This is essentially the typology of Weber's The City, although Weber is very explicit in the fact that he is using ideal types rather than precise descriptions of reality (a nuance that is often ignored). Weber essentially viewed the city as parasitic on the countryside--a reasonable perspective given that cities do not feed themselves. Ancient cities, from his argument, did not return value from the resources it sucked in, while medieval cities did. Early modern Italian and Germanic cities, incidentally, he considered merchant cities.

The work should be in public domain and the introduction is worth flipping through. Just bear in mind that he did not have access to the data we do today, particularly regarding cities outside Europe, so he occasionally imposes barriers where they should not exist.

This being said, the idea of the ancient city as being entirely parasitic has largely fallen out of favor. Certain cities, like Rome, were absolutely parasitic as they used their political power to extract resources. Others, though, had a more symbiotic relation. Ian Morris' paper "The growth of Greek cities in the first millennium BC" does a good job of covering the theoretical ground.

The idea the any "Roman" city behaved the same as all the others is a statement made for dramatic impact, not based in fact. I don't know of any archaeologist or historian who would make a sweeping claim like that. Even between the Eastern and Western Empire, and then in different provinces and regions within those provinces, there is a great deal of variety between different settlements, both urban and rural, and a great lack of any textual or other evidence that would define without a doubt such relationships.

As /u/Tiako mentioned, the cities would not normally produce essentials such as grain, which tended to be supplied by the rural areas nearby, or of course sometimes imported from further afield (e.g. Egypt, North Africa). With the denizens of the countryside mainly (in very general) raising livestock and working the land, they would need to obtain essentials they did not produce themselves from the cities (e.g. pottery, glass, clothing, tools). With that in mind, it's easy to see how this situation could be symbiotic, given that both parties have needs that the other can meet.

In the end, there's no simple answer as to how cities and the countryside interacted—especially when considering the vastness and variety of the Roman-controlled territories—and not one that everyone would necessarily agree on. Perhaps of interest would be Ward-Perkin's "Land, labour and settlement" in The Cambridge Ancient History Volume 14: Late Antiquity: Empire and Successors, AD 425–600, which provides an archaeology-based analysis of life and economy in the Roman/Byzantine countryside and explores questions of self-determination and exploitation.

The dichotomy's perhaps overstated in the heading, but it's indeed surfaced repeatedly in the literature: Bert Hoselitz in the 1950s distinguished "parasitic" and "generative" urban growth, the one fuelled by tribute, tax or agrarian rent, the other obliged to pay its way through production and trade, thereby stimulating exchange and growth. That's probably overdrawn too: as others observe, even inhabitants of the "tribute city" produced for part of their upkeep and bought rural produce unavailable through other means, while the later trading city welcomed agricultural rentiers - indeed its own wealthier inhabitants were often eager to join them.

The medieval city can itself be subdivided into those which enjoyed seigneurial or territorial jurisdiction over their hinterlands, those - usually lesser towns - which themselves owed dues to a lord, and those outside the traditional structure - free or royal cities: but even while a Venice of Florence might lord it over their regions or Paris or London might enjoy part of the royal tax receipts, it would be difficult to portray them as merely living off rural labour except to the extent that their food was grown in the countryside. And of course the greatest beneficiaries of manorial exploitation preferred their own secure dwellings in the countryside.

A complication is that even "parasitism" has been considered "generative" to the extent that it prompts cultivators to grow more to pay off their burden. The downside here is that it also strips away wealth that might be used for improvement. But cities tended to be surrounded by zones of rural wealth given the higher prices they offered for local produce (competing as it did against costlier goods from afar).

The model's perhaps more useful in inviting us to look at the sources of urban growth rather than its impact: some cities were indeed artificial creations supported by expropriation, others had to work their way up and it becomes interesting to consider the geographical factors that contributed to their success or failure. All may be imagined to have fuelled increased local production to some extent, but some contributed more than others.