The "Neoclassical production function" post-dates Leontief by at least 20 years, so the wikipedia article is an attempt looks like an attempt at deducing the Leontief model from general neoclassical principles, with partial derivatives and the whole works. It looks just like the kind of dreadful stuff Paul Samuelson would do. I'm looking at it from a different perspective. The article does say

The inimitable book by Leontief himself remains the best exposition of input-output analysis. See bibliography.

Maybe I should try to get ahold of that. Drew should have easier access to it through his university library.

The wikipedia article also says

Despite the clear ability of the input-output model to depict and analyze the dependence of one industry or sector on another, Leontief and others never managed to introduce the full spectrum of dependency relations in a market economy.

in other words, the program is too ambitious to be practical. For instance

Irving Hock at the Chicago Area Transportation Study did detailed forecasting by industry sectors using input-output techniques. At the time, Hock's work was quite an undertaking, the only other work that has been done at the urban level was for Stockholm and it was not widely known. Input-output was one of the few techniques developed at the CATS not adopted in later studies. Later studies used economic base analysis techniques.

This economic base analysis sounds like a simplification of the Leontief model for tractability, though it appears to have been developed independently. That Leontief was too ambitious to be practical did not prevent people from coming up with even more general (and presumably more intractable) models

In 2003, Mohammad Gani, a pupil of Leontief, introduced Consistency Analysis in his book 'Foundations of Economic Science', which formally looks exactly like the input-output table, but explores the dependency relations in terms of payments and intermediation relations.

...

In a technical sense, input-output analysis can be seen as a special case of consistency analysis without money and without entrepreneurship and transaction cost.

Anyway, it does seem like economics is indeed much harder than chemistry or ecology as I discussed with TBG on a parallel thread.

Those whom the Gods wish to destroy They first make mad. -- Euripides

OK, I'm de-puzzled. I was reading the relationship backward, and the reverse direction makes perfect sense. It's saying that the amount of each kind of stuff required is the sum of the amounts required for each of the consumers of that stuff. (The flow of stuff is in the opposite direction from the arrows you showed, which corresponded instead to the direction of demand.)

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