Lesson M17.L02: IS-LM-PC Dynamics: Demand Shocks

Module: From Short to Medium Run: The IS-LM-PC Model Level: intermediate Duration: 30 minutes Learning Objective: Trace the dynamic adjustment path after a positive demand shock in the IS-LM-PC model. Data as of: 2024 Provenance: RBA Research Discussion Papers | MIT OCW 14.02

Explanation

A demand shock (e.g., a fiscal stimulus, surge in consumer confidence, or housing boom) shifts the IS curve rightward. In the IS-LM-PC model, this triggers a multi-period dynamic adjustment. Understanding this sequence is central to macroeconomic policy analysis.

Notation: - Yₙ = potential output; Y₀ = initial (= potential) output - i = nominal interest rate (central bank instrument) - πᵉₜ = expected inflation in period t (formed adaptively: πᵉₜ = πₜ₋₁) - β = PC slope w.r.t. output gap (≈ 0.5–1.0) - π* = central bank inflation target - ε = supply shock (= 0 here)

Five-step dynamic sequence after a positive demand shock:

1. IS shifts right: Higher autonomous demand raises Y above Yₙ. The output gap (Y − Yₙ)/Yₙ > 0. The central bank may also allow i to rise somewhat (moving up the LM or reaction function).

2. Inflation rises: The positive output gap feeds into the PC: π₁ = πᵉ₁ + β × gap₁ > πᵉ₁. Actual inflation exceeds expected inflation.

3. Expectations adjust upward: With adaptive expectations, πᵉ₂ = π₁ > π*. The PC shifts up. The same output gap now produces even higher inflation.

4. Central bank tightens: Observing π > π*, the central bank raises i. The LM shifts left (or equivalently, the policy rate rises along the reaction function), pushing Y back toward Yₙ. The output gap narrows.

5. Return to medium-run equilibrium: As Y → Yₙ, the output gap closes. Inflation stabilises, but at a higher level (π + inflation overshoot). Eventually, if the CB holds firm, πᵉ re-anchors at π and π → π*.

Credibility shortcut: If the central bank has perfect credibility and commits credibly to π, rational agents keep πᵉ = π throughout. The PC does not shift up. Adjustment is faster and less costly in output terms.

Australian example (2021–2023): Post-COVID reopening boom created a large positive demand shock. The RBA began raising the cash rate in May 2022 (from 0.1% to 4.35% by late 2023), working to close the output gap and return inflation to the 2–3% target band.

Worked Example

Setup: - Yₙ = 100; β = 0.5; π* = 2%; πᵉ initially = 2%; ε = 0 - Demand shock: IS shifts so Y₁ = 102 in period 1 (output gap = +2%) - CB response (periods 2–3): raises i to partially close gap. Assume CB reduces gap by 50% per period. - Adaptive expectations: πᵉₜ₊₁ = πₜ

Period 0 (pre-shock baseline):

Y₀ = 100 = Yₙ, gap₀ = 0%, πᵉ₀ = 2%, π₀ = 2%

Period 1 (demand shock hits):

Y₁ = 102,  gap₁ = (102 − 100)/100 = 2%
πᵉ₁ = π₀ = 2%
π₁ = πᵉ₁ + β × gap₁ = 2% + 0.5 × 2% = 3.0%

Period 2 (CB tightens, gap halves):

Y₂ = 101,  gap₂ = (101 − 100)/100 = 1%
πᵉ₂ = π₁ = 3.0%
π₂ = πᵉ₂ + β × gap₂ = 3.0% + 0.5 × 1% = 3.5%

Period 3 (CB tightens further, gap closes):

Y₃ = 100 = Yₙ,  gap₃ = 0%
πᵉ₃ = π₂ = 3.5%
π₃ = πᵉ₃ + β × 0 = 3.5%

Summary table:

Period	Y	Gap	πᵉ	π
0	100	0%	2.0%	2.0%
1	102	+2%	2.0%	3.0%
2	101	+1%	3.0%	3.5%
3	100	0%	3.5%	3.5%

Key insight: Even after the output gap closes, inflation is 3.5% — 1.5 pp above the original target. The demand shock has permanently raised the inflation expectation (given adaptive expectations). To return π to 2%, the CB would need to push Y below Yₙ to generate a negative output gap, creating a painful disinflationary recession. This is the inflation legacy of delayed tightening.

Common Misconception

Misconception: "Once the central bank raises interest rates and closes the output gap, inflation will immediately fall back to target."

Correction: With adaptive expectations, πᵉ rises each period that π > π. Even when the output gap returns to zero, the PC has shifted up. Inflation is now anchored at the new higher level of πᵉ. Returning to π requires either a prolonged period of Y < Yₙ (a deliberate recession) or a credible commitment that shifts πᵉ down quickly. This is why early and decisive central bank action — before expectations de-anchor — is far less costly than delayed action.

Practice Prompts

1. Conceptual: Why does a central bank with credibility achieve a faster and less costly return to equilibrium after a demand shock than one without? → Answer: A credible CB anchors πᵉ = π* regardless of short-run inflation outcomes. The PC does not shift up when π temporarily rises above target, because agents believe the CB will close the output gap. As a result, the inflation overshoot is smaller, and no disinflationary recession is needed. Credibility reduces the sacrifice ratio — the output loss per unit of inflation reduction.

2. Numerical: Using β = 0.5, Yₙ = 500, and the adaptive expectations rule πᵉₜ₊₁ = πₜ, trace the adjustment path for 2 periods given: π = 2%, period-1 demand shock raises Y₁ = 510, and the CB closes the gap entirely by period 2 (Y₂ = 500). → Answer:
   Period 1: gap₁ = (510−500)/500 = 2%, πᵉ₁ = 2%
   π₁ = 2% + 0.5×2% = 3%
   Period 2: Y₂ = 500, gap₂ = 0%, πᵉ₂ = π₁ = 3%
   π₂ = 3% + 0.5×0% = 3%*
   Inflation stays at 3% even after the gap closes. The CB must push Y below 500 to drive π back to 2%, or re-anchor expectations through forward guidance.

3. Application: In 2022, the RBA increased the cash rate from 0.10% to over 3% within 12 months. Using the IS-LM-PC adjustment framework, explain why rapid rate increases were preferred to gradual ones. → Answer: Rapid rate increases act earlier in the adjustment path to close the positive output gap before expectations can drift significantly above π*. In the IS-LM-PC model, the longer a positive output gap persists, the more πᵉ rises (under adaptive expectations), the higher the PC shifts, and the more painful the eventual disinflation. Fast tightening minimises the number of periods during which πᵉ adjusts upward, reducing the total inflation overshoot and the eventual output cost of returning to target.

Visual — Dynamic IS-LM-PC Adjustment After a Demand Shock

Figure: A positive demand shock first shifts IS right and pushes output above potential. Inflation then rises above expected inflation, expectations adjust upward, and the central bank tightens to bring output back to potential — but at a higher inflation level unless expectations are re-anchored.

Further Resources

• 📺 Macroeconomics: The IS-LM-PC Model — Academic Macro Channel (12 min)
• 📺 The IS-MP-PC Model — Macro Lectures (15 min)
• 📚 RBA Research: Monetary Policy Transmission — RBA research discussion papers on policy effectiveness