Algorithms for subelliptic multipliers in $\mathbb{C}^2$

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.