chapter 8 slutsky equation. two decompositions slusky decomposition: keeping the consumption bundle...
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![Page 1: Chapter 8 Slutsky Equation. Two Decompositions Slusky Decomposition: keeping the consumption bundle constant Hicksian Decomposition: keeping utility constant](https://reader035.vdocuments.net/reader035/viewer/2022081421/56649e735503460f94b72938/html5/thumbnails/1.jpg)
Chapter 8
Slutsky Equation
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Two Decompositions
• Slusky Decomposition: keeping the consumption bundle constant
• Hicksian Decomposition: keeping utility constant
• Total price effect=pure substitution effect + income effect
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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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08.01
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08.02
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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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08.03
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Case 3: Leontief Function
• If U=min[x,y], the substitution effect is zero.
• Total Price Effect=Income Effect
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08.04
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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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08.05
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08.06
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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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08.07
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08.09
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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.
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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.