The link is to a summary of the work in progress "The Cambridge Economic History of the Greco-Roman World." My first thought about when reading the part on economic growth are the words of M.I. Finley quoted in the last blog entry: "in early societies, war was the basic factor in economic growth and consequently in social structure" (Ancient History: Evidence and Models, p. 74).
The answer to the question "How significant was growth?" is quite different if you are willing to envision economic growth as something quite different, whatever a state gained from expansionary warfare during this era, the economic technologies, products, and labor not being what we normally envision. Warfare was also basically a zero-sum game, redistributing wealth extracted out of the food surplus of competing polities, rather than adding value to raw materials, so if all the competing polities are taken together, it is not growth per se, but as the summary above remarks even raising this question "reflects the preoccupations of the twentieth century."
The way this sub-domain of ancient history is characterized is interesting to compare with mainland Southeast Asian history. The discipline must move, "beyond the current framework of polarities that weigh down the broad debates about the nature of the ancient economy. Since the publication of Moses Finley's The Ancient Economy in 1973, these debates have too often been framed in terms of a contest between the 'primitivists' and the 'modernists.' The 'primitivist' position, associated with Karl Polanyi and Finley, has been represented as one of a subsistence agricultural economy with autarkic households, an economy of no growth, no markets, insignificant trade, and non-rational economic actors. In supposedly polar opposition, the 'modernist' view, associated with M. I. Rostovtzeff, is credited with interpreting the ancient economy in capitalist terms of significant growth, vital markets, long-distance trade, and rational actors in pursuit of profits".
"...Very few historians today would subscribe to a fully primitivist or modernist position, even though many more are ready to attribute one or the other to their opponents. In fact, neither Rostovtzeff nor Finley should be characterized as modernist or primitivist. There is more than a little irony in the facts that Rostovtzeff, not Finley, used the words very primitive to describe the living conditions of the peasants, who were an enormous majority of the population of the Roman Empire (1957: 346), and that Finley broke with Polanyi precisely over the latter's denial of markets."
There's a lot of interesting material on the homepage of professor Walter Scheidel at Stanford University.