Proof:
The structure of this proof is, that we determine the accounting equations, find the reduced form tax relations that are implicit in these, and then deduce the critical tax parameters that determine the regime switch.
Looking at the BHL concept, the only possibility for variation is in category (iii-b). The recipients in that class all move together, and thus there are only two regimes (in or out of benefit dependency). Given that gross productivity has been fixed, the only possible variation concerns net income. We assign the term “tax regime” to the possible states in net income. We find, in other words, that these regimes are implicit in the BHL concept. Let t be the index for tax regime 0 (unemployment) or 1 (full employment).
Given BHL-ness, we thus have: t is 0 or 1, and:
b permanent benefit recipients;
h persons with gross productivity H and net N(t);
l persons with gross productivity L << H, and net K(t).
The regimes are characterized by net income conditions K(0) < B and K(1)
B: