Measuring_Mobility_Theory_October_8

Measuring_Mobility_Theory_October_8 - Economics 140a...

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Economics 140a Measuring Mobility – theory Michael Rothschild October 8, 2009
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Measuring Mobility – theory 2 Measuring Mobility 1 The problem 1.1 Natural questions What are plausible measures of mobility? What corresponds to the Gini coefficient – a single number to measure how mobile a society is? Is there anything comparable to Lorenz domination – a clear and convincing test for saying when one society is more mobile than another? 1.2 Answers are complex To see why must look at how we describe mobility Economics 140a
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Measuring Mobility – theory 3 2 A language for describing mobility. 2.1 Positive or transition matrices 2.1.1 Some Notation For x;y 2 R N x y iff x i y i for all i x > y iff x y and x 6 = y x ± y iff x i > y i for all i The positive orthant is R N + = f x 2 R N j x & 0 g The set of probability vectors is Q = f x 2 R N + j P x i = 1 g Economics 140a
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Measuring Mobility – theory 4 2.2 Examples 2.2.1 Markov matrices Consider a world with N distinct states, i = 1 ;:::;N: Suppose that the probability of moving from one state to another is constant over time. Then (probabilistic) dynamics are described by an N N matrix T with typical element p ik = probability of going from state k to state i Since the probability of going somewhere from state k (where going somewhere in- cludes staying in k ) must be one we have P i p ik = 1 ; letting = (1 ; 1 ;::; 1) ; the last equation can be written in matrix form as 0 T = 0 (Markov) so that is an eigenvector of T with corresponding eigenvalue 1 . A matrix with positive elements satisfying ( Markov ) is called a Markov matrix. Economics 140a
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Measuring Mobility – theory 5 Let x t = ( x t 1 ;:::;x t n ) 2 Q with. x t i = probability of being in state i at time t . x t +1 = Tx t x t + w = T w x t T w is a transition matrix whose ik th element gives the probability of going from state k to state i in w steps. Assume, largely for convenience that
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This note was uploaded on 03/15/2010 for the course ECON ECON 140A taught by Professor None during the Fall '09 term at UCLA.

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Measuring_Mobility_Theory_October_8 - Economics 140a...

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