Chapter 8

Slutsky Equation

Two Decompositions

• Slusky Decomposition: keeping the consumption bundle constant

• Hicksian Decomposition: keeping utility constant

• Total price effect=pure substitution effect + income effect

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.

08.01

08.02

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 (+).

08.03

Case 3: Leontief Function

• If U=min[x,y], the substitution effect is zero.

• Total Price Effect=Income Effect

08.04

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)

08.05

08.06

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).

08.07

08.09

Slutsky Equation: One-line Proof

• Let (x10, x2

0) the original bundle. The compensated demand at (p1, p2) is x1

s=x(p1, p2, x1

0, x20), which is equal to the ordinary demand

at (p1, p2) and income p1 x10+p2x2

0.

• That is, x1s=x(p1, p2, x1

0, x20)= x1(p1, p2, p1

x10+p2x2

0).

• Partial differentiation: dx1s /dp1 =dx1/dp1+x1

dx1/dm, which yields the Slutsky equation.

Slutsky Equation: One-line Proof (Hicksian Substitution)

• Let U0 the original utility level. The compensated demand at (p1, p2) is x1

s=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.