(1) society is willing to classify them as (iii-a)
(2) like the Swedish/Japanese approach, they may keep on working with some employer subsidy Z; in that case L = L’ + Z
(3) society lowers B to B = S or B = L’, and reconsiders the problem
(4) if regulations are the bottleneck, then changing these regulations redefines ‘given’ productivity L’. Similarly, if Keynesian methods solve unemployment, then only if people’s effective productivity is restored. So the reduced form applies anyhow. (In that case the regulation or lack of a policy measure is a tax in terms of the reduced form, and ‘real productivity’ is higher than L’.)
(5) they get charity, steal or die, and hence there is no welfare state.
Hence BHL-ness implies that these cases can be ‘averaged out of the discussion’ or be left out for expository reasons.
Q.E.D.
Remark: In other words, BHL-ness is sufficient for discussing employment in the welfare state (but not necessarily for other topics, for example, how regulations affect productivity).
The theorem
Theorem BHL.1: For a BHL economy, both full employment and unemployment are possible.