Slutsky Theorem - Mathematical Presentation:
We already know from the first order conditions of utility Maximisation that,
where D_{ij} is the co-factor of the ith row and jth column of the determinant |D|. For income effect we know dp_{1}=dp_{2}=0, therefore we have from equation (h),
Now for own price effect we have dM=dp2=0. So from equation (h) we get,
Lastly, to find out own substitution effect we consider utility is constant in terms of income so, -dM+x_{1}dp_{1}+x_{2}dp_{2}=0 and dp_{2}=0. We have from equation