I get nervous when I, armed only with my undergraduate degree, question the economic analysis of real economists, but there are times I just can't resist. Hopefully I won't get myself into too much trouble here.
Advocates of globalization love to argue that free trade lowers prices, and the argument seems sensible enough. Think of all the cheap goods from China that we can buy at Wal-Mart. But anyone who understands comparative advantage knows that free trade affects relative prices, not the price level (the latter being the province of macro and monetary factors). When a country opens up to trade (or liberalizes its trade), it is the relative price of imports that comes down; by necessity, the relative prices of its exports must go up! Consumers are better off to the extent that their consumption basket is weighted towards importables, but we cannot always rely on this to be the case.
As an example of prices going up, he notes:
... when the U.S. gets better market access abroad for its agricultural exports (a key demand under the Doha round), you can be sure that this will raise domestic prices for these goods, not lower them.
So the argument is that with free trade, prices will come down on goods for which the country is a net importer. However, prices will go up on goods for which the country is a net exporter. My understanding of the reason for the price rise on net exported goods is that there is a reduced supply of those goods in the domestic market. As more goods are exported, the domestic supply falls and the price goes up.
The impact of all this is that while prices might fall for some people, they will rise for others, depending on the basket of goods they consume. Thus, according to Rodrik, there is no general decline in prices due to free trade.
In an update on the post, Rodrik did acknowledge one possible reason prices might come down more generally: scale economies. In addition to that point, though, it seemed to me there are some other reasons to think the price decline might be broader than he indicates.
First, while it is true that an increase in exports might trigger a domestic price rise for those products in the short term, over the longer term won't producers increase investment in the goods at issue in order to meet the additional export demand, thereby increasing supply and bringing prices back down to the old level?
Second, doesn't free trade lead to more competition in all markets, which should bring prices down? This seems to be the standard thinking in antitrust/competition policy, for example, that fewer firms in an industry often leads to higher prices. If we can increase the number of competitors in a market by letting in imports, I would think that could help lower prices in many industries.
And finally, given that tariffs are a tax on goods, and thus a cost that is passed along to consumers, wouldn't the reduction of tariffs lower producers' costs generally and therefore lead to lower prices?