Lesson M17.L01: Linking IS-LM to the Phillips Curve

Module: From Short to Medium Run: The IS-LM-PC Model Level: intermediate Duration: 30 minutes Learning Objective: Integrate the IS-LM model and the Phillips curve into the IS-LM-PC framework. Data as of: 2024 Provenance: RBA Explainer: Inflation | MIT OCW 14.02

Explanation

The IS-LM-PC framework bridges short-run output determination and medium-run inflation dynamics. It combines three building blocks.

IS curve: Goods market equilibrium. Output Y is a decreasing function of the real interest rate r:
Y = C + I(r) + G, where higher r reduces investment and hence Y.

LM curve (or central bank reaction function): The central bank sets the nominal interest rate i. In the modern interpretation, the LM is replaced by a monetary policy rule, but for this framework we treat i as the central bank's instrument.

Phillips Curve (PC): Links inflation π to the output gap and expectations:

π = πᵉ + β × (Y − Yₙ)/Yₙ + ε

Notation: - π = actual inflation rate - πᵉ = expected inflation (formed last period) - Yₙ = potential (natural) output — the level consistent with the natural rate of unemployment - (Y − Yₙ)/Yₙ = output gap (positive when boom, negative when recession) - β = sensitivity of inflation to output gap (≈ 0.5–1.0 for Australia); derived as α × Okun coefficient, where α is the PC slope in unemployment–inflation space - ε = supply shock (positive for cost-push shock)

Derivation link: The original PC is π = πᵉ − α(u − uₙ) + ε. Okun's Law states that (u − uₙ) ≈ −λ × (Y − Yₙ)/Yₙ, where λ ≈ 0.5. Substituting: π = πᵉ + αλ × (Y − Yₙ)/Yₙ + ε = πᵉ + β(Y − Yₙ)/Yₙ + ε, where β = αλ.

Three-panel logic: The IS-LM panel (top) determines Y and i. The output gap (Y − Yₙ)/Yₙ is read off relative to the vertical line at Yₙ. The PC panel (bottom) maps the output gap to inflation given πᵉ. In the medium run, the central bank adjusts i so that Y → Yₙ and π → π*.

Australian context: The RBA targets π* = 2.5% (midpoint of 2–3% band). When the economy is above potential — as in 2022 — the positive output gap drives π above πᵉ, prompting the RBA to tighten.

Worked Example

Setup: Potential output Yₙ = 100. Actual output Y = 102 (positive demand shock). Expected inflation πᵉ = 2%. PC parameter β = 0.5. No supply shock (ε = 0).

Step 1 — Compute the output gap:

(Y − Yₙ)/Yₙ = (102 − 100)/100 = 2/100 = 0.02 = 2%

Step 2 — Apply the Phillips Curve:

π = πᵉ + β × (Y − Yₙ)/Yₙ + ε
π = 2% + 0.5 × 2% + 0
π = 2% + 1% = 3%

Step 3 — Interpret: Actual inflation (3%) exceeds expected inflation (2%) by 1 percentage point. This 1 pp gap is generated entirely by the positive output gap. If expectations adapt (πᵉ → 3%), inflation will rise further in the next period unless the central bank intervenes to close the output gap.

Step 4 — Policy implication: To restore π = πᵉ = π* = 2%, the central bank must raise i to shift IS leftward until Y = Yₙ = 100, eliminating the output gap.

Common Misconception

Misconception: "The Phillips Curve says lower unemployment always causes higher inflation — so the central bank can permanently lower unemployment by accepting higher inflation."

Correction: The PC in the IS-LM-PC framework is the expectation-augmented version. Any attempt to hold Y permanently above Yₙ causes πᵉ to drift upward each period. The PC shifts up, and inflation accelerates without bound. There is no permanent inflation–unemployment trade-off; there is only a short-run trade-off along a given PC (fixed πᵉ). In the long run, the economy returns to Yₙ at whatever π the central bank has allowed expectations to settle on.

Practice Prompts

1. Conceptual: What is the role of πᵉ in the expectation-augmented Phillips Curve, and why does it shift the entire PC? → Answer: πᵉ determines the intercept of the PC. When expected inflation rises, wage and price setters build higher inflation into contracts even before any output gap exists. This shifts the entire PC upward: for any given output gap, inflation is higher. Without anchored expectations, repeated positive output gaps cause ever-rising πᵉ and a spiral of inflation.

2. Numerical: Suppose Yₙ = 500, Y = 490, πᵉ = 3%, β = 0.8, ε = 0. Calculate actual inflation π. → Answer:
   Step 1: Output gap = (490 − 500)/500 = −10/500 = −0.02 = −2%
   Step 2: π = πᵉ + β × gap + ε = 3% + 0.8 × (−2%) + 0 = 3% − 1.6% = 1.4%
   Interpretation: A 2% negative output gap (recession) reduces inflation 1.6 pp below expectations. Actual inflation of 1.4% is below the 3% expectation, consistent with a recessionary environment putting downward pressure on prices.

3. Application: In 2022, Australian CPI inflation reached approximately 7.8% while the RBA's inflation target midpoint is 2.5%. Using the IS-LM-PC framework, explain what must have been true about the output gap and/or supply shocks. → Answer: If πᵉ ≈ 2.5% (well-anchored expectations), β = 0.5, and actual π = 7.8%, then: 7.8 = 2.5 + 0.5 × gap + ε. The remaining 5.3 pp must reflect a combination of a large positive output gap and a positive supply shock ε (global energy and supply-chain disruptions). RBA analysis suggests both factors were present: the post-COVID reopening boom created a positive output gap, while global commodity price shocks shifted the PC upward (ε > 0).

Visual — Linking IS-LM to the Phillips Curve

Figure: The IS-LM block determines output and the interest rate. If equilibrium output lies above potential, the positive output gap feeds into the Phillips curve and pushes actual inflation above expected inflation.

Further Resources

• 📺 Macroeconomics: The IS-LM-PC Model — Academic Macro Channel (12 min)
• 📺 The Phillips Curve — Macro Topic 5.2 — Jacob Clifford (7 min)
• 📚 MIT OCW 14.02 Lecture Notes — Full intermediate macro course materials