as I understand Marx, the labor that the King commands produces no
value unless it produces commodities for sale. Public-school teachers
don't produce value. (That's not a bad thing, by the way: among other
things, teachers are "indirectly productive," in Jim O'Connor's
phrase, in a lot of ways. That is, their work helps others produce
value and surplus-value.)  The value of their wages would be the value
of their labor-power, but that's different from the value they
produce.

In the US National Income and Product Accounts, the problem of
public-school teachers not producing commodities is "dealt with" by
saying that their product = their wages. That's a bit doubtful.

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Well that is the problem you arrive at when you apply the categories of a 
single mode of production to a social formation with a combination of modes of 
production. It is clear that the teachers produce no exchange value, so their 
labour does not assume the capitalist value form. But their labour is still 
socially necessary.

The problem I think comes from approaching it from the standpoint of the profit 
making capitalist. To him it appears that labour 'produces' value. But this is 
a superficial view, since for the capitalist value is only apparent as exchange 
value.
If we follow your basic argument earlier, which I accept, that value is 
measured in hours of socially necessary labour time, then there is no doubt 
that the labour of teachers constitutes part of the socially necessary labour 
time and thus in social accounting terms is value, even if it is not manifest 
in the characteristic capitalist form of the  exchange value of an output 
commodity.

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