X(1) < H. A prime example is X(1) = B. Hence (iii-b) is empty.
Unemployment: L is given as the market clearing wage for low productivity persons. If X(0) < XS, then taxes on these persons are increased, and their net income drops below B. Given that K(0) < B, they are eligible for benefits, and apply. Hence (iii-b) is not empty.
It has been shown that both cases are possible. Q.E.D.
Remark: This exposition may seem an overly complex translation of the Cohen Stuart 1889 quote (above) to the welfare state situation. The proof might have said “self-evident” after the first paragraph. Given the record of unnecessary unemployment, this author may however be excused for driving the point home. The usefulness of the BHL concept may be, that officials now can report, “we have diagnosed l people on benefit who should be able to earn L > B on the market, so let’s try to find out how we are stopping them from doing so”.
Remark: A more didactic exposition may start with a structural tax relation, e.g. with R(t) replaced by r in (2-t); see for example the Bentham tax. Then one can show that a ceteris paribus reduction of the tax exemption will increase unemployment. Hence, for the return of full employment it is necessary (but not sufficient) to increase income tax exemption - or something from the ceteris paribus part. Then, the second step in the exposition (as we have done here) is to rename the axis into compounded variables (including VAT, regulations, subsidies, excises, charity, etcetera), and then consider (2-t) as the reduced form. Then we find necessary and sufficient conditions. This however only works satisfactorily for an accepted model of a real economy.
Remark: The theorem doesn’t establish that unemployment has only one cause. Various kinds of unemployment have various causes. But, when various causes are mapped into the world of BHL-ness, then the theorem applies. For example, a long term unemployed academic would be categorised as unskilled labour, even though his employed colleagues earn much more. (The BHL concept thus is drastic. The reasons for applying it have been explained elsewhere.)
Remark: The theorem is strongest in the t = 1
t = 0 part. Given full employment, it is easy to mess it up; and it is easy to see that you can mess it up. The other way around is less obvious. Here, both the requirement L
