Mobility

Examining the Transition between states.

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.

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).

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.

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.

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.

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).

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).

Mobility and the Transition Matrix

Let f(y) be the distribution of a characteristic in the ultimate state and let f(x) be the distribution of a characteristic in the initial state, x and y have a joint distribution f(y,x) so that f(y) and f(x) are the respective marginal distributions. At one extreme there is a sense of no relationship when f(y,x) = f(y)f(x) (x and y independent) at the other there is a completely deterministic environment whereby y = a + bx. Partitioning y and x into k mutually exclusive and exhaustive regions where p(y) and p(x) are respectively vectors of the marginal probabilities of falling into those regions such that p(y) = p(x), we are interested in the elements of the square matrix T defined by p(y) = T(y,x)p(x) = J(y,x)M(x)-1p(x) where J(y,x) is a square matrix of joint probabilities. T is of course the matrix of conditional probabilities formed by the product of the two square matrices in the equation:

1

1 11 12 1 1 1

2 21 22 2 2 2

1 2

( ) ( , ) ( , ) . ( , ) ( ) 0 0 ( )

( ) ( , ) ( , ) . ( , ) 0 ( ) 0 ( )

. . . . . . . . . .

( ) ( , ) ( , ) . ( , ) 0 0 . ( ) ( )

k

k

k k k kk k k

p y p y x p y x p y x p x p x

p y p y x p y x p y x p x p x

p y p y x p y x p y x p x p x

Which is a matrix of conditional probabilities i.e. T = ||p ij(y,x)/pj(x)|| i, j = 1, ..,k.

The Transition Matrix

When x and y are independent the elements of J will be given by pij(y,x) = pi(y)pj(x) and T will be of the form:

1 1 1

2 2 2

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )k k k

p y p y p y

p y p y p yT

p y p y p y

The test of the null “T has identical columns” is the standard contingency table or independence test based upon the matrix J. When y = h(x) where h() is monotonic non-decreasing and the relationship becomes deterministic, T becomes the identity matrix (J is a diagonal matrix with p(x) as the diagonal vector) T = Ik The hypothesis of dependence can be examined via a Wald test on J for example.

Non-Square Transition Matrices and Complete Dependence

It may not possible to partition X and Y spaces in such a way that p(x) = p(y), the notion of transition still exists but it is no longer between commonly defined quantiles, we shall call such a T matrix miss-aligned. We can construct what the J and T matrices would look like from the marginals, just as we do in the case of independence or contingency table tests. The resultant T for such a case can be computed given the marginal values p(y) and p(x). For example imagine that x is partitioned at x1 and x2 (where x2 > x1) and y is partitioned at y1, y2 and y3 (where y1<y2<y3) further more suppose F(y1) < F(x1) < F(y2) and F(y2) < F(x2) < F(y3), where F( ) are the corresponding cumulative densities. Then the joint density matrix JImm can be shown to be of the form:

1

1 1 2 1Im

2 2 3 2

3

( ) 0 0

( ) ( ) ( ) ( ) 0

0 ( ) ( ) ( ) ( )

0 0 1 ( )

m

F y

F x F y F y F xJ

F x F y F y F x

F y

In this case TImm the corresponding transition matrix will be of the form:

1

1

1 1 2 1

1 2 1Im

3 22 2

2 1 2

3

2

( )0 0

( )

( ) ( ) ( ) ( )0

( ) ( ) ( )

( ) ( )( ) ( )0

( ) ( ) 1 ( )

1 ( )0 0

1 ( )

m

F y

F x

F x F y F y F x

F x F x F xT

F y F xF x F y

F x F x F x

F y

F x

Just as T with identical columns corresponds to the null of complete independence, a T of the form TImm corresponds to the null of complete dependence (notice again the columns sum to one). Examining the coherence of this hypothesis with the data is not so straightforward as examining independence, largely because of the problem of null cells (the corresponding Pearson test would have zero divisors in some cells) however the likelihood ratio for the non-null cells could be computed or a Wald test could be performed.

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.

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.

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.

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).

The Class Transition Structure

The generational income class transition structure: Parental incomes x = {1, 2, 3, 4} transit to child incomes y = {1, 2, 3, 4}. Marginal child and adult income probability vectors are c(y) and p(x) respectively with Joint distribution of parent - child incomes J so that:

11 12 14

21 22 24

41 42 44

1 1

4 42 2

1 13 3

4 4

.

.Pr( ) ( , ) ;

. . . .

.

( ) ; ( ) ; , 1,..,4i ij j ijj i

j j j

j j jy i x j J y x

j j j

c p

c pc y p x with c J and p J i j

c p

c p

.

Letting P = Diag(p(x)), the conventional transition matrix T may be written as T=J(y,x)P-1 whose i,j’th element is P(y=i|x=j) yielding the child income class vector c(y) from the equation c = Tp. An equal opportunity environment J* would be one where J* = c(y)p(x)’ (i.e. independence between child and adult outcomes which yields a transition matrix with the common columns c(y)).

The Planners Problem with no growth

A move toward J* that preserves the child’s income distribution (i.e. maintains the child’s marginal income probabilities), makes the children of one parental income group worse off while making the children of another better off. This may be seen by letting J* = cp’ and noting that T*p = Tp and suppose J ≠ J* in that Jij > cipj then Jkj < ckpj for some k ≠ i, J il < cipl, for some l ≠ j and J kl >ckpl. If k < i then:

1

1

( * ) / 0 1,..,4

( * ) / 0 1,..,4

m

ij ij ji

m

il il li

J J p for m and

J J p for m

So that children from j’th parental income group will be worse off (in the sense that their income distribution at least second order dominates that of the equal opportunity distribution) and children from l’th parental income group will be better off by the move to equal opportunity (in the sense that their income distribution is at least second order dominated by that of the equal opportunity distribution).

The Planners Solution with Growth and a “Paretian” income constraint (That children from any parental class should not

be made worse off).

Then the planner’s problem may be written as:

1

4 4 41 1 2

1 1 1

41 0 1 0

1 1

4 41 1 1

1 1

( ) :

( ) 0 1,.,4 ( )

, 1,..,4 , , 1,..,4 0 , .

ij ij jJ i j j

k

i i i ii i

ij j ij i iji j

Min j j p such that

c c k and c c i g

where j p j j c i and j for all i j

The Lagrangian in terms of the unconstrained jij’s for this problem may be

written as:

:

3 3 4 3 31 1 2 0 1

1 1 1 1 1 1

3 3 31 0 1 0

41 1 1

1

( ) ( )

( ( ( ) 4( ( ))

0 , .

k

ij il j k ij iji j l k i j

ij ij ij iji j j

ij

L j j p j j

g i j j j j

where j for all i j

The Solution

The Kuhn Tucker conditions:

41 1

411 1

41 0

1 1

3 3 31 0 1 0

41 1 14

2(1 )( ) (4 ) 0, , 1,.,3 [1]

( ) 0 0, 1,.,3 [2]

( ( ) 4( ( )) 0 0 [3]

i

j ij il j kl kij

k

ij ij ki jk

ij ij ij iji j j

Lp j j p i i j

j

Lj j and k

Lg i j j j j and

When the constrants do not bind (all λ’s=0) we have the equal

opportunity solution. As the constraints successively bind the

equal opportunity outcome is successively compromised:

41 1 13 3

[1] 1,.,3

2(1 )

i

kk

j jj

Solution for for i

j c pp

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.

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