Parallel session IARIW 2014

1. Equalization of opportunity: De

2. nitions, implementable
conditions and application to early-childhood policy
evaluation
Francesco Andreoli
CEPS/INSTEAD and University of Verona
Tarjei Havnes
ESOP and University of Oslo
Arnaud Lefranc
THEMA, University of Cergy-Pontoise
Discussant: Flaviana Palmisano
University of Luxembourg
33th IARIW Conference
Rotterdam, the Netherlands, August 24-30, 2014
Flaviana Palmisano Rotterdam, the Netherlands August 24-30, 2014 1 / 24

3. The EOp model to evaluate public policies
Public policies are usually aimed at promoting equal opportunities.
To assess their ecacy possibly use the equality of opportunity theory
(inter alia Roemer 1998 and Lefranc et al. 2009):
y = fp(c, e, l ) (1)
y individual outcome; c circumstances; e eort; l luck; p social state.
c and e are illegitimate respectively legitimate sources of inequality. l
legitimate source of inequality as long as it aects individual outcomes in a
neutral way given c and e (see on this Dworkin 1981, Roemer 1998,
Fleurbaey 2008).
A type is a set of individuals having same c. An opportunity set is a set of
feasible outcomes for each type.
Fp(jc, e) outcome prospects in state p for individuals with same
circumstances and eort, with associated quantile function F1
p (pjc, e)
In a given state p equality of opportunity is satis

4. ed i 8(c, c0) and 8e:
Fp(jc, e) = Fp(jc0, e) (2)
Flaviana Palmisano Rotterdam, the Netherlands August 24-30, 2014 2 / 24

5. The EOp model to evaluate public policies (2)
(2) can be used to compare social states, distinguishing between states
where EOP prevails and states where EOP fails. (2) is a strong condition
and in most empirical applications, it turns out to be violated ) very
partial ranking.
No insight on the severity of the departure from EOP ) policy evaluation
remains silent on policies that fail to completely eradicate existing
inequality of opportunity or that reduce the existing level of EOP.
Alternative approach: indices of inequality of opportunity (see on this
Checchi and Peragine 2010, Bourguignon et al. 2007, Ramos and Van de
Gaer 2012). They are, however, fragile to the precise speci

6. cation of the
welfare function. They require summarizing the opportunity sets faced by
each type in a scalar, which may mask important dierences between policy
states .
Flaviana Palmisano Rotterdam, the Netherlands August 24-30, 2014 3 / 24

7. Research question and aims
How to compare and rank, robustly, social states in situations where EOP is
not satis

8. ed?
Develop a robust criterion to rank social states that is:
I consistent with conventional theories of equality of opportunity;
I consistent with recent developments in measuring social welfare;
I readily implementable in policy evaluation.
Evaluation of child care expansion in Norway.
Flaviana Palmisano Rotterdam, the Netherlands August 24-30, 2014 4 / 24

9. Contribution
Contribute to the EOp literature by providing set of conditions to order
social states on the base of the EOp criterion which:
I is robust to the speci

10. c individual or social welfare functions.
I is based on the extent of the economic advantage enjoyed by the
advantaged types in society. It posits that social state 1 is better than
social state 0, if the unfair advantage attached to favorable
circumstances is lower in state 1 than in state 0.
Contribute to the literature on public policy evaluation by providing a new
theoretical framework for the ranking of policy states coherently with the
theory of EOp.
Flaviana Palmisano Rotterdam, the Netherlands August 24-30, 2014 5 / 24

11. The framework
0p
0p
0pTwo types c and c0 with common level of eort e. Their respective outcome
prospect in state p is Fp for c and Ffor c0
W(Fp) (W(F)) individual utility from outcome prospect Fp (F)
DW(Fp, F0p
) = W(Fp) W(F0p
) economic advantage of type c relative to
type c0 in social state p
jDW(Fp, F0p
)j economic distance between types
Yaari's (1987) rank-dependent model to represent individual preferences
under risk:
W(F ) =
Z 1
0
w(p)F1(p)dp,w(p) 0 (3)
Let R denote the class of rank-dependent expected utility
Flaviana Palmisano Rotterdam, the Netherlands August 24-30, 2014 6 / 24

