Skilled or unskilled? To work or not?
Female Decisions under Imperfect Information∗
Alessandra Casarico
Department of Institutional Analysis and Econpubblica,
Università Bocconi, Milan, Italy; CEPR, London, UK;
CESifo Munich, Germany.
Paola Profeta
Department of Institutional Analysis and Econpubblica,
Università Bocconi, Milan, Italy; CESifo Munich, Germany;
First draft. Very preliminary: October 2009
∗
Alessandra Casarico,
Università Bocconi,
Via Roentgen 1,
20136 Milan,
Italy;
alessan-
[email protected]; Paola Profeta, Università Bocconi, Via Roentgen 1, 20136 Milan, Italy,
[email protected]. We thank Carlo Devillanova and Vincenzo Galasso for their comments.
1
Abstract
We develop a two-period model to jointly analyze the female decisions of investing
in education and of participating in the labor market. In the first period women, who
are heterogeneous in talent, decide whether to invest in education, or remain unskilled
and start working immediately. In the second period they have a kid and they make
their participation decision: the skilled decide whether to enter or not the labor market;
the unskilled whether to continue or drop out. Kids are costly, and the cost is divided
among parents. This cost may be of a high or a low type, and the true type is revealed
only after the child’s birth. Female education decisions are thus taken under imperfect
information on the cost of the kid. We solve the model backward and we characterize the
participation and education decisions. We find that, due to the imperfect information,
there are instances where women who have invested in education find it profitable not to
work. We also find that imperfect information may be associated with a higher output
than what would result in the case of perfect information, since in the former case highability women with a high cost kid may find it convenient to invest in education and work.
We then study a tax-transfer scheme targeted on working women as a specific example
of government’s intervention to sustain both female decisions of investing in education
and work. In the presence of this policy it’s more likely that more women participate to
the labor market when compared to the case where the government does not subsidize
the child cost. Their expectation to participate may in turn induce them to invest in
education. The policy also reduces the amount of unused human capital related to the
presence of educated women who decide not to participate.
Keywords: child costs, imperfect information, tax-transfer scheme.
JEL Classification: J16, J24.
2
1
Introduction
Industrialized countries are characterized by large cross-country differences in terms of
female participation and employment rates. The same can be said if one focuses on
education and looks at the share of women with upper secondary education. While for
instance in Italy only 48% of women in the 25-64 cohort had at least upper secondary
education in 2004, in Sweden they were 85%. If we focus on younger cohorts this gap is
smaller but not yet closed. Not only are there differences in terms of the share of educated
women, but also in terms of the percentages of educated women who work, indicating
that female human capital is often underutilized. Among women with at least upper
secondary education, 61% of Spanish women works, while more than 74% of women works
in countries such as Denmark, Sweden, the Netherlands and the UK.
Figure 1 and 2 suggest that countries where the number of educated women and the
percentages of educated women who work are higher are those where public subsidies to
home produced goods (child care for instance) are larger, or where expenditures on family
policies account for a greater share of GDP or where measures of flexibility on the labor
market are more used (part-time for instance).1 Scandinavian countries are an example.2
Previous contributions have studied the impact of cross-country differences in institutions on the female labour market participation and on the fertility decision (see Del Boca
1
In Figure 1a we plot the share of female with at least upper secondary education in 2004 (OECD,
2006) and the availability of part-time jobs (same year) based on Eurostat data. In Figure 1b we plot the
share of female with at least upper secondary education and the final child care score for children of age 0-2
taken from De Henau et al. (2007). In Figure 1c we plot the share of female with at least upper secondary
education and family expenditure as a percentage of GDP taken from the OECD Family Database and
referred to 2003. In Figure 2a, 2b and 2c on the vertical axis we have the share of educated women who are
employed in 2004 (OECD, 2006), instead of the share of educated women. These are simple correlations
and do not imply any causal relationship.
