For answering this question, you are supposed to study lecture notes on measuring gains from trade, as well as the paper entitled "An Empirical Assessment of the Comparative Advantage Gains from Trade: Evidence from Japan" by Bernhofen and Brown. The authors' goal is to measure a nation's welfare gains from free trade compared to autarky. Note. You do not need to read this paper in its entirety. Study the Introduction of the paper (pages 1-3) and only the parts related to the following questions. Suppose the economy consists of N goods. Let pª = [p.p²,...,.p] be the vector of prices in autarky, and p = [p. p....pl be the vector of prices under free trade. Let x² = [x1,x2,...,x] be the vector of production quantities in autarky, and cª = [c1, c²,...,c] be the vector of consumption quantities in autarky. Similarly, define x and c as production and consumption vectors under free trade. The change to welfare between autarky and free trade evaluated at autarky prices, is given by AW = pi (cc) + + p (c − c ) = pª• (cf - cª) (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 AW = p² (c-x) - pª• (x²-x²) b) What does vector c/ -x/ refer to? What does it mean if the nth element, c-x, 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 pª. (x − x/) is non- negative. Explain why their claim is correct. e) Why do the authors measure only an upper bound on AW instead of AW 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 21 trillion dollars, and U.S. population is roughly 330 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?