The most important aspect of specialization and trade is that we specialize in just a few tasks but we enjoy the product of millions of tasks. This fact was noticed by Adam Smith, but it has not been "formalized" in any useful way that I can think of.One reason that urban agglomeration has become so important in today's more prosperous regions, is that specialization and trade now extends to organizational patterns which were not considered particularly important on economic terms, in Smith's time. In other words, he would have called many of today's time based services "unproductive" - vital though they may appear, otherwise. While high skill human capital holds considerable value, it has yet to be organized in ways which directly contribute to wealth creation. As a result, one's skills capacity which is not associated with tradable product, is mostly arbitraged in a secondary market of already existing wealth, which is only indirectly associated with primary wealth formation.
Unlike the forms of specialization Smith spoke of which contributed to tradable sector productivity, much of today's specialization in terms of time value has an entirely different purpose: offering sets of knowledge use in the most "concentrated" form possible, via extensive investment in one's early years (normally), to be followed by skills repetition in high population density environments which can reward the investment risk involved. Even though skills repetition leads to greater efficiency, this sacrificial investment approach - which includes long initial years of uncompensated effort - has led to a dearth of important time based services capacity in regions not capable of providing a similar level of compensation for high investment risk. It's an efficiency result which detracts from the coordination efficiency potential of total factor productivity.
Kling's post provided a helpful reference point for models that take organizational divisions of labor into account. One could say that specialization and trade was relatively "pure" in Adam Smith's time, since so many divisions of labor contributed to the actual wealth of tradable product. Indeed, one of the most overlooked aspects of models in general - given services dominance over tradable sectors in developed nations - is that time based services have yet to be organized in ways that contribute to wealth. Alas, budget deficit formation for time and knowledge value is not the same thing. As a result, the fiat monetary structure which supports so much knowledge use and time based endeavor in the present, is not trusted by many political constituents.
Smith - and other early classical economists - recognized themselves as standing "outside" of the actual wealth creation model which they highlighted. Only remember that Adam Smith referred to those who were hired for activity in which time was the only discernible product, as "unproductive". In this framework, one could say that Adam Smith became "productive" not when he was hired to mentor or lecture, but when the separately existing product of his books sold in the marketplace! What would Adam Smith have thought about the changed nature of today's economic realities? One has to wonder, especially when entire markets are built which present problems because they were not conceived as wealth creation structure in the first place.