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

The Zero Lower Bound & Deflation Trap

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The nominal rate cannot fall below zero (approximately) because households can always hold cash at zero return. This means r = i − πe ≥ −πe. If rn < −πe, the CB cannot achieve Y = Yn. Output stays below potential, deflation sets in, πe falls, and the real rate rises further — a self-reinforcing deflation trap. Policy responses: forward guidance, QE, helicopter money, fiscal policy.

Derivation

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The ZLB Constraint

Cash yields zero. So no financial asset can sustainably trade below zero:

i \geq 0 \iff r \geq -\pi^e

The real-rate floor is $-\pi^e$ , not zero. With $\pi^e = 2\%$ , the CB can push $r$ down to $-2\%$ — but no further via conventional means.

When the ZLB Binds

r_n < -\pi^e \iff \text{ZLB binds, } Y < Y_n

If the natural real rate is sufficiently negative (deep recession, secular stagnation, demographics), the CB cannot achieve potential output with $i = 0$ .

The Deflation Trap

If the ZLB binds, $Y < Y_n$ and the Phillips curve drives inflation down. Adaptive expectations then lower $\pi^e$ , raising the real rate $r = -\pi^e$ back up:

\downarrow \pi \to \downarrow \pi^e \to \uparrow r \to \downarrow Y \to \downarrow \pi \to \cdots

This self-reinforcing spiral is the deflation trap — Japan 1990s, Europe 2014–15 flirted with it.

Escape Tools

| Tool | Mechanism | |------|-----------| | Forward guidance | Raise $\pi^e$ or lower $r^{e'}$ by committing to future easing | | Quantitative easing | Compress term and risk premia ( $\downarrow x$ ) | | Fiscal expansion | Shift IS right; full multiplier at the ZLB | | Helicopter money | Direct transfers — bypass the banking system | | Negative rates | Possible below zero up to cash-storage cost (~−0.75%) |

Why the Fiscal Multiplier Is Larger at the ZLB

Away from ZLB, a fiscal expansion may trigger the CB to raise $i^T$ if it targets a Taylor-type rule. At the ZLB, $i$ stays at zero — no crowding out. The pure multiplier $1/(1-c_1)$ applies fully.

Worked Example

Yn = 900. IS: Y = 800 − 2000r (after autonomous-demand fall). πe = 2%.

Solve for rn: 900 = 800 − 2000rn → rn = −100/2000 = −5%.
ZLB real rate floor: r = −πe = −2%. CB can only achieve r = −2%, not −5%.
At r = −2%: Y = 800 − 2000×(−0.02) = 840. Gap = 840 − 900 = −60.
Δπ = (α/L)·(Y − Yn). For α = 0.5, L = 100: Δπ = (0.5/100)·(−60) = −0.3 pp/year. Deflation begins; πe will fall, raising r further.

rn = −5%, ZLB binds. At i = 0: Y = 840, gap = −60. Without intervention, deflation dynamics deepen the gap — a classic liquidity trap.

Common Mistakes

—Confusing the ZLB on i with a floor on r — the real-rate floor is −πe, not zero.
—Assuming fiscal policy also hits a ZLB-like constraint — it does not; the fiscal multiplier is actually larger at the ZLB.
—Treating QE and forward guidance as identical — QE moves x (risk premium); FG moves πe and r^e.
—Missing that negative rates are possible up to the cost of cash storage (~−0.75% in practice).

Exam Cues

→ZLB binds iff rn < −πe. Core Mock Q3/Q4 structure.
→Output gap at ZLB: Y_ZLB = IS(r = −πe) < Yn when constraint binds.
→Deflation trap: ↓π → ↑r → ↓Y → ↓π. Self-reinforcing.
→Policy: fiscal has full multiplier at ZLB; monetary works only through expectations (FG) and premia (QE).