Slutsky Equation - Lecture notes 10 PDF

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When is compensated demand (Hicksian demand) and regular uncompensated demand (Marshallian demand) equal to each other?

x c ( Px , Py , U )=x (Px , Py , E(Px , Py ,U )) In other words, when consumer has just enough income to achieve U. Goal: To decompose the effect of a price change on Marshallian demand into substitution and income effects Take partial derivative with respect to Px of both sides ∂ xc ∂ x ∂ x ∂ E + = ∂ Px ∂ Px ∂ E ∂ Px ∂ x ∂ xc ∂ x ∂ E = − ∂ Px ∂ Px ∂ E ∂ Px The term on the left hand side of the above equation is the effect of a price change on Marshallian demand. ∂ xc is t he substituion effect ∂ Px −∂ x ∂ E −∂ x ∂ E = ∂ E ∂ Px ∂ I ∂ Px ∂E =x ∂ Px The last equation is due to Envelope theorem or Shephard’s lemma: Intuitively, a 1 dollar increase in Px raises the necessary expenditures by x dollars, because 1 extra dollar must now be paid for each unit of x purchased. Slutsky decomposition of a price change the becomes:

c ∂x x ∂x =∂x − ∂ Px ∂ Px ∂ I

∂ xc is negative when MRS of x for y decreases with x (as we move ∂ Px down the indifference curve towards more x) −∂ x −∂ x ∂x >0 then x if x is a normal good x will be negative ! Income effect: ∂I ∂I ∂I Substitution effect:

So for a normal good both terms are negative. ∂x ∂ xc . Graph of Marshallian demand will be is going ¿ be higher−¿ absolute value than ∂ Px ∂ Px flatter than graph of Hicksian demand (Price on vertical axis) Bottom line: Marshallian demand is more responsive to changes in price than Hicksian demand for a normal good.

Example: U ( x , y )=x 0.5 y 0.5 We found Marshallian demand functions as: x ( Px , Py , I )= y (Px , Py , I ) =

0.5 I Px

0.5 I Py

a. Find the Hicksian demand b. Decompose the effect of a change in price on Marshallian demand into substitution effect and the income effect. a. Plug in the Marshallian demand function in to the utility function to find V the indirect utility. 0.5 I Py ¿ ¿ 0.5 0.5 I V =U ( x , y) =( ) ¿ Px In indirect utility function, olve for I I =2V Px 0.5 Py 0.5 Then solve for x to find Hicksian demand function (compensated demand function) 0.5 V Py c x= 0.5 Px ∂ x c −0.5V Py 0.5 = b. Substitution effect: ∂ Px Px 1.5

If we plug for V V =

0.5 I Px0.5 Py 0.5

we can write the substitution effect in terms of

income and prices. Income effect:

−0.5 0.5 I −0.25 I −∂ x x= = 2 ∂I Px Px Px

Let’s check that the two add up to the effect of a price change on Marsahllian demand: 0.5 I x ( Px , Py , I ) = Px The total effect of price change on Marshallian demand

∂ x −0.5 I = ∂ Px Px 2 Income effect: −0.5 0.5 I −0.25 I −∂ x = x= Px Px ∂I Px 2

Substitution effect: ∂ x c −0.5V Py 0.5 = ∂ Px Px 1.5 ¿ compensated demand function Plug for V :V =

c

∂x = ∂ Px

−0.5

0.5 I 0.5 0.5 Px Py

0.5 I 0.5 Py 0.5 −0.25 I Px Py = 1.5 Px 2 Px 0.5

0.25 I 0.25 I ∂ x −0.5 I =¿− − = 2 2 2 ∂ Px Px Px Px In this example substitution effect and income effect are exactly equal and work in the same direction. If price increases both substitution and income effect decrease (Marshallian) quantity demanded. If price decreases both substitution and income effect increase (Marshallian) quantity demanded....