Differences between Hicksian and Slutskian approaches 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!
The income effect (IE) is about assessing purchasing-power impacts of a price change, while the substitution effect (SE) is about the impact of that price change on the relative attractiveness of the different goods. In reality these effects are not observable - when a price changes, your consumption choices will change for both reasons. But we can conceptually decompose the overall effect into IE/SE. In the SE we will be looking to isolate the impact of relative prices changing, without the purchasing-power effect.

Now, there are two ways of thinking about this:

-Slutsky: what if price changes but my purchasing power were (literally) to remain constant (i.e. I could still buy the exact same bundle as before)? How would the relative price change by itself affect my decision?

-Hicks: what if the price changes but my purchasing power were adjusted so I am still able to achieve my initial level of utility (i.e. I would be as well-off as before)? How would the relative price change by itself affect my decision?

Either way, the SE is always negative, that is, a higher price for one good will tend to make consumption of that good lower.

After you understand the SE, the IE is just the income adjustment needed in either case to get us to the actual final decision. It can reinforce the SE or contradict it, depending on whether the good is normal or inferior.

Hope this helps.