Q.E.D.
Remark: Theorem BHL.2 thus gives an existence proof for this general theorem, i.e. shows that it is not vacuous.
Remark: The value of the theorem is that it focusses our attention on the perceptions that we have to deal with when judging the arguments in this book. Some questions to be answered are: (1) Do we still believe in full employment (only friction unemployment), or do we think that there are serious bottlenecks - or do we even think that we live in a probabilistic universe ? (2) Do we seriously believe that governments have done their best, or at least a reasonable effort, for (a) using available information, (b) finding additional solutions ? (3) Do we really think that the BHL-concept is useless, and that governments have been right to neglect the papers on them ? (4) Do we seriously believe that the PO-changes that seem so likely, are not POC ?
[On the interaction of the reduced form theorems]
Our analysis has not provided complete statistics on existing welfare states, and it can neither replace the need for more study, especially with the cornucopia of applied general equilibrium modelling. The analysis here does however fit in with the stylized facts. It is good strategy to apply logic to circumvent the uncertainty of parameter estimates. There is sufficient reason as well to accept that the two propositions forwarded here give main results in a nutshell.
The first proposition is that both unemployment and full employment are possible for the (BHL) welfare state. The second proposition is that unemployment follows from either conscious choice or wrong co-ordination caused by (deliberate) lack of knowledge, and full employment from choice or chance.
It may be emphasized that the logical force of the argument derives from the undeniables both that one can take subgroup averages and that two points share a line. That line finds its translation, in economic vocabulary, of a social welfare function with a power interpretation.
Above discussion on information is a small step in formalising rather well-known insights. Formalisation, how small the step may be, can be crucial to get the statistics going, and in helping to establish what the state of the world actually is. Apparently we need statistics on what economic advisers and policy makers believe.
Above discussion provides a foundation for a policy conclusion, that it would be good for many welfare states with declared objectives on full employment to improve on informational procedures.