1. Chapter 8 Slutsky Equation

2. Two Decompositions
Slusky Decomposition: keeping the consumption bundle constant
Hicksian Decomposition: keeping utility constant
Total price effect=pure substitution effect + income effect

3. Case 1:
Both substitution and income effects move in the same direction.
Lower price induces consumers to substitute x for y. Income effect encourages them to buy more, thus reinforcing the effect.

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6. Case 2: Two effects offset each other
Panel A: The income effect (-) is stronger than the substitution effect (+). [note: sign is labeled in terms of the change in x]
Panel B: The income effect (-) is weaker than substitution effect (+).

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8. Case 3: Leontief Function
If U=min[x,y], the substitution effect is zero.
Total Price Effect=Income Effect

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10. Case 4: Linear and Quasi-linear Function
Linear: Perfect substitution between x and y
case 1:Total Effect=Substitution Effect (switching from one good to another corner)(Fig. 8.5).
Case 2: Total Effect=income effect (consuming the same good), which can be zero or non-zero. (not drawn)
Quasi-linear: imperfect substitution between x and y, but its income effect is zero. (Fig.8.6)

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13. Tax and Rebate Original bundle (x, y), yielding U(x, y).
Tax reduces U (to a level lower than U(x’,y’) (not shown))
Rebating a tax will not bring U back to its original level, i.e., U(x’, y’)< U(x, y).

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16. Slutsky Equation: One-line Proof
Let (x10, x20) the original bundle. The compensated demand at (p1, p2) is x1s=x(p1, p2, x10, x20), which is equal to the ordinary demand at (p1, p2) and income p1 x10+p2x20.
That is, x1s=x(p1, p2, x10, x20)= x1(p1, p2, p1 x10+p2x20).
Partial differentiation: dx1s /dp1 =dx1/dp1+x1 dx1/dm, which yields the Slutsky equation.

17. Slutsky Equation: One-line Proof (Hicksian Substitution)
Let U0 the original utility level. The compensated demand at (p1, p2) is x1s=x(p1, p2, U0), which is equal to the ordinary demand at (p1, p2) and income m (=p1 x1+p2x2).
That is, x1s=x(p1, p2, U0)= x1(p1, p2, m)
Partial differentiation: dx1s /dp1 =dx1/dp1+x1 dx1/dm, which yields the Slutsky equation.