Dynamic equality of opportunity. John E. Roemer Yale University. Burak Ünveren Yıldız Teknik Üniversitesi

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1 June 2012 Dynamic equality of opportunity by John E. Roemer Yale University Burak Ünveren Yıldız Teknik Üniversitesi

2 * We apply an EOp policy at time t, which presumably will change the distribution of types at time t+1. 2 *In particular, agents will respond optimally to the EOp policy *At date t+1 we again apply (another )EOp policy. *What is the equilibrium of such a process? It is a stationary state. * A stationary state is a policy, a distribution of types, and actions by all individuals, which reproduce the same distribution of types and actions in the next generation

3 3 * Thus, the foresighted planner must be concerned with how good the stationary states are, from the viewpoint of equalizing opportunities. * In general, there will be many stationary states, and the planner s objective is to choose the policy that induces the best stationary state, according to the EOp ethic *We study this problem in the simplest interesting model we can think of

4 Model 4 * Population of families; fraction f R are Rich and fraction f P = 1! f R are Poor. Each family has one child. The incomes of R family and P family are fixed at w R > w P. * Both the State and the family may invest in the education of child. Call these investments s J,i J for J!{P, R}. The probability that a R child becomes a R adult is:! R (z P,z R ) = where z j = s j + i j ; e z R e z R + e z P

5 5 The prob that a P child becomes a R adult is! R (z P,z R ) = ae z P e z R + e z P, some 0 < a < 1. * a reflects the culture/nbhd effects of living in a P family. * the standard of living of a family of type j is y j = (1! t)w j! i j * the utility of a type j adult at date τ is U j! = y j! + "(# j y j!+1 + (1$ # j )y % j!+1 ) where! reflects intergenerational altruism.

6 6 Def n stationary state A stationary state is a policy (t *,s * R,s * P ) and a pair of private investments (i * P,i * * R ) and a pop n distribution f R such that: 1) ( ) = f R * s R * + 1! f R * t * µ * = t * f * R w R + ( * 1! f R )w P 2) i R * maximizes (over i): ( 1! t * )w R! i ( )s P * ( ) + ( " # R s * * ( R + i,z P )(( 1! t * * )w R! i ) R + ( 1! # R s * * ( R + i,z P ))(( 1! t * * )w P! i )) P $ & % P & ' 3) i * maximizes over i : P ( 1! t * )w P! i ( ) + ( " # P ( z * R,s * P + i) (( 1! t * * )w R! i ) R + ( 1! # P ( z * R,s * P + i) )(( 1! t * * )w P! i )) P 4) f * R! R z * * * ( R,z P ) + ( 1" f R )! P z * * * ( R,z P ) = f. R $ & % P & '

7 Define E J =! J (( 1" t)w R " i ) R + 1"! J ( ) 1" t ( ) ( )w P " i P at a stationary state. We define the EOp ordering of stationary states by:!! " ˆ! " min(e R (!), E P (!)) # min(e R ( ˆ!), E P ( ˆ!)). Our task is to find the best stationary state (and associated policy) according to ". That is:! max! min j=p,r (E j (!)). Analytically difficult problem, b/c the IC constraints render it a non-concave optimization problem. We do some analysis of the problem & based on this, we compute by simulation the optimal EOp policy for 7

8 a large set of randomly simulated economies. Easy to show: In all stationary states f R * = a. 1+ a The poor are always with us. 8 An economy is specified by a tuple (a,w R,w P,!). These are the possible kinds of ss: case s R s P i R i t E P R -E P The State never invests in R children, and the P never invest privately.

9 9 State 4 is laissez-faire. Only in state 2 is full EOp achieved. In 76% of cases, EOp optimal ss is laissez-faire! Disturbing result. The intuition: If the state invests in the P children, the Rich undo the effect by investing privately in their own children. For a better understanding, we now fix ( w R,w P ) = ( 80,40). We randomize over (a,!) and can now plot the results in the (a,!) plane.

10 10 Red: Laissezfaire; Blue: Case 2 (full EOp) Summary: If! > 0.2, soln is laissez-faire; only if! < 0.2 does the state invest in children. The state acts in loco parentis only if parents don t care about their children. To drive point home: Suppose we are in the red zone but state invests in P children. Eventually the dynamics will lead to a ss where P children have a lower expectation of becoming R adults than if the State had not invested in them!

11 Expanding the policy space 11 How might we do so? The state could invest in P parents, thus increasing a. We assume a m ( ) = 1! 1! a 0 ( )exp!m ( ). Note a(0) = a 0, lim a(m) = 1. m!" Now a state policy is (t,s P,s R,m). There are now 10 possible kinds of ss: case m t E s R s i i P R P R E P [0,1]

12 12 1a a [0,1] 3a a a

13 13 A simulation on the same support as before now produces these results: Case percentage no solution found 1.45% % % 5 0 1a 0 2a 6.15% 3a 0 4a 80.65% 5a 0 Only 11.7% of cases are laissez-faire: the predominant case 4a is that m > 0,s P = 0,i R > 0. The State only invests in families, not in children directly.

14 14 As before, we now fix (w R,w P ) = (80,40) and simulate on (a 0,!) "[0,1] 2. For normal parameter values, yellow (case 4a). Only for very small values of! does state invest in both P families and children (red); only for a 0 close to 1 laissez-faire the optimal policy (blue).

15 Concluding remarks 15 * We don't know how general our results are. We have analyzed an example. *It appears more attention should be paid to family culture. Perhaps this is due our functional forms. * EOp policies may have limited effects, unless there is a social ethos, in which people care about other peoples children. A pessimistic conclusion.