2
While the focus here is on the role of institutions, how good the labour market is in terms of rewarding
human capital investments can contribute to explain the cross-country differences in female education and
employment. This explanation is however silent on the existing cross-country differences in the gaps in
employment rates of women with/without children: while the employment rates of women with children are
typically lower than the ones of women without children, for instance in France or Scandinavian countries
this gap is smaller than in Italy or Germany.
3
et al., 2007 and De Henau et al., 2007 for instance). The relationship between institutions
and educational choices is instead less explored. Including educational choices into the
analysis of female employment is however essential to shed a light on what can be done to
build further human capital and to reduce the unused human capital of educated women
who do not work.
Why are there women who invest in education but do not participate in the labor
market? Is there a policy which can support the education and participation decisions?
In this paper we build a two-period model to jointly analyze the female decisions of
investing in education and of participating in the labor market. For simplicity, we consider
that these decisions are both binary, i.e. a woman has to decide whether to become skilled
or remain unskilled and whether to work or not. In the first period of time women,
who are heterogeneous in talent, decide whether to invest in education or not. If they
invest, they devote the entire first period to education. If they do not invest, they remain
unskilled and they start working. At the beginning of the second period, all women have
a child. Kids are costly in terms of time that they require. The cost of the kid is divided
among the parents. If they both work, each of them has to buy the own share of required
child care time on the market. However, the mother can also live on home production,
differently from the father, and thus she can decide whether to stay at home and provide
personally her due time to the kid, or to work and buy it on the market. We assume
that the cost of the kid may be of two types, high or low, and the true cost of the kid
is revealed only after her birth. This heterogeneity captures the idea that some kids are
more demanding than others, due for instance to her health status or good nature, or due
to the fact that grandparents or informal networks help taking care of the kid. Female
education decisions are taken under imperfect information on the cost of the kid: women
only know the probabilities that the kid is either a high or a low-cost type. At the child
birth, women make their participation decision: the skilled decide whether to enter or not
the labor market; the unskilled whether to continue or drop out. Their decision will be
based on the comparison between their consumption possibilities when participating to
the labor market and when staying at home, assuming that in this latter case consumption
depends on a given level of home production.
4
We solve the model backward starting from the participation decision: we identify
the cut-off level of ability such that skilled women find it convenient to participate and
the minimum unskilled wage required for the unskilled to find it convenient to continue
working after child birth. As expected, we find that, for a given level of child cost, if an
unskilled woman works, so does a skilled one. We also find that, for a given level of talent,
if a woman with a high-cost kid works, so does a woman with a low-cost kid. Knowing
the probabilities to participate, given the ability and cost types, we examine the education
decision and we calculate the cut-off levels of ability such that women find it profitable
to invest in education. We find that, due to the imperfect information, there are indeed
instances where women who have invested in education find it profitable not to work. We
also find that imperfect information may be associated with a higher output than what
would result in the case of perfect information, since in the former case high ability women
with a high-cost kid may find it convenient to invest in education and work.
The government can intervene to sustain both female decisions. We study a taxtransfer scheme targeted on working women as a specific example of government’s intervention to sustain both female decisions of investing in education and work. In particular,
the government provides a subsidy which is proportional to the kid’s cost if the mother
works, and which is financed by a proportional tax on wages paid by all workers. In the
presence of this policy more women participate to the labor market when compared to
the case where the government does not subsidize the child cost. Their expectation to
participate may in turn induce them to invest in education. By working as an insurance
device against the risk of having a high-cost kid, the tax-transfer scheme also reduces the
amount of unused human capital related to the presence of non-working educated women.
