Yes, prices of exportables rise and prices of importables fall [as a result of free trade]. But importables have a higher weight in consumption than in production (that's why we import them). And exportables have a lower weight in consumption than in production (that's why we export them). So if the average price of production stays the same--and incomes stay the same, because everything paid to buy our production winds up as somebody's income--then the average price of consumption falls.
I think we do agree on the economics, which is not rocket science after all. Under the usual conditions, trade does increase aggregate real incomes, and whether that is because the numerator (nominal income) rises or the denominator (price level) falls is immaterial--and in fact something that micro-economic theory cannot shed light on.
But this discussion has never been about the aggregate gains from trade, and that is why Brad is confused I think. I started on this particular thread to make a point about distributional effects. The argument on prices is often presented as if it is an add-on to the income-side effects of trade, and one that can override otherwise important distributional consequences. "Yes, these workers might lose their jobs, but look how much the rest of us benefit on account of lower prices." See for example the post by Dan Drezner that started me off. My point was that the "but..." part of the argument is irrelevant or need not hold. In particular:
(a) In our workhorse model of trade (the HOS model), certain groups will lose out from trade, regardless of price movements.
(b) And in other models (such as those with some sector-specific factors), whether any particular group of workers gains from the price effects of trade depends on how their consumption basket relates to traded goods. If you are a worker who consumes a lot of the export good (my Argentine example), you will be worse off, regardless of the aggregate relationships discussed by Brad (i.e., even though exportables may have a smaller share in consumption than in production in aggregate).
I should add that because of these distributional effects, the median (i.e., typical) consumer can also end up worse off, to the extent that her income sources and budget shares differ from those of the average. So before you take too much comfort about all the cheap prices at Wal-Mart, these are some of the things you have to worry about.
So my earlier summary of these issues is still pretty good, I think.