# What is the point of Slutsky and Hicksian Compensation. And What to they have to to with WARP?

I am currently working through a Advanced Microeconomics textbook by Muñoz-Garcia for a upcoming class. I am very confused about the idea of compensation in which we take away wealth in price decreases and give wealth in price increases. Is there a reason to get back to the original state. The book mentioned something about not violating WARP but that doesn't make sense to me either.

We study substitution effect of price by keeping income effect to be zero. Since a price decrease is equivalent, let's discuss the scenario where there is a price increase from $$p$$ to $$p'$$. Ideally, we should compare $$h(p,u)$$ vs $$h(p',u)$$, keeping the same utility level $$u$$. This allows us to say any differences between these two demand are caused by substitution effect. Since there is a price increase, presumably the agent is worse off, so we need to compensate her with some money to restore her utility to the original level $$u$$, and the amount of money needed is called Hicksian compensation. This approach is especially handy when you have an indifference curve-- you can just find a (compensated) budget line under $$p'$$ that is tangent to the same indifference curve to which the original budget line with $$p$$ is tangent.
However, indifference curve or utility function is an abstract notion. In practice, we don't know how to compensate the agent toward some certain utility level. But at least we know that we can compensate the agent so that she can just afford the original consumption bundle, and by WARP, she is not worse off than the previous case. This is the compromise we must give. That is, we are going to compare $$x(p,w)$$ vs $$x(p',w')$$, where $$w' = p' \cdot x(p,w)$$. This kind of compensation is called Slutsky compensation.