A recent growing literature analyzes how gender gaps in wage and in participation can
arise when the distribution of ability between men and women is the same (Francois, 1998;
Albanesi and Olivetti, 2008; Bjerk and Hahn, 2007, Lommerud and Vagstad, 2007). These
contributions share a common assumption, namely, skills are given and cannot be accumulated. While the focus of this paper is on explaining female education and participation
decisions rather than on gender gaps, we complement the previous studies by including
5
the education decision. The role of education is the focus of a very recent literature which
analyses the effects of pre-marital schooling decisions on the marriage market: educational
attainment may influence the marital surplus share that can be extracted in the match, by
increasing the prospects of marrying an educated partner and by affecting the competitive
strength in the bargaining within the couple (Iyigun and Walsh, 2007b; Chiappori et al.,
2008). We here abstract from gains in the marriage market as a motivation to acquire
education: we will however perform some comparative statics exercise to see how a change
in the share of the child rearing costs which are borne respectively by the mother and the
father influences education and participation decisions.
There is also a literature on the effects of child care subsidies or public child care provision on female employment (see for instance Ohlsson and Lundholm, 1998; Rosen, 1997).
However the effects that these policies can have on the decision to invest in education is
never addressed, as far as we know.
Our paper is also to some extent related to the literature that analyzes the role of
imperfect information in education investments. Bernhard and Zilcha (2009) find that, on
the one side, better information increases the efficiency of aggregate human capital formation, while on the other side, by decreasing the risk sharing opportunities, if agents are
very risk averse, welfare may be higher under a less informative system. In our framework
imperfect information may have the positive effect of inducing high ability types, who turn
out to have a high-cost kid, to invest in education given their expectation of having an
average-cost kid. They would have not found it convenient had they known the true cost.
If they end up working as skilled individuals, this will represent a gain of output.
The paper is organized as follows: the next Section develops the model, Section 3
studies the implications of imperfect information. Section 4 introduces the tax-transfer
scheme and it analyses the role of government and Section 5 concludes.
6
2
The model
2.1
General features
We develop a two-period model. Women are heterogeneous in talent  , which is distributed on the interval [1 ] with a distribution function  and a continuous density function
 (·) and is known to women, firms and government.
In the first period of time women decide whether to invest in education or not. If they
invest, they devote the entire first period to education and they become skilled. If they
do not invest, they remain unskilled and they start working and receive a salary equal to
 , which is transferred to the second period at an exogenous interest rate .
At the beginning of the second period, all women have a child and they decide whether
to work or not. The participation decision is again binary: the skilled decide whether to
enter or not the labor market; the unskilled whether to continue or to stop. Their decision
will be based on the comparison between their consumption possibilities when participating
to the labor market and when staying at home. Consumption takes place at the end of
the second period of life when agents consume all their lifetime income.
Kids are costly, in terms of the time that they require. We assume that the time
required by a kid may be of two types, high or low:    =  . This is to capture the
idea that some kids are more demanding than others, due for instance to her health status
or good nature, or to the presence of grandparents or informal networks that help taking
care of the kid, thus reducing the amount of time to be financed by parents. The cost of
the kid is divided among the parents: the mother is responsible for the amount  (1 − )
and the father for the amount  , with  ∈ (0 1) indicating the division of time of child
care in the couple. To start with, we assume that  is exogenous and we analyze the
implications for our results of different values of . Each parent may decide to provide
personally the required child care time for the kid or to buy it on the market. Thus, for
the mother and the father the child time requirements are, respectively:
7
 (1 − ) =  + 
(1)
  =  + 
where  and  are the time that the female and male adult decide respectively to
spend personally with the kid and  and  denote the time bought on the market. We
assume that a unit of time bought on the market costs .
For each adult, the total amount of time of their second period of life, whose length
is 1, may be allocated to labor  with  =   , to home production   and to time
spent with the kid  , so that the following second period time constraints are satisfied:
1 =  +   + 
(2)
We start with the allocation of time of the mother. If she decides to work,  = 1
and thus   =  = 0. Thus, the mother has to buy all the time of child care under her
responsibility on the market:  (1 − ) =  at the cost of  (1 − ). Let  =  ; then
the cost of a kid for a working woman is  (1 − ).
