The IS–LM Model

Deriving the IS Curve

coreExam · highTA · PS2-Q1Mock · Q1

The IS curve is the locus of (Y, i) pairs at which the goods market clears. It is downward-sloping because a higher interest rate reduces investment, which reduces demand and equilibrium output. Fiscal policy shifts the IS curve; the interest rate moves the economy along it.

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 IS Curve

The IS curve maps every $(Y, i)$ pair at which the goods market clears. It extends the simple goods-market model by allowing investment to depend on the interest rate.

The IS curve is a summary of the goods market: for any interest rate $i$ the central bank might choose, the IS curve tells you what output $Y$ will emerge from goods-market equilibration.

Why Downward Sloping?

\uparrow i \;\Rightarrow\; \downarrow I(Y,i) \;\Rightarrow\; \downarrow Z \;\Rightarrow\; \downarrow Y

Higher borrowing costs depress investment. The multiplier amplifies the demand shortfall into a larger output decline.

The IS Equation

With $C = c_0 + c_1(Y-T)$ and $I = I_0 - d_1 i$ :

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

IS Shifts vs Movements Along IS

Movements along IS: the central bank changes $i$ (via the LM / interest-rate target).

IS shifts: fiscal policy ( $\Delta G$ , $\Delta T$ ), consumer confidence ( $\Delta c_0$ ), or investment expectations ( $\Delta I_0$ ).

The horizontal shift from $\Delta G > 0$ is $\Delta G/(1-c_1)$ — the full spending multiplier.

Worked Example

c₁=0.6, d₁=500, T=200, G=200, c₀=100, I₀=150. Write the IS equation and find Y* when i=4%.

Autonomous demand: A = 100 − 0.6×200 + 150 + 200 = 330.
Multiplier: 1/(1−0.6) = 2.5.
IS: Y = 330×2.5 − 500×2.5×i = 825 − 1250i.
At i=0.04: Y* = 825 − 1250×0.04 = 825 − 50 = 775.

IS equation: Y = 825 − 1250i. At i = 4%, equilibrium output Y* = 775.

Common Mistakes

—Drawing IS upward sloping — it is always downward sloping in (Y, i) space.
—Confusing shifts in IS (fiscal policy) with movements along IS (interest rate changes).
—Forgetting to multiply the fiscal shock by the multiplier when computing the IS horizontal shift.
—Using d₁ alone as the slope; the correct slope coefficient on i is d₁/(1−c₁).

Exam Cues

→IS shifts right: ↑G, ↓T, ↑c₀, ↑I₀ (any autonomous demand increase).
→IS shifts left: ↓G, ↑T, ↓c₀, ↓I₀.
→Movement along IS: the central bank changes the interest rate target.
→Horizontal IS shift = fiscal change × 1/(1−c₁).