The IS-LM Model

Derivation · Deriving the IS curve

1. 01

   Goods-market equilibrium: Y = C(Y − T) + I(Y, i) + G.

   $$Y = c_0 + c_1(Y - T) + I_0 - d_1\,i + G$$
2. 02

   Collect Y on the left.

   $$Y(1 - c_1) = c_0 - c_1 T + I_0 - d_1 i + G$$
3. 03

   Divide by (1 − c₁).

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

   Predict

   What does ∂Y/∂i look like (slope of IS in (Y, i) space)?

   • Positive — IS slopes up.
   • Zero — IS is flat.
   • Negative — IS slopes down.
   • Vertical — IS doesn't depend on i.
4. 04

   Horizontal shift: a fiscal expansion ΔG > 0 shifts IS right by ΔG · k where k = 1/(1−c₁).

   $$\Delta Y_{IS} \;=\; \frac{\Delta G}{1-c_1}$$

Worked example · Compute Y* with default parameters

c₀=100, c₁=0.6, T=200, I₀=150, G=200, d₁=500, iᵀ=0.04.

1. 1

   Autonomous demand A = c₀ − c₁T + I₀ + G.

   $$A = 100 - 0.6\cdot 200 + 150 + 200 = 330$$
2. 2

   Multiplier k = 1/(1 − 0.6) = 2.5.

   $$k = 2.5$$
3. 3

   IS intercept (i = 0): a = k · A = 2.5 · 330 = 825.

   $$a = 825$$
4. 4

   Slope coefficient: b = k · d₁ = 2.5 · 500 = 1250.

   $$b = 1250$$
5. 5

   Y* = a − b · iᵀ = 825 − 1250 · 0.04 = 825 − 50 = 775.

   $$Y^* = 775$$

✓ Y* = 775.