12. Result: Necessary condition for equalization of opportunity
01
De

13. nition (EZOP 00
equalization of opportunity between two types): Moving
from policy state p = 0 to p = 1 equalizes opportunity between circumstances c
and c0 at eort e on the set of preferences R if and only if for all preferences
W 2 R, we have: jDW(F0, F)j jDW(F1, F)j.
Here jDW(Fp, F0p
)j = j
R 1
0 w(p)G(F, F 0, p)dpj;
G(F, F 0, p) = F1(p) F 01(p) is the gap curve;
jG(F, F 0, p)j is the absolute gap curve.
Proposition (1)
If EZOP is satis

14. ed over the set of preferences R then for all p 2 [0, 1] :
jG(F0, F00
, p)j jG(F1, F01
, p)j (4)
Flaviana Palmisano Rotterdam, the Netherlands August 24-30, 2014 7 / 24

15. Results: Necessary and sucient conditions for EZOP
0p
0p
If individuals agree on the ranking of types:
Fp F8W 2 R () Fp ISD1 FProposition (2)
If 8p Fp ISD1 Fp then: EZOP is satis

16. ed over the set of preferences R if
and only if for all p 2 [0, 1] :
G(F0, F00
, p) G(F1, F01
, p) (5)
Flaviana Palmisano Rotterdam, the Netherlands August 24-30, 2014 8 / 24

17. Results: Restricted consensus on EZOP
If Fp ISD1 F0p
does not hold, consider R2 R such that Fp ISD2 F0p
.
This is the set of risk-avers rank-dependent preferences.
Proposition (3)
If 8p Fp ISD2 Fp then: EZOP is satis

18. ed over the set of preferences
R2 R if and only if for all p 2 [0, 1] :
Z p
0
G(F0, F00
, t)dt
Z p
0
G(F1, F01
, t)dt (6)
Flaviana Palmisano Rotterdam, the Netherlands August 24-30, 2014 9 / 24

19. Results: The general case
If Fp ISD2 F0p
does not hold, consider Rk Rk1 ... R such that
Fp ISDk F0p
. This is the set of of rank-dependent preferences putting
increasing restrictions on the cumulative weighting scheme ˜w
(p) such that
Rk =
n
W 2 Rj(1)i1 di ˜w
(p)
dpi 0, di ˜w(1)
dpi = 08p 2 [0, 1]andi = 1, ..., k
o
let k denote the minimal order at which Fp and F0p
can be ranked. Let
L2
p(p) =
R p
0 F1
p (u)du and Lk
p(p) =
R p
0 L(k1)
p (u)du
p,L0k
p , p) = Lk
p(p) L0k
p (p) the cumulative distribution gap
let G(Lk
integrated at order k 1
If for all p Fp ISDk F0pthen for all set of preferences W 2 Rk, DW in p is
an increasing function of G(Lk
p,L0k
p , p)
Proposition (4)
EZOP is satis

20. ed over the set of preferences Rk , 8p 2 [0, 1]:
jG(Lk0
,L0k
0 , p)j jG(Lk1
,L0k
1 , p)j (7)
This represents a necessary condition if we look at the whole set of
preferences R
Flaviana Palmisano Rotterdam, the Netherlands August 24-30, 2014 10 / 24

21. Comments (1)
If I understand well, Proposition 4 is a generalization of proposition 2 and 3.
Then I would expect the following condition:
G(Lk0
,L0k
0 , p) G(Lk1
,L0k
1 , p) (8)
rather then:
jG(Lk0
,L0k
0 , p)j jG(Lk1
,L0k
1 , p)j (9)
Flaviana Palmisano Rotterdam, the Netherlands August 24-30, 2014 11 / 24

22. Comments (2)
I think you could generalize this and the following result by considering not
only situations where individuals do not agree on the ranking of types, but
also to situations in which even though people agree there is no gap curve
dominance. Therefore, Proposition 3 and 4 can also be de

