This paper identifies the free boundary arising in the two-dimensional monotone follower, cheap control problem. It proves that if a region of inaction A is of locally finite perimeter (LFP), then A can be replaced by a new region of inaction A whose boundary is locally C1 (up to sets of lower dimension). It then gives conditions under which the hypothesis (LFP) holds. Furthermore, under these conditions even higher regularity of the free boundary is obtained, namely C2,alpha, except perhaps at a single comer point.