Suppose the economy consists of goods. Let be the vector of prices in autarky, and be the vector of prices under free trade. Let be the vector of production quantities in autarky, and be the vector of consumption quantities in autarky. Similarly, define and as production and consumption vectors under free trade.

The change to welfare between autarky and free trade evaluated at autarky prices, is given by

(1)

where • is the inner product. (Note: The authors of the above paper drop • as the inner product with the understanding that they always use inner product.)

a) Show that

(2)

b) What does vector refer to? What does it mean if the th element, , is positive?

c) Why did the authors use the more complicated equation (2) instead of the simpler equation (1)?

d) Our equation (2) here is the same as equation (4) in the paper "An Empirical Assessment ..." in page 212. In the same page, the authors claim is non-negative. Explain why their claim is correct.

e) Why do the authors measure only an *upper bound* on instead of itself? What upper bound do they use?

f) The authors report 9% as the change to income from autarky to trade relative to GDP for Japan in the nineteenth century. U.S. GDP is roughly 18 trillion dollars, and U.S. population is roughly 320 million people. Suppose the change to U.S. GDP from a hypothetical move to autarky is 9% of its current GDP. How many dollars would on average an American lose from a move to autarky?