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Does Free Trade Bring Lower Prices?

I'm confused. Dani Rodrik says that free trade does not bring lower prices:

Dani Rodrik's weblog: Does Free Trade Bring Lower Prices?: 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.

Consider your typical Argentinian for example, who consumes a lot of wheat and beef. Since these are export products for Argentina, free trade implies a rise in the relative price of the Argentine consumption basket. (The gains from trade are still there, of course, but they derive from the usual allocative efficiency improvements, not from lower prices across the board.) And in the U.S., the Wal-Mart effect has to be qualified to take into account the fact that the relative price of the goods that the U.S. exports (including for example agricultural commodities) is higher than it would have been absent trade.  Similarly, 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. 

Yes, prices of exportables rise and prices of importables fall. But importables have a higher weight in consumption than in production (that's why we import them). And exportables have a lower weight in consumptio 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.

It seems perfectly fine to me to give the following shorthand description of trade: Trade on average raises the prices of what you make and sell, and lowers the prices of what you buy and consume. A better description might be: Trade raises your income relative to the prices you pay if you are a net buyer of importables, and lowers your income relative to the prices you pay if you are a net buyer of exportables--but as a country, on average, we are net buyers of importables (that's why we import them) and net sellers of exportables (that's why we export them). But the first shorthand description seems to me to be correct, if incomplete.