Lecture 5 - Lecture 5 Utility Maximization II But seek ye...

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Lecture 5 Utility Maximization II But seek ye first the kingdom of God, and his righteousness; and all these things shall be added unto you. Matthew 6:33 Wherefore, seek not the things of this world but seek ye first to build up the kingdom of God, and to establish his righteousness, and all these things shall be added unto you. Matthew 6: 38 JST The Economics Outline 1. Hicksian Demand 2. Indirect Utility Function 3. Expenditure Function 4. Shephard’s Lemma 5. Roy’s Identity The Mathematics Outline 1. Homogeneity 2. Duality 3. Envelope Theorem Homogeneity A function is homogeneous of degree k if: , for t > 0 ) ,... ( ) ,..., ( 1 1 n k n x x f t tx tx f = Demand functions are homogeneous of degree 0. For example, if you double all prices and income, demand is unchanged. Mathematically, we write this result as follows: , where t can be any positive number. ) , ,..., ( ) , ,..., ( 1 * 1 * I P P x tI tP tP x n i n i = Demonstrate this is true for the Marshallian demand functions from a general Cobb- Douglas utility function. Duality The problem: Maximize: ) , x U = β α 2 1 2 1 ( x x x Subject To: I x P x P x x = + 2 1 2 1 has a dual in the following problem: Minimize: 1 P x P 2 1 2 x x x + Subject To: 2 1 x x U = Dr. Steven Waters Econ 380 Page 1 of 7

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L5: Utility Maximization II In both problems we are choosing a set of x ’s that will optimize the objective function. The answers to the maximization problem are Marshallian Demand functions (uncompensated demand functions) that are functions of price and income. The answers to the minimization problem are Hicksian Demand functions (compensated demand functions) that are functions of price and utility. Marshallian Demand: These are “regular” demand functions. When you drew demand functions in your principles class, you were drawing (inverted) Marshallian demand. These are functions of price and income and are sometimes called uncompensated . They are “uncompensated” in the sense that income is fixed and utility changes along the curve – i.e., there is no compensation (in the form of income) when prices change to keep you at the same level of utility. The utility change along a Marshallian demand curve can be thought of intuitively by thinking about a price decrease that allows you to attain a higher level of utility
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This note was uploaded on 11/30/2011 for the course STAT 380 taught by Professor Stevens during the Spring '11 term at Brigham Young University, Hawaii.

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Lecture 5 - Lecture 5 Utility Maximization II But seek ye...

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