).

Need 16 pp-years of u above un. Fast 2-year path: u = 13% for 2 years. Slow 4-year path: u = 9% for 4 years. Total u-cost is the same under adaptive expectations.

Common Mistakes

—Using SR = α instead of 1/α — inverse relationship.
—Assuming the sacrifice ratio is constant across regimes — it depends on how expectations adjust.
—Forgetting the stock-vs-flow nature: SR is cumulative pp-years, not a single-year effect.
—Treating disinflation as painless under rational expectations — it's only costless if the announcement is fully credible.

Exam Cues

→Sacrifice ratio formula: 1/α. Typical α = 0.5 gives SR = 2 (2 pp-years per 1 pp disinflation).
→Slow vs fast disinflation: same total cost under adaptive, different distribution.
→Credibility reduces SR — forward-looking wage setters moderate demands.
→Historical values: US Volcker ~5; UK Thatcher ~4; Chile (credible reform) ~1.