Mobility

1. Mobility Examining the Transition between states.

2. Welfare as an intergenerational Issue • Problem of “Dynastic” Rich and Poor, the children of the poor (rich) become the poor (rich) group when they are adult. • Thought to be a “bad thing” • Children’s “life chances” should be independent of parental circumstances.

3. The Policy Imperative • “The conception of social justice held by many, perhaps most, citizens of the Western democracies is that of equality of opportunity. Exactly what that kind of equality it requires is a contested issue, but many would refer to the metaphor of ‘leveling the playing field’, or setting the initial conditions in the competition for social goods as to give all, regardless of their backgrounds an equal chance of achievement. A central institution to implement that field leveling is education, meaning education that is either publicly financed or made available to all at affordable costs….” • John Roemer (2006).

4. Background and Motivation • Equal Opportunity’s foundations are to be found in the Egalitarian Political Philosophy of the 1980’s (Arneson, Dworkin...et. Al.) which holds agents responsible for their effort but not for circumstances beyond their control (e.g. their parents). • Thus “pure” EO policies modify the joint distribution of outcomes and circumstances promoting independence between them. • Picketty (1990) observed that “Mobility” is an aspiration of both the political left and right, their debate is how much is there of it. But do the left and right really want the same thing? • Starting from perfect immobility “pure” EO policies will induce increased downward mobility for the richly endowed and increased upward mobility of the poorly endowed. • However casual empiricism suggests EO policies in “liberal” societies appear to promote the latter rather than the former. • Conjecture: Unlike Marxists Liberal’s aspire to a qualified notion of EO. Has implications for Measurement and Policy Evaluation.

5. Intergenerational Regressions • Solon Model looks at the regression of child’s log income when adult (y) on parents log income (x) i.e. y = α+βx+ε. • β = Intergenerational income elasticity, β→0 implies intergenerational mobility β→1 implies intergenerational immobility • Attempts to use permanent income measures. • Need to deal with possibility that the relationship is non-linear. • Quantile regressions, relate the income quantile that the child is in (when adult) to the quantile that the parent is in.

6. Transition Models and Markov Chains. • We’ve met the transition matrix model in the parametric analysis of income distributions. Champernowne used the idea to rationalize the Pareto formulation for the income distribution. • Let p(xt) be the k long vector of probabilities of being in the k intervals of the range of x in period t and let p(xt-1) be the corresponding vector for period t-1. Let M be the k x k matrix whose typical element mij i=1,..,k, j=1,..,k is the probability of transiting from element j in period t-1 to element I in period t. • The process p(xt)=M p(xt-1) constitutes a Markov Chain model with M being the transition matrix. • Axiomatic Development of Mobility Indices based on M in terms of the extent to which such indices reflect desirable transformations M.

7. Axioms regarding “desirable” transformations of M. • MOV (Moving probability mass away from the diagonal). • EOP (Moving probability mass to equalize the opportunities of children of different descent without changing the orderings between descent categories.) • FP (focus on probabilities so that prizes do not matter) • ELC (Equalization of child life chances between parental categories.) • AN (Anonymity here means that parental status does’nt matter) • I (Defines the perfectly immobile matrix.) • PM (Defines perfect mobility).

8. Notes • Not all axioms are mutually compatible (MOV and PM, MOV and AN, MOV and PP, MOV and FP, EOP and FP). • Analysis only useful when the child outcome states are the same as the parental outcome states and when they are both single indexed (about which more later). • The Axioms are very much a matter of taste and some proponents of mobility have trouble with AN (An inferior outcome for child of rich parents relative to a child from poor parents is as unequal an opportunity as an inferior outcome for child of poor parents relative to a child from rich parents).

9. Mobility and the Transition Matrix

10. The Transition Matrix

11. Non-Square Transition Matrices and Complete Dependence

12. Some Transition Matrix Based Mobility Indices • (Trace(T)-1)/(n-1) (ignores off diagonals) • |T|1/(n-1) (attains perfect mobility with just 2 common columns) • Second Largest Eigenvalue of T • Σipi(x)ln(pi(x)/pi(y)) (where common states rather than common quantiles are used) • All relate to square transition matrices.

13. A Mobility Index for non-square multi dimensional transition matrices • Work with joint density matrix and examine its proximity to independence • M = ΣiΣkmin(pik,(pi.p.k)) • Monte Carlo evidence that M is asymptotically normal so can easily test for improvement or deterioration in mobility.

14. Multi-dimensional mobility analysis • The foregoing is amenable to analysing situations in which child’s and parents states are characterized by more than one variable • e.g. child’s state defined by w and x with joint density f(w,x) and parent’s state defined by y and z with joint density g(y,z) overal joint density h(w,x,y,z). • Mobility index given by: ∫∫∫∫min[h(w,x,y,z),(f(w,x)g(y,z))]dwdxdydz • Can also look at the notion of Conditional Mobility or Qualified Equal Opportunity.

15. Conditional Mobility • Consider a perfectly immobile society (with M=I) where human capital is the focus of endowment. • Movement toward equal opportunity requires diss-endowing the richly endowed as well as improving the lot of the poorly endowed. • Some societies would not approve of this and would only wish to do the latter. i.e. make more equal the life chances of the poorly endowed (contradicting the anonymity axiom).

16. The Class Transition Structure

17. The Planners Problem with no growth

18. The Planners Solution with Growth and a “Paretian” income constraint (That children from any parental class should not be made worse off).

19. The Solution

20. The Planners Adjustments To “J” and “Conditional Mobility” • Suppose the initial state is complete immobility (jii = pi for all I = 1,..,4) and g > 0. The social planner would reallocate the j1j’s to the extent that [3] does not bind and [2] does not bind for k = 1. • Thus mobility will be improved for the poorest children (note increased mobility for the richest children would involve increased downward mobility making them worse of and conflicting with the dominance condition [2]). • Should there still be capacity for change the j2j’s would next be reallocated and so on until the growth constraint is exhausted or complete equality of opportunity is achieved. • To the extent that the constraints bind only conditional mobility rather than complete mobility will be observed.

21. Generational Regressions • Yic = ac+b1cXic+b2cX2ic+eic where i =1,..,n (agents) in c=1,..,C (cohorts) with Y corresponding to child outcome and X corresponding to parental outcome. • E(ln(e2ic)=h1c+h2cXic • Qualified Mobility hypotheses: • Convexification b2m > b2n m < n • Heteroskedasticity h2m < h2n m < n