The Slutsky Decomposition

This scene splits a price rise into two effects to see why demand falls.

A price change brings two effects at once. The substitution effect pushes you away from the pricier good. The income effect comes from feeling poorer than before. You can compensate by the old bundle or by the old utility. The two ways split the formula below differently.

For example, if 🍎 apples get pricier, switching to 🍌 bananas is the substitution effect. If you then buy less fruit overall because 💰 money feels tight, that is the income effect.

Δxl=ΔxlS+ΔxlI\Delta x_l = \Delta x_l^{S} + \Delta x_l^{I}
∂xl∂pk=∂hl∂pk−∂xl∂wxk\frac{\partial x_l}{\partial p_k} = \frac{\partial h_l}{\partial p_k} - \frac{\partial x_l}{\partial w} x_k

Part of these learning paths

Examples & Exercises

Let u(x1,x2)=x11/2x21/2u(x_1, x_2) = x_1^{1/2} x_2^{1/2}, p2=1p_2 = 1, w=60w = 60.

When p1p_1 rises from 11 to 22, verify by direct computation the signs of the substitution and income effects under Slutsky compensation.

A=x(p,w)=(awp1,(1−a)wp2)=(30,30)A = x(p, w) = \left(\frac{aw}{p_1}, \frac{(1-a)w}{p_2}\right) = (30, 30)
w′=p′⋅A=2⋅30+30=90,B=x(p′,w′)=(22.5,45)w' = p' \cdot A = 2 \cdot 30 + 30 = 90, \quad B = x(p', w') = (22.5, 45)
C=x(p′,w)=(15,30)C = x(p', w) = (15, 30)
a=1/2a = 1/2, p1:1→2p_1: 1 \to 2, w=60w = 60 — Slutsky compensation.
Show proof

  1. Cobb-Douglas demand is x1=aw/p1x_1 = aw/p_1. The original optimum is A=(30,30)A = (30, 30).

  2. Slutsky compensated income lets the consumer afford the old bundle at the new prices: w′=2⋅30+1⋅30=90w' = 2 \cdot 30 + 1 \cdot 30 = 90. The compensated optimum is B=(90/4,90/2)=(22.5,45)B = (90/4, 90/2) = (22.5, 45).

  3. The substitution effect is B1−A1=22.5−30=−7.5<0B_1 - A_1 = 22.5 - 30 = -7.5 < 0 — negative, as the compensated law of demand requires.

    The new optimum is C=(15,30)C = (15, 30), so the income effect is C1−B1=15−22.5=−7.5<0C_1 - B_1 = 15 - 22.5 = -7.5 < 0.

    Cobb-Douglas goods are normal, so the income effect points the same way.

    Δx1=−7.5⏟substitution+(−7.5)⏟income=−15\Delta x_1 = \underbrace{-7.5}_{\text{substitution}} + \underbrace{(-7.5)}_{\text{income}} = -15

In the same numerical example, show that the Hicks compensated point BhB^h lies exactly on the original indifference curve, while the Slutsky point BB yields higher utility.

u0=30⋅30=30u_0 = \sqrt{30 \cdot 30} = 30
e(p′,u0)=u0(p1′a)a(p21−a)1−a=30⋅41/2⋅21/2=308e(p', u_0) = u_0 \left(\frac{p_1'}{a}\right)^a \left(\frac{p_2}{1-a}\right)^{1-a} = 30 \cdot 4^{1/2} \cdot 2^{1/2} = 30\sqrt{8}
a=1/2a = 1/2, p1:1→2p_1: 1 \to 2, w=60w = 60 — Hicks compensation.
Show proof

  1. Original utility is u0=30⋅30=30u_0 = \sqrt{30 \cdot 30} = 30. Hicks compensated expenditure is e(p′,u0)=308≈84.85e(p', u_0) = 30\sqrt{8} \approx 84.85, and Hicksian demand is Bh=(e/4,e/2)≈(21.21,42.43)B^h = (e/4, e/2) \approx (21.21, 42.43).

  2. u(Bh)=21.21×42.43=30=u0u(B^h) = \sqrt{21.21 \times 42.43} = 30 = u_0 — on the original indifference curve by construction. But the Slutsky point B=(22.5,45)B = (22.5, 45) gives u(B)=22.5×45≈31.82>30u(B) = \sqrt{22.5 \times 45} \approx 31.82 > 30. Slutsky compensation affords the old bundle, so re-optimizing raises utility — which is why the dashed indifference curve sits outside the solid one.

4. Classical Demand Theory