This is an update on this previous post. There are a few papers in the literature that explore the consequences, in sticky-price Keynesian models, of having more price flexibility in some sectors of the economy relative to others. Here are a couple of early papers by Lee Ohanian and coauthors, in the JMCB and the Richmond Fed EQ. Also see a more recent paper by Barsky, House, and Kimball, "Sticky Price Models and Durable Goods." The upshot of this work is that it is important to think in general equilibrium terms. Sometimes prices that are actually flexible can behave like sticky prices, and sometimes there are counter-intuitive effects of monetary policy shocks, or effects which appear inconsistent with evidence. I'll leave it to others to sort out whether this literature has anything to say about current observations.
A recent paper that has some bearing on these issues is this one, by Head, Liu, Menzio,and Wright. Basically, this takes a Burdett/Judd search model of price dispersion, and uses it to think about the implications of monetary intervention for the distribution of prices across sellers. The idea is that, when the equilibrium object is a distribution of prices, then shocks that change this distribution do not require that all sellers change their prices. Indeed, we could observe prices changing very infrequently in a world where all prices are flexible and money is neutral in the short run. The first time I saw this paper, I thought of it as just a clever example, but now I think the idea is quite deep.
While it is useful to have the characterization of the price data that we get from work by Bils/Klenow, etc., as macroeconomists we should not be too focused on the behavior of individual prices. We observe price dispersion because of search and information frictions, and this can help explain observable price behavior, and yet have no implications for the short run effects of monetary policy. Further, most sellers are sellers of multiple goods, and the prices of those goods are interdependent. For example, the Target store prices particular items to bring shoppers into the store, and then uses physical placement of goods in the store and the prices of those goods to extract as much revenue from the buyer as possible. Online shopping is obviously different, as physical location and travel costs are irrelevant but, for example, Amazon uses a set of techniques to bring people to its site, and then uses information and prices to direct you to things they hope you will buy. None of this looks much like incurring a menu cost for each individual price posted, and then meeting whatever demand arises at that price.