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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
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 ² 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
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 T is indecomposable There exists w > 0 such that T w >> 0 : (Indecomposable) This assumption means that it is possible to get from any state to any other state in a finite number of steps.

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