07

The Phillips Curve

The Phillips Curve & Inflation Expectations

coreExam · highTA · PS4-Q1

The Phillips curve: πt = πte + (m+z) − α·ut. Natural rate un = (m+z)/α is where π = πe. Pre-1970: anchored expectations (θ=0) gave a level inflation–unemployment trade-off. Post-1970: adaptive expectations (θ=1) turned it into Δπt = −α(ut − un). Below-natural unemployment continuously raises inflation.

Derivation

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Adaptive (post-1970)πₜ = θ·πₜ₋₁ + (1−θ)·π̄ − α(u − uₙ)
tππᵉπ
Adaptive expectations (θ≈1): a persistent negative unemployment gap creates accelerating inflation — no long-run trade-off.
Controls
θ (adaptive weight)1.00
u (unemployment)3.0%
π₀ (initial)2.0%
u − uₙ-2.0 pp
π₁₂13.0%
Drift+11.0 pp

θ = 0 (anchored): stable level trade-off. Fixed πᵉ = π̄.

θ = 1 (adaptive): Δπ depends on gap. Accelerationist.

Sacrifice ratio: 1/α = 2.0 pp·years of unemployment per pp of disinflation.

The Phillips Curve

The Phillips curve emerges from combining wage and price setting. Workers bargain based on expected prices; firms mark up over wage costs. When unemployment is below the natural rate, workers have bargaining power — wages and prices rise faster than expected.

Core equation: πt=πte−α(ut−un)\pi_t = \pi_t^e - \alpha(u_t - u_n). Below-natural unemployment (ut<unu_t < u_n) generates positive inflation surprises. Above-natural generates negative surprises (disinflation).

Two Eras

Pre-1970 (anchored expectations, θ=0): Inflation was non-persistent so πte=πˉ\pi_t^e = \bar{\pi}. The result was a stable level trade-off: choose lower unemployment, accept higher inflation.

Post-1970 (adaptive expectations, θ=1): Inflation became persistent so πte=πt−1\pi_t^e = \pi_{t-1}. The level trade-off vanished; a new relation appeared:

πt−πt−1=−α(ut−un)\pi_t - \pi_{t-1} = -\alpha(u_t - u_n)

The Natural Rate

Setting πt=πte\pi_t = \pi_t^e (no acceleration):

un=m+zαu_n = \frac{m + z}{\alpha}

The natural rate rises with the product-market markup mm and labour-market catchall zz (unemployment insurance, employment protection, minimum wage). It falls with α\alpha (stronger wage sensitivity to unemployment).

Disinflation

Reducing inflation requires ut>unu_t > u_n — above-natural unemployment — for multiple periods. The sacrifice ratio =1/α= 1/\alpha measures the cumulative unemployment gap cost per percentage point of inflation reduction.

Worked Example

α=0.5, un=5%. Unemployment is held at ut=3% for two years. Initial π₀=2%. Find π₁ and π₂.

  1. Year 1: Δπ₁ = −0.5×(3%−5%) = +1 pp. π₁ = 2% + 1% = 3%.
  2. Year 2: Δπ₂ = −0.5×(3%−5%) = +1 pp. π₂ = 3% + 1% = 4%.
  3. Each year of below-natural unemployment adds 1 pp to inflation.
π₁ = 3%, π₂ = 4%. Sustained below-natural unemployment continuously accelerates inflation.

Common Mistakes

  • —Using the pre-1970 level Phillips curve when θ=1 — post-1970 the relation is between Δπ and u, not π and u.
  • —Confusing un with zero unemployment — un is typically 4–6%, not 0%.
  • —Getting the sign wrong: ut < un → Δπ > 0 (inflation rises, not falls).
  • —Forgetting that un shifts when m or z change — labour market reforms that lower z reduce un.

Exam Cues

  • →Post-1970 form (used in Grassi): Δπt = −α(ut − un). This appears in all TA and mock problems.
  • →Disinflation requires ut > un for multiple periods. Sacrifice ratio = 1/α pp of unemployment per 1 pp Δπ.
  • →un rises with higher m (less competition) or higher z (more generous UI, stronger employment protection).
  • →IS-LM-PC model (Ch 8): central bank targets ut = un by adjusting iᵀ → stable inflation.

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