23. ned for
additional restriction on the weight of (3).
In the discussion of the advantages of your model you say '...Third, our
criterion does not even require a priori that individuals agree in their ranking
of the various types...' (page 11). However you need to reach agreement (at
order k) on the ranking of types in order for Proposition 2, 3 and 4 to hold.
When agreement (at any k) on the ranking of types is not required, you have
only a necessary condition (Proposition 1), hence you cannot rank social
states, but you can just state that if there is violation than EZOP is ruled
out, a limitation similar to that of other existing models described above.
Flaviana Palmisano Rotterdam, the Netherlands August 24-30, 2014 12 / 24

24. Results: generalization to multiple circumstances (1)
Let C = c1, ..., ci , ..., cT the set of all possible circumstances.
De

25. nition 2 (Non-anonymous EZOP between multiple types) Moving from
policy state p = 0 to p = 1 equalizes opportunity over the set of circumstances
C at eort e on the set of preferences R if and only if for all preferences W 2 R,
for all (i , j ) 2 1, ...,T we have:
jW(F0(jci , e) W(F0(jcj , e)j jW(F1(jci , e) W(F1(jcj , e)j
Generalizing proposition 4, integrated gap curve dominance for each pair of
types ci and cj provides a necessary and sucient condition for EZOP over
the subclass Rkij , where kij is the minimal order at which F (jci , e) and
F (jcj , e) can be ranked according to ISD. This condition is, however, only
necessary when looking at the whole class R
Proposition (5)
If EZOP between multiple types over the set of preferences R then
8(i , j ) 2 1, ...,T, 8p 2 [0, 1]:
jG(Lki ,j
0 (pjci , e),Lki ,j
0 (pjcj , e))j jG(Lki ,j
1 (pjci , e),Lki ,j
1 (pjcj , e))j (10)
Flaviana Palmisano Rotterdam, the Netherlands August 24-30, 2014 13 / 24

26. Results: generalization to multiple circumstances (2)
De

27. nition 2 makes the identity of each type relevant, since the advantaged
between any pair ci and cj under p = 0 is confronted with the advantage
between the same two types under p = 1.
It could be the case that only the magnitude of the gaps (and not the
identity of the types involved) is relevant.
let rW
p (c) be the rank function assigning to circumstance c its rank in policy
state p, according to preferences W.
De

28. nition 2 (Anonymous EZOP between multiple types) Moving from
policy state p = 0 to p = 1 equalizes opportunity over the set of circumstances
C at eort e on the set of preferences R if and only if for all preferences W 2 R,
for all (i , j , h, l ) 2 1, ...,T such that rW
0 (ci ) = rW
1 (ch) and rW
0 (cj ) = rW
1 (cl ) we
have: jW(F0(jci , e) W(F0(jcj , e)j jW(F1(jch, e) W(F1(jcl , e)j
Flaviana Palmisano Rotterdam, the Netherlands August 24-30, 2014 14 / 24

29. Results: generalization to multiple circumstances (3)
Let Rkmax , where kmax = maxi ,j2f1,...,Tg fki ,j g, be the intersection of all
the sets Rki ,j
Proposition (6)
If EZOP between multiple types over the set of preferences R then
8(i , j , h, l ) 2 1, ...,T such that rW
1 (ch) and rW
0 (cj ) = rW
0 (ci ) = rW
1 (cl )
and 8p 2 [0, 1]:
jG(Lkmax
0 (pjci , e),Lkmax
0 (pjcj , e))j jG(Lkmax
1 (pjci , e),Lkmax
1 (pjcj , e))j
(11)
Comment: Could you also extend also the this case proposition 2 and 4?
Comment: does population share play a role in your model?
Flaviana Palmisano Rotterdam, the Netherlands August 24-30, 2014 15 / 24

30. An aggregate index of inequality of opportunity
Let pc the relative frequency of type c, an index of inequality of opportunity
is obtained aggregating the abslute welfare gap across all the pairwise
comparisons :
IO(p) =
Tå
i=1
Tå
j=i+1
pci pcj jW(Fp(jci , e)) W(Fp(jcj , e))j (12)
IO always allows to rank policy states, although the conclusion is not robust
with respect to the evaluation of advantage and it is consistent with the
anonymous EZOP.
Flaviana Palmisano Rotterdam, the Netherlands August 24-30, 2014 16 / 24

