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Section-2-Utility-Possibilities-Frontier-slides

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The Utility Possibilities Frontier Todd Sarver Northwestern University Econ 310-2 – Fall 2011 Todd Sarver (Northwestern University) The UPF Econ 310-2 – Fall 2011 1 / 28

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Refresher: Notation Todd Sarver (Northwestern University) The UPF Econ 310-2 – Fall 2011 2 / 28
Notation The symbol (called Sigma ) denotes a summation. Example: If there are 35 units of good x to be divide between 6 individuals, the feasibility constraint can be written as either x 1 + x 2 + x 3 + x 4 + x 5 + x 6 35 or 6 i =1 x i 35 . Todd Sarver (Northwestern University) The UPF Econ 310-2 – Fall 2011 3 / 28

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Notation The symbol (called Sigma ) denotes a summation. Example: If there are 35 units of good x to be divide between 6 individuals, the feasibility constraint can be written as either x 1 + x 2 + x 3 + x 4 + x 5 + x 6 35 or 6 i =1 x i 35 . The symbol Q (called Pi ) denotes a product. Example: Q n i =1 x i = x 1 × x 2 × · · · × x n Todd Sarver (Northwestern University) The UPF Econ 310-2 – Fall 2011 3 / 28
Notation The symbol (called Sigma ) denotes a summation. Example: If there are 35 units of good x to be divide between 6 individuals, the feasibility constraint can be written as either x 1 + x 2 + x 3 + x 4 + x 5 + x 6 35 or 6 i =1 x i 35 . The symbol Q (called Pi ) denotes a product. Example: Q n i =1 x i = x 1 × x 2 × · · · × x n The notation [ a, b ] denotes an interval : the set of all numbers between a and b . Example: The interval [1 , 3] contains all points between 1 and 3, not just 1, 2, 3. For instance, 1 . 5 , 4 3 , and 2 are all in this interval. Todd Sarver (Northwestern University) The UPF Econ 310-2 – Fall 2011 3 / 28

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Notation The notation ( x 1 , . . . , x n ) denotes a vector . The term x i is called the i th coordinate of this vector. Todd Sarver (Northwestern University) The UPF Econ 310-2 – Fall 2011 4 / 28
Notation The notation ( x 1 , . . . , x n ) denotes a vector . The term x i is called the i th coordinate of this vector. The notation means implies . For two statements a and b , a b means a implies b . That is, any time a is true, so is b . Example: x > 5 x 2 > 25 . Todd Sarver (Northwestern University) The UPF Econ 310-2 – Fall 2011 4 / 28

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Notation The notation ( x 1 , . . . , x n ) denotes a vector . The term x i is called the i th coordinate of this vector. The notation means implies . For two statements a and b , a b means a implies b . That is, any time a is true, so is b . Example: x > 5 x 2 > 25 . The notation means if and only if . For two statements a and b , a b means a is true exactly when b is. That is, both a b and b a . Example: x 2 > 25 ; x > 5 since ( - 6) 2 > 25 , and therefore x > 5 < x 2 > 25 . However, x > 5 x 3 > 125 . Todd Sarver (Northwestern University) The UPF Econ 310-2 – Fall 2011 4 / 28
Refresher: Preferences and Utility Besanko & Braeutigam, Chapter 3 Todd Sarver (Northwestern University) The UPF Econ 310-2 – Fall 2011 5 / 28

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Preferences Preferences: Consider two different alternatives A and B .
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