I am studying about learning methods in statistics and the regression section explains that the main difference between Lasso and Ridge regressions is the formulation of the regularization term, which for Lasso is the ℓ1 norm, and for Ridge the ℓ2 norm.

Given that Lasso regression shrinks some of the coefficients to zero and Ridge regression helps us to reduce multicollinearity, I could not gain a grasp of the effects of these regularization methods on variance and bias.

I am looking for a possible mathematical or intuitive explanation of how variable elimination affects model variance and bias.