From Short to Medium Run (IS-LM-PC)

The IS–LM–PC Model (Short to Medium Run)

coreExam · highMock · Q3, Q4

The IS-LM-PC model combines IS (goods market), flat LM (monetary policy), and the output-gap Phillips curve π − π₋₁ = (α/L)(Y − Yn). The central bank adjusts r to close the output gap. Medium-run equilibrium: Y = Yn, π stable, r = rn (natural real rate). ZLB binds when rn < −πe — the central bank cannot achieve Y = Yn.

Derivation

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IS–LM Shock Lab

Flat LM (interest-rate targeting). Drag sliders to shock the model.

IS curve: Y = a − b·i. LM: i = iᵀ (central bank target). Equilibrium Y* = a − b·iᵀ.

Fiscal policy

G (govt spending)200

T (taxes)200

Monetary policy

iᵀ (rate target)4.0%

c₀ (autonomous C)100

Structure

c₁ (MPC)0.60

d₁ (invest. sensitivity)500

The Three Building Blocks

| Block | Equation | Role | |-------|----------|------| | IS | $Y = C(Y-T) + I(Y,r+x) + G$ | Goods market: $r \downarrow \Rightarrow Y \uparrow$ | | LM | $r = i^T - \pi^e$ | Real rate set by CB | | PC | $\pi - \pi_{-1} = \frac{\alpha}{L}(Y - Y_n)$ | Output gap → inflation change |

Short Run → Medium Run Dynamics

Short run: CB sets $i^T$ ; IS gives $Y$ ; PC translates gap into $\Delta\pi$ .
Adjustment: CB raises/lowers $i^T$ to bring $Y \to Y_n$ .
Medium run: $Y = Y_n$ , $\Delta\pi = 0$ , $r = r_n$ .

The Natural Real Rate

r_n: \quad Y_n = IS(r_n) \quad \Rightarrow \quad r_n = \text{solve from IS at } Y = Y_n

Zero Lower Bound

i \geq 0 \implies r \geq -\pi^e \quad \Rightarrow \quad \text{ZLB binds if } r_n < -\pi^e

At the ZLB the economy is stuck below potential. Deflation can worsen the trap: lower $\pi^e$ raises the real rate further, depressing $Y$ even more.

Mock Q3–Q4 Template

Find $r_n$ : substitute $Y = Y_n$ into IS, solve for $r$ .
Compute $i_n = r_n + \pi^e$ . If $i_n < 0$ : ZLB binds.
At ZLB ( $i=0$ ): $r = -\pi^e$ , compute $Y$ from IS, then output gap $= Y - Y_n$ .
Demand shock (Mock Q4): new IS shifts → new $r_n$ → new medium-run $i^T$ .

Worked Example

IS: Y = 1000 − 2000r. Yn = 900. πe = 2%. (a) Find rn. (b) Find nominal target in = rn + πe. Does ZLB bind? (c) Now autonomous consumption falls: IS becomes Y = 800 − 2000r. Repeat.

(a) Yn = 900: 900 = 1000 − 2000rn → rn = 100/2000 = 5%.
(b) in = 5% + 2% = 7%. No ZLB — central bank sets iᵀ = 7% and achieves Y = 900.
(c) New IS: 900 = 800 − 2000rn → rn = −100/2000 = −5%. in = −5% + 2% = −3% < 0.
ZLB binds. At i=0: r = −πe = −2%. Y = 800 − 2000×(−0.02) = 800+40 = 840. Gap = 840−900 = −60.

Demand shock: rn = −5%, ZLB binds, Y = 840, output gap = −60. Deflation risk — Δπ = (α/L)×(−60) < 0.

Common Mistakes

—Confusing real rate r with nominal rate i — ZLB applies to i, not r.
—Using π directly in IS when the model uses r = i − πe.
—Saying medium-run equilibrium requires π = 0 — it requires Δπ = 0 (stable, not zero inflation).
—Forgetting that a demand shock changes rn permanently, requiring a new iᵀ in the medium run.

Exam Cues

→Mock Q3 steps: (1) find rn from IS at Yn, (2) compute in = rn + πe, (3) check if in < 0.
→Mock Q4: ↑c₀ shifts IS right → rn rises → medium-run iᵀ must rise to keep Y = Yn.
→ZLB formula: ZLB binds iff rn < −πe. With πe=2%, ZLB binds iff rn < −2%.
→Output gap Phillips: Δπ = (α/L)(Y − Yn). Y = Yn → Δπ = 0 → π stays at current level.