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slide10-sept14 - ECON 401: Theorem of Lagrange: examples...

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ECON 401: Theorem of Lagrange: examples Siyang Xiong Rice University September 20, 2011 Xiong (Rice University) ECON 401 September 20, 2011 1 / 16
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examples: consumption Vs leisure agents get utility from consumption goods x and leisure l : u . x , l / ; for example, there are two goods x 1 , x 2 and u . x 1 , x 2 , l / D p x 1 C p x 2 ± l agents are endowed with H units of labor; for example H D 24 (hours) agents can use labor to earn wage, with w being wage rate, i.e., if he worked for k units of labor, he earns a wage of wk ; for example w D 2 p 1 , p 2 are the prices for conumption goods, and agents can use his wage to buy consumption goods, for example p 1 D 1 and p 2 D 3 Xiong (Rice University) ECON 401 September 20, 2011 2 / 16
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examples of optimization continued: consumption Vs leisure the consumption-leisure problem is: max p x 1 C p x 2 ± l s.t. p 1 x 1 C p 2 x 2 D w . H l / i . e ., x 1 C 3 x 2 D 2 . 24 l / Xiong (Rice University) ECON 401 September 20, 2011 3 / 16
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examples of optimization continued: consumption Vs leisure Step 1: write down the Lagrange: L D p x 1 C p x 2 ± l C 1 [ x 1 C 3 x 2 48 C 2 l ] Step 2: take derivatives and set them to zero @ L @ x 1 D 1 2 l p x 1 C 1 D 0 @ L @ x 2 D 1 2 l p x 2 C 3 1 D 0 @ L @ l D p x 1 C p x 2 ± C 2 1 D 0 @ L 1 D x 1 C 3 x 2 48 C 2 l D 0 Xiong (Rice University) ECON 401 September 20, 2011 4 / 16
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slide10-sept14 - ECON 401: Theorem of Lagrange: examples...

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