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Unformatted text preview: x 2 and less x 1 . Technical aside. Many students tried to use the ﬁrst order conditions MRS = p 1 p 2 to ﬁnd the optimal bundle. What they failed to realize was that the preferences were not convex. To ﬁnd the original optimal bundle, we integrate over the marginal utilities of both goods, separately. U ( x 1 ,x 2 ) = Z MU 1 dx 1 = Z 2 x 1 dx 1 = x 2 1 + c 1 U ( x 1 ,x 2 ) = Z MU 2 dx 2 = Z 3 x 2 dx 2 = 3 2 x 2 2 + c 2 So, we conclude that: U ( x 1 ,x 2 ) = x 2 1 + 3 2 x 2 2 + c Where c , c 1 , c 2 are arbitrary constants. The indiﬀerence curves of this utility function are elliptical, centered around the origin. It is clear then that preferences are not convex, so the ﬁrst order conditions MRS = p 1 p 2 cannot be used. (Nonconvex preferences imply that points of tangencies are local minima, not maxima.) 2...
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 Fall '08
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 Economics, Microeconomics, Utility

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