Difference between Hicks and Slutsky Decomposition

The Slutsky decomposition and the Hicks decomposition are both useful tools for analyzing the impact of price changes on consumer behavior and welfare, but they approach the problem from slightly different perspectives. The choice between the two depends on the specific research question being addressed and the data and assumptions available.

The Hicks decomposition separates the impact of a price change into an income effect and a substitution effect. The income effect reflects how changes in the consumer's purchasing power impact their demand for the good, while the substitution effect reflects how changes in the relative price of the good affect the consumer's willingness to substitute between that good and other goods.

The Slutsky decomposition, on the other hand, takes a more general approach and separates the impact of a price change into a substitution effect and a change in the consumer's budget constraint. The substitution effect is similar to the Hicks substitution effect, reflecting the impact of changes in the relative price of the good on the consumer's willingness to substitute. The change in the budget constraint reflects how changes in the price of the good affect the consumer's overall purchasing power, which may in turn impact their demand for all goods, not just the one in question.

Slutsky decomposition gives advantage to consumers as they can move to higher indifference curve if a consumer substitute one good for another, which is not the case of Hick decomposition. Likewise, Hicks decomposition prevails higher compensation variation, such as tax or subsidy, which yields  benefits for government if purchasing power of a consumer rises and yields benefits to consumer if purchasing power of a consumer decreases. 

Overall, both the Hicks and the Slutsky decompositions are useful tools for analyzing the impact of price changes on consumer behavior and welfare, and the choice between the two will depend on the specific research question and the data and assumptions available. In general, the Hicks decomposition is considered a more straightforward and intuitive approach, while the Slutsky decomposition provides a more general and comprehensive framework for analyzing the impact of price changes. Hence, they both are better one or the other way.