Section-4-Social-Welfare-Functions-for-Utilities-slides

Section-4-Social-Welfare-Functions-for-Utilities-slides -...

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Unformatted text preview: Social Welfare Functions that Aggregate Utilities Todd Sarver Northwestern University Econ 310-2 Fall 2011 Todd Sarver (Northwestern University) Social Welfare Functions for Utilities Econ 310-2 Fall 2011 1 / 26 1 Definition 2 Examples 3 Connection with Pareto Efficiency 4 Relating Arrows Theorem and Social Welfare Functions for Utilities Todd Sarver (Northwestern University) Social Welfare Functions for Utilities Econ 310-2 Fall 2011 2 / 26 General Setting Instead of aggregating preferences, we now consider aggregating utility functions in order to make a social decision. Then, later in the lecture, we will discuss how this approach relates to Arrows theorem. The setting we use is the same as before, except we now use utility functions to represent preferences. Environment Individuals: 1 , 2 , 3 ,...,n Alternatives: Denoted by capital letters such as A,B,C,... Utility Functions: Each individual i has a utility function u i for the alternatives. Utility Profile: The vector of utilities that the individuals obtain from a particular alternative, for example, ( u 1 ( A ) ,u 2 ( A ) ,...,u n ( A )) . Todd Sarver (Northwestern University) Social Welfare Functions for Utilities Econ 310-2 Fall 2011 3 / 26 Social Welfare Functions for Utilities Definition We refer to the mechanism for aggregating the individuals utilities as a social welfare function for utilities (SWFU) . The SWFU is a function W that takes the utilities of all individuals as input and gives a single social utility (or social welfare ) as output. Profile of Utilities ( u 1 ( A ) ,...,u n ( A ))- Social Welfare Function for Utilities- Social Welfare W ( u 1 ( A ) ,...,u n ( A )) Todd Sarver (Northwestern University) Social Welfare Functions for Utilities Econ 310-2 Fall 2011 4 / 26 Social Welfare Functions for Utilities Definition We refer to the mechanism for aggregating the individuals utilities as a social welfare function for utilities (SWFU) . The SWFU is a function W that takes the utilities of all individuals as input and gives a single social utility (or social welfare ) as output. Profile of Utilities ( u 1 ( A ) ,...,u n ( A ))- Social Welfare Function for Utilities- Social Welfare W ( u 1 ( A ) ,...,u n ( A )) A SWFU gives a social ranking in the obvious way: For any two alternatives A and B , A * B W ( u 1 ( A ) ,...,u n ( A )) > W ( u 1 ( B ) ,...,u n ( B )) Todd Sarver (Northwestern University) Social Welfare Functions for Utilities Econ 310-2 Fall 2011 4 / 26 Illustration If we know the equation for the UPF, then maximizing social welfare is similar to a standard utility-maximization problem for a single individual. Todd Sarver (Northwestern University) Social Welfare Functions for Utilities Econ 310-2 Fall 2011 5 / 26 Illustration If we know the equation for the UPF, then maximizing social welfare is similar to a standard utility-maximization problem for a single individual....
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This note was uploaded on 01/03/2012 for the course ECON 310-2 taught by Professor Sarver during the Spring '08 term at Northwestern.

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Section-4-Social-Welfare-Functions-for-Utilities-slides -...

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