W12 MIC 05 Indifference Curves

# W12 MIC 05 Indifference Curves - INDIFFERENCE CURVES(B&B...

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Northwestern University ECON 310-1: Microeconomic Theory Profs Jim Hornsten & Ron Braeutigam Winter 2012 INDIFFERENCE CURVES (B&B Chapter 3) Notes for Lecture #03b Modeling Consumer Choice Recap of Utility Cobb-Douglas Utility Quasi-Linear Utility Perfect Substitutes Perfect Complements A Non-Valued Product A Bad The Lesser of Two Evils A Bliss Point Graphing Indifference Curves

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Northwestern University ECON 310-1: Microeconomic Theory Profs Jim Hornsten & Ron Braeutigam Winter 2012 Modeling Consumer Choice IDEA: A person chooses the best combination of products s/he can afford SYN: combination, bundle, basket How do we determine what is best ? How do we determine what is affordable ? A special ECON problem! Actor = Consumer Objective = max Utility Choice = bundle of goods Constraints = budget Useful insights for those shopping for groceries, courses, extracurricular activities, significant others, houses The foundation of demand curves
Northwestern University ECON 310-1: Microeconomic Theory Profs Jim Hornsten & Ron Braeutigam Winter 2012 Recap: Utility Functions To compare various baskets of Products X and Y , we run these quantities (or graphical coordinates) through a Utility Function, U[X,Y] , which generates a number (or index) expressed in utils , allowing us to make a cardinal ranking (order + magnitude). A 3-D plot of Total Utility function U[X,Y] reminds us that utility changes when we move in the X -dimension or the Y -dimension. There are many (X,Y) combinations that yield the same altitude (utility), and then all rest on one iso-utility curve or indifference curve . The Marginal Utility of U[X,Y] with respect to X is the partial derivative with respect to X , or the slope of U[X,Y] in the X -dimension holding Y constant: a 2-D slice of the 3-D hill. The Marginal Rate of Substitution of X for Y is the ratio of the marginal utilities (with the MU of the 1st variable in the numerator: MU X ); it is also the negative of the slope of the indifference curve (with the 1st variable on the X -axis). B&B 3/e, Figure 3-4, p.77 MRS x , y = MU x MU y = " dy dx = " slope of the indifference curve U [ x , y ] " MU x = # U [ x , y ] # x and MU y = # U [ x , y ] # y

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Northwestern University ECON 310-1: Microeconomic Theory Profs Jim Hornsten & Ron Braeutigam Winter 2012 E.g., A Utility Function with Two Goods: Finding the MU’s Let U [ x , y ] = xy = x y = x 1 2 y 1 2 . (Many prefer to use powers rather than radical signs.) Using the Power Rule, dAx b dx = bAx b " 1 , find MU x = # U # x TIP : When thinking in the x dimension, we hold all y terms constant. To compute the partial derivative " U " x using the function U [ x , y ] = x 1 2 y 1 2 , we treat the y 1 2 part as a constant - call it A - so it's as if the function were U [ x ] = x 1 2 A . Then " U " x = d dx x 1 2 A # \$ % % & ( ( = A d dx x 1 2 # \$ % % & ( ( = A 1 2 ) * + , - .
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