FS07 - Economics 104A Solution for Final Exam Winter 2007 1...

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Economics 104A Solution for Final Exam Winter 2007 1. Jack lives in SB and is considering a move to LA because prices for some goods are lower there. In all other respects, he is indifferent between the two cities. Jack consumes two goods: good 1 and good 2. Good 1 and good 2 are priced in SB twice as high as they are priced in LA. Jack has a utility function u ( x 1 , x 2 ) = ln x 1 + ln x 2 and his weekly income is I , which will not change if he moves. a) (4pt) At least by how much must Jack’s income increase in SB to make him willing to remain there? Answer: The optimal bundle is always an interior bundle regardless of what the prices and income are because of the logarithmic form of the utility function. Thus, given prices p 1 , p 2 and given income I , the optimal bundle is determined by x 2 x 1 = p 1 p 2 (1) and p 1 x 1 + p 2 x 2 = I. (2) Solve (1) and (2), we have x 1 ( p, I ) = I/ 2 p 1 and x 2 ( p, I ) = I/ 2 p 2 (3) as the demand functions for Jack. Let p L 1 and p L 2 denote the prices in LA and p B 1 and p B 2 those in SB. Then, using demand functions in (3), Jack’s utility level in LA is u L = ln I 2 2 1 p L 1 p L 2 . Denote by I S the least income that makes Jack willing to remain in SB. Then, we must have u S = ln I S 2 2 1 p S 1 p S 2 = ln I 2 2 1 p L 1 p L 2 . Since p S 1 = 2 p L 1 and p S 2 = 2 p L 2 , the above equation implies I S = 2 I . So, Jack’s income must be doubled in order to make him willing to stay in SB. 1
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b) (4pt) What would his income in LA have to be to make him indifferent between moving to LA and staying in SB?
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