Before I turn to your enumerated remaining points of disagreement, let me first restate my general take on all of this:
==> Say himself was exploding the common “man on the street” explanation for recessions. Someone who has not first thought through the matter as a classical economist would (in which you typically abstract away from money), would benefit greatly from this. To think about the fact that, say, a doubling of all production would mean we were richer, and not that we were impoverished, is obviously a good first step. It shows how silly the typical businessman’s complaint about a “scarcity of money” is.
==> JS Mill and other free-market economists since have appreciated the great wisdom of what Say wrote, and still think it’s critical for the layperson to read it as Say intended. But then they get really anal with it and say, “OK, if you are talking about a barter economy, then it is mathematically true. If you’re talking about a monetary economy, then in principle we could have a glut of all commodities and a shortage of money, but that would be due to prices not adjusting for some reason. Say was aware of all of this, and I still don’t think you need government deficit spending to boost AD, because why won’t prices just adjust?”
==> However, someone who wanted to disparage Say’s Law could say “Duh it just assumes away the problem. Of course we’re dealing with a monetary economy and prices/wages are sticky for various reasons.” I think this is a very unfair response, which (a) gives too much credit to the original “man on the street” diagnosis of recessions and (b) is bad policy advice, because the Phillips Curve will simply shift once you pump in more money.
Now on to your three points. I’ll handle them out of order.
“3) Whether the best defense of the Law of Markets is the mathematically-demonstrable equality between (ex post) aggregate supply and demand in a barter economy, with necessary modifications to allow for money”
It depends what you mean by “best defense.” Someone who has been trained in economics can think in terms of various markets for goods, and so yes I would have to couch Say’s Law in that framework to show how it is still true (with flexible prices).
But if you mean, “When I first explain Say’s Law to a regular person,” then I give the historical context and quote Say himself. I.e. just what I did in the lecture.
“1) Whether Mill interpreted Say’s analysis as only being necessarily true in a barter economy”
Yes, of course he did. He said so explicitly in “Some Unsettled Questions…”:
============[QUOTE FROM JS MILL BEGINS]===========
<i>There can never, it is said, be a want of buyers for all commodities; because whoever offers a commodity for sale, desires to obtain a commodity in exchange for it, and is therefore a buyer by the mere fact of his being a seller. The sellers and the buyers, for all commodities taken together, must, by the metaphysical necessity of the case, be an exact equipoise to each other; and if there be more sellers than buyers of one thing, there must be more buyers than sellers for another.
This argument is evidently founded on the supposition of a state of barter; and, on that supposition, it is perfectly incontestable. When two persons perform an act of barter, each of them is at once a seller and a buyer. He cannot sell without buying. Unless he chooses to buy some other person’s commodity, he does not sell his own.
If, however, we suppose that money is used, these propositions cease to be exactly true. It must be admitted that no person desires money for its own sake, (unless some very rare cases of misers be an exception,) and that he who sells his commodity, receiving money in exchange, does so with the intention of buying with that same money some other commodity. Interchange by means of money is therefore, as has been often observed, ultimately nothing but barter. But there is this difference—that in the case of barter, the selling and the buying are simultaneously confounded in one operation; you sell what you have, and buy what you want, by one indivisible act, and you cannot do the one without doing the other. Now the effect of the employment of money, and even the utility of it, is, that it enables this one act of interchange to be divided into two separate acts or operations; one of which may be performed now, and the other a year hence, or whenever it shall be most convenient. Although he who sells, really sells only to buy, he needs not buy at the same moment when he sells; and he does not therefore necessarily add to the immediate demand for one commodity when he adds to the supply of another. The buying and selling being now separated, it may very well occur, that there may be, at some given time, a very general inclination to sell with as little delay as possible, accompanied with an equally general inclination to defer all purchases as long as possible. This is always actually the case, in those periods which are described as periods of general excess. And no one, after sufficient explanation, will contest the possibility of general excess, in this sense of the word. The state of things which we have just described, and which is of no uncommon occurrence, amounts to it.</i>
===============[QUOTE FROM JS MILL ENDS]==============
So I don’t know how you can deny that JS Mill did in fact make that concession (if that’s what you want to call it), which is why even sympathetic people like Thomas Sowell said he did.
Now to be sure, in full context, JS Mill isn’t throwing Say under the bus. Rather, he’s trying to show that the fans and critics of Say’s Law ultimately agree on the facts of the matter of a recession, and he’s trying to show why they are talking past each other.
“2) Whether such an interpretation is in any way justified”
This is ambiguous. If you mean, “Was Say thinking in terms of a Walrasian model and assumed that the money commodity was just like any other?” then no, of course not.
But if you mean, “Are we allowed to think about individual markets clearing, to test whether it is literally impossible in a monetary economy to have a general glut in everything but money coupled with a shortage of money?” well why can’t we? That is consistent with modern Misesian price theory.
I think it helps isolate exactly what the difference is between Say and his critics.