The IS–LM Model

The LM Curve — Modern Interest-Rate Targeting

coreExam · highTA · PS2-Q2Mock · Q2

In the modern view, the central bank sets a target interest rate iᵀ and supplies whatever money is needed. The LM curve is therefore horizontal at i = iᵀ. Grassi Ch 5: "The central bank chooses the interest rate and adjusts the money supply so as to achieve it."

Derivation

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The Modern LM

In Grassi's treatment, the central bank is an interest-rate targeter: it announces $i^T$ and stands ready to buy or sell bonds to enforce it. Money supply adjusts passively.

This matches modern central banking practice: the ECB, Fed, and Bank of England all operate by setting a policy rate — not by targeting M1 or M2.

LM in the Diagram

The LM curve is a horizontal line at $i = i^T$ in the $(Y, i)$ plane:

\text{LM:} \quad i = i^T

Any output level $Y$ is consistent with this interest rate — the central bank creates whatever money is needed.

IS–LM Equilibrium

Y^* = \frac{c_0 - c_1 T + I_0 + G}{1 - c_1} - \frac{d_1}{1-c_1} \cdot i^T

The equilibrium is at the intersection of the downward-sloping IS curve and the horizontal LM line.

Monetary Policy

The central bank shifts the LM curve by changing $i^T$ :

Easing: $\downarrow i^T$ → LM shifts down → higher $Y^*$ (more investment)
Tightening: $\uparrow i^T$ → LM shifts up → lower $Y^*$ (less investment)

Worked Example

IS: Y = 825 − 1250i (from the IS derivation atom). Central bank sets iᵀ = 4%. (a) Find Y*. (b) Central bank lowers iᵀ to 2%. Find new Y*.

(a) Y* = 825 − 1250×0.04 = 825 − 50 = 775.
(b) Y* = 825 − 1250×0.02 = 825 − 25 = 800.
Monetary easing raises Y* by 25.

(a) Y*=775 at iᵀ=4%. (b) Y*=800 at iᵀ=2%. LM shifts down by 2 pp.

Common Mistakes

—Drawing LM upward sloping — in the Grassi model LM is always horizontal.
—Saying the central bank controls M — in the modern view M is endogenous; iᵀ is the instrument.
—Confusing a shift in LM (change in iᵀ) with movements along the IS curve.

Exam Cues

→Grassi's LM is flat. If the question says 'central bank sets i', draw horizontal LM.
→Monetary expansion = ↓iᵀ → LM shifts down → Y* rises.
→Fiscal expansion under flat LM: IS right, Y* rises, i unchanged — no crowding-out.
→Mock Q2: restoring Y₀ after fiscal shock requires ↑iᵀ so LM shifts up and Y* falls back.