Differences between Hicksian and Slutskian approaches?

Question

When deriving the substitution effect for both Slutskian and Hicksian definitions, a 'phantom' budget line is drawn.However, for a Slutskian definition, the 'phantom' budget line is drawn parallel to the new budget line(change in price) and **through** the point of tangency for the original budget line and indifference curve.On the other hand, for a Hicksian definition, the phantom budget line is drawn parallel to the new budget line(change in price) and lies on the original indifference curve on a different point of tangency.Is there any significance to this inherent difference between the Slutskian and Hicksian approaches when deriving the substitution effect?I'm familiar with the definition of the Slutskian and Hicksian approaches but am unable to reconcile the definitions of the approaches with the differences in drawing the phantom budget line, and subsequently deriving the substitution effect.Any clarifications or attempts to enlighten me would be much appreciated.Thanks!

Answer 1

use the elasticity form of the Slutsky Equation

e(marshall) = e(hicks) +b(income elasticity)

for example. Let U = XY. MRS= Y/X = px/py or pxX=pyY substitute into Budget constraint and get X= M/2px and y = M/2py bx=by=1/2 e(x) =e(y)=-1 Both goods are unit income elastic so. ex(hicks) =-1 -1/2 or a 1% increase in price leads to a 1% decrease in quantity. Half is the substitution effect and half the income effect.