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# lect3 - Utility Maximization and Expenditure Minimization...

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Utility Maximization and Expenditure Minimization Intermediate Microeconomics Amy Brown University of California, Los Angeles August 11, 2008 A. Brown (UCLA) Econ 11 Lecture 3 08/11/08 1 / 32

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Outline of Lecture 1 Review of Indirect Utility Function 2 Normal/Inferior Goods 3 Income/Substitution E±ects 4 Break 5 Expenditure Minimization 6 Compensated Demand This should cover most of chapter 5 in Nicholson. A. Brown (UCLA) Econ 11 Lecture 3 08/11/08 2 / 32
Indirect Utility Function The indirect utility function describes a consumer±s utility as it is depends indirectly on the prices of goods and the individual±s income. 1 Solve for the demand functions: x ( p x , p y , M ) , y ( p x , p y , M ) 2 Plug them into the utility function: U ( x , y ) = V ( p x , p y , M ) Example Bud±s indirect utility function is: U ( s , c ) = s ± c = M 2 p s ± M 2 p c = M 2 4 p s p c = V ( p s , p c , M ) A. Brown (UCLA) Econ 11 Lecture 3 08/11/08 3 / 32

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E/ect of Income Changes x , y change as income M changes? Is it always the case that demand increases with income? A. Brown (UCLA) Econ 11 Lecture 3 08±11±08 4 ± 32
Engel Curve The Engel curve traces out the relationship between demand for a good x and a consumer±s income M . A. Brown (UCLA) Econ 11 Lecture 3 08/11/08 5 / 32

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Inferior Goods A. Brown (UCLA) Econ 11 Lecture 3 08/11/08 6 / 32
Normal and Inferior Goods Demand is increasing in income for normal goods . The curve of optimal consumption bundles is increasing in income. (Engel curve is increasing.) Demand is decreasing in income for inferior goods in some price levels. The curve of optimal consumption bundles is locally decreasing or backwards bending. Example Demand for bus tickets. Notice that not all goods can be inferior at all times! A. Brown (UCLA) Econ 11 Lecture 3 08/11/08 7 / 32

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Example Suppose Jennifer has preferences U ( x , y ) = x 2 ( 15 y ) for purchases of x and y . These give her demand functions (check this for an exercise): x ( p x , p y , M ) = 2 M 30 p y 3 p x y ( p x , p y , M ) = M + 30 p x 3 p y Are these goods normal or inferior? x
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lect3 - Utility Maximization and Expenditure Minimization...

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