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slide15-sept26 - ECON 401: Theorem of Khun and Tucker...

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ECON 401: Theorem of Khun and Tucker Siyang Xiong Rice University September 25, 2011 Xiong (Rice University) ECON 401 September 25, 2011 1 / 18
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Another example A consumer with a utility function given by u . x 1 , x 2 / D x 1 x 2 has an income of 100. The unit price of x 1 , x 2 are 4 , 5 respectively. (a) compute the utility maximization bundle, if consumption must be non-negative. (b) Now suppose the consumer were o/ered the following option. By paying a lump sum of a to the government, the conusmer can obtain coupons which will enable him to purchase commodity 1 at a price of 3. (the price of commodity 2 remains unchanged.) For what values of a will the consumer accept this o/er? For what values of a is he indi/erent between accepting the o/er and not accepting it? Xiong (Rice University) ECON 401 September 25, 2011 2 ± 18
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Another example: part (a) max [ x 1 x 2 ] s.t. 4 x 1 C 5 x 2 100 I x 1 ± 0 x 2 ± 0 . Step 1: write down the Lagrange: L D [ x 1 x 2 ] C 1 . 100 ² 4 x 1 ² 5 x 2 / C 2 x 1 C 3 x 2 Xiong (Rice University) ECON 401 September 25, 2011 3 / 18
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Another example: part (a) Step 2: take derivatives @ L @ x 1 D x 2 4 1 C 2 D 0 (1) @ L @ x 2 D x 1 5 1 C 3 D 0 (2) . 100 4 x 1 5 x 2 / ± 0 I 1 ± 0 I 1 . 100 4 x 1 5 x 2 / D 0 . (3) x 1 ± 0 I 2 ± 0 I 2 x 1 D 0 . (4) x 2 ± 0 I 3 ± 0 I 3 x 2 D 0 . (5) Xiong (Rice University) ECON 401 September 25, 2011 4 / 18
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Another example: part (a) Step 3: solve the system derived in Step 2:
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slide15-sept26 - ECON 401: Theorem of Khun and Tucker...

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