  We give examples of pseudoconvex domains of finite type in $\mathbb{C}^2$
where the Kohn algorithm for subelliptic estimates fails to yield an effective
lower bound for the order of subellipticity in terms of the type. We show how
to modify the algorithm to obtain an effective procedure to prove
subellipticity on domains of finite type in $\mathbb{C}^2$ with real analytic
boundary satisfying a condition slightly stronger than pseudoconvexity. We
close with a generalization to higher dimensions.
