Slutsky equation

The Slutsky equation (or Slutsky identity) in economics, named after Eugen Slutsky (1880–1948), relates changes in Marshallian (uncompensated) demand to changes in Hicksian (compensated) demand, which is known as such since it compensates to maintain a fixed level of utility. The equation demonstrates that the change in the demand for a good, caused by a price change, is the result of two effects:

The Slutsky equation decomposes the change in demand for good i in response to a change in the price of good j:

where ${\displaystyle h(\mathbf {p} ,u)}$ is the Hicksian demand and ${\displaystyle x(\mathbf {p} ,w)}$ is the Marshallian demand, at the vector of price levels ${\displaystyle \mathbf {p} }$, wealth level (or, alternatively, income level) ${\displaystyle w}$, and fixed utility level ${\displaystyle u}$ given by maximizing utility at the original price and income, formally given by the indirect utility function ${\displaystyle v(\mathbf {p} ,w)}$. The right-hand side of the equation is equal to the change in demand for good i holding utility fixed at u minus the quantity of good j demanded, multiplied by the change in demand for good i when wealth changes.

...
Wikipedia