31. Empirical application
Assess whether the expansion of child care in Norway, through the
Kindergarten Act in 1975, equalized opportunity among children.
This reform assigned responsibility for child care to local municipalities and
was followed by large increases in federal funding.
They use the Norwegian registry data, covering the population of Norwegian
children born to married mothers in the relevant cohorts.
S.e. obtained using a non-parametric bootstrap with 300 replications.
Circumstances: parental earnings decile during early childhood. Outcome:
individual earnings at age 30-36.
Flaviana Palmisano Rotterdam, the Netherlands August 24-30, 2014 17 / 24

32. Implementation (1)
DiD approach, exploiting that the supply shocks to subsidized child care
were larger in some areas than others.
Compare the adult earnings of children aged 3 to 6 years old before and
after the reform, from municipalities where child care expanded a lot
(treatment group) and municipalities with little or no increase in child care
coverage (comparison group).
Order municipalities according to the percentage point increase in child care
coverage rates over the expansion period; separate the sample at the
median, the upper half constituting the treatment municipalities and the
lower half the comparison municipalities.
Pre-reform group: children born 1967-69, they enter school before the
expansion period. Post-reform group: children born 1973-76, they are in
child care age after the expansion period has ended.
The EZOP criterion rests on the comparison of the eects of the reform at
quantiles of the earnings distribution conditional on circumstances.
Flaviana Palmisano Rotterdam, the Netherlands August 24-30, 2014 18 / 24

33. Implementation (2)
DiD estimation:
I fyit yg = gt (y ) + [b0(y ) + b1(y )Pt + b2(y )Ti + b3(y )Pt Ti ] g(xit ) + I is the indicator function; yit average earnings in 2006-2009 of child i born
in year t; y is a threshold value of earnings (How is it de

34. ned?); Ti is a
dummy indicating the treatment municipality; Pt is a dummy equal
indicating post-reform cohorts. gt is a birth cohort

35. xed eect; eit is the
error term.
g(xit ) is a
exible function of average yearly earnings of the child's parents
when the child was in child care age, denoted xit . b3(y ) DiD estimates of
how the reform aected the earning distribution of aected children.
Flaviana Palmisano Rotterdam, the Netherlands August 24-30, 2014 19 / 24

36. Implementation (3)
QTE estimation:
QTE(pjc) =
E[b3(y1)(pjc) g(xit )jCit = c]
f (y1(pjc)jCit = c)
(13)
The counterfactual quantiles in the absence of treatment (Q0(pjc)) are
estimated as:
Q0(pjc) = Q1(pjc) QTE(pjc) (14)
Q1(pjc) is the quantile conditional on circumstances in the actual distribution of
earnings among treated children. are obtained through quantile treatment eect
estimation; QTE(pjc) is an estimate of the quantile treatment eect at quatile p
for children with circumstances c.
Flaviana Palmisano Rotterdam, the Netherlands August 24-30, 2014 20 / 24

37. IST1 across types: actual and countefactual distribution
Ten types: children whose parents had earnings in each decile of the
parental earning distribution.
Flaviana Palmisano Rotterdam, the Netherlands August 24-30, 2014 21 / 24

38. Proposition (2) with ten deciles
Flaviana Palmisano Rotterdam, the Netherlands August 24-30, 2014 22 / 24

39. Other Comments
You are only using one circumstance, i.e. parental income deciles during
childhood, which makes the analysis more interpretable in terms of
intergenerational mobility.
Have you thought about including other circumstances?
You should link better the empirical application to the theoretical
framework. For instance, discussing the role of eort.
Flaviana Palmisano Rotterdam, the Netherlands August 24-30, 2014 23 / 24

40. Other Comments (1)
Really interesting and well-written paper.
It represents an original theoretical contribution to the literature with a very
nice empirical application.
Flaviana Palmisano Rotterdam, the Netherlands August 24-30, 2014 24 / 24