5. Research Question
In the Hopenhayn model, firm size distribution (or equivalently productivity) is stochastically increasing in the age of the cohort. And new entrants are set to start with a relatively lower productivity hence a smaller size compared with older firms. However, in \citet{Foster_2008}, they find that earlier literature (including the Hopenhayn's model) understates new producers' productivity advantages since previous work usually did no distinguish between the revenue productivity and physical productivity. They show that physical productivity is inversely correlated with price while revenue productivity is positively correlated with price. And furthermore, young producers charge lower prices than incumbents.
To capture this feature, a model of demand differentiation could be added into the Hopenhayn's model. Specifically, in addition to the firm-specific productivity, new entrants start with a demand of a smaller consumer base (e.g. local markets). Once they become incumbents, they can try to move to new markets (e.g. state market) and expand their consumer bases by invoking marketing costs and draw a new productivity from the same distribution as the initial one where they drew to become an entrant.
Equivalently, it can be thought as a combination of markets of different sizes with different entering costs that incumbents could sequentially choose in the Hopenhayn's model. Under this setting, higher entering cost markets imply higher market prices and higher revenues on average (if the result of Q4 applies), but does not imply a growing physical productivity over time or across the markets, which is what \citet{Foster_2008} describes.