Working on the market, all unskilled women receive the same wage, equal to  , while
a skilled woman with ability  receives a salary equal to   . Thus, the consumption
possibilities for a skilled working woman and an unskilled working woman whose kid costs
 are, respectively:
 


 =   −  (1 − )
(3)



 =  −  (1 − )
(4)
and
If, on the contrary, she decides not to work,  = 0. The kid’s time requirement borne
by the mother is supplied by her directly and she does not buy any child care on the
market, that is  (1 − ) =  . Thus, given the time constraint, she can devote to home
production an amount of time equal to   = 1 −  (1 − ). Indicating with  ∈ (0 1) the
amount of home production that a woman can deliver if   = 1 when a woman decides
¤
£
to stay at home, her consumption possibilities are equal to 1 −  (1 − ) , which can be
8
rewritten as  − (1−)   Assume for simplicity that  = 1; the consumption possibilities
for non working women are therefore:


 =  −  (1 − )
(5)
Differently from the mother, we assume that the father does not produce anything at
home, i.e.  = 0 for him and thus he always works on the market:3  = 1 and thus
  =  = 0 He receives a wage equal to  if he is unskilled and equal to   if he
is skilled and his ability is equal to  .4 Thus, the consumption possibilities for a skilled
and an unskilled man whose kid cost is  are, respectively:


=   −  


=  −  
(6)
At the time of the child’s birth, the woman decides whether to work or not knowing
the true cost of her kid  and comparing her consumption possibilities in the two cases.
The education decision instead is taken under imperfect information on the cost of the
kid: we assume that with probability  the cost is high,   1, and with probability
(1 − ) the cost is low,   1, where    . Women and firms know   and  
Firms operate in a perfectly competitive environment.
2.2
Individual problem
We focus on female education and participation decisions. Figure 3 summarizes the 2stage game and pay-offs: in the first step a woman decides whether to become skilled or
not, forming expectations on the cost of her kid; in the second step, the kid is born, the
true cost is revealed and the woman decides whether to work or not.
3
This is equivalent to assuming that women have a comparative advantage in home production. The
assumption is instrumental to obtaining that men always work, which is in line with the evidence that on average - male participation rates are higher than female ones. A similar result would be obtained if
we assumed that male wages are always such that their consumption possibilities are greater when they
participate rather than when they stay at home.
4
The distribution of market abilities is assumed to be the same across males and females.
9
We solve the model backward starting from the participation decision. A skilled woman
with ability  and with a kid of cost  (with  =  ) decides to work if her consumption
possibilities in the case she works are larger than those in the case she does not work.
Comparing (3) and (5) we obtain:
  −  (1 − ) ≥  −  (1 − )
(7)
which delivers the following threshold level of ability:
 ≥
 +  (1 − )(1 − )

(8)
Notice that, when the cost of the kid is high, a higher level of ability is required to
work, with respect to the case where the cost of the kid is low. Moreover, the higher the
time of child care under the responsibility of the mother, represented by 1 − , the higher
the level of ability such that a woman find it convenient to work.
Similarly, comparing (4) and (5) the unskilled continue working in the second period
if the following condition is satisfied:
 −  (1 − ) ≥  −  (1 − )
(9)
i.e. if the wage of the unskilled is larger than the following threshold:
 ≥  +  (1 − )(1 − )
(10)
As expected, this condition is more easily satisfied for women with a low-cost kid than
for those with a high-cost one. Again, it is also more easily satisfied when men participate
more in taking child-care responsibilities within the couple (higher levels of ).
Since   ≥  for all  and    we have three possible cases depending on
the level of wage of the unskilled:
• Case 1:    +  (1 − )(1 − )
• Case 2:    +  (1 − )(1 − )
• Case 3:  +  (1 − )(1 − )     +  (1 − )(1 − )
We now analyze separately the participation and education decisions in these three
scenarios.
10
2.2.1
Case 1: All women work
If the wage of the unskilled workers is large enough to induce unskilled women to continue
working after the child’s birth, i.e.    + (1−)(1−) then all skilled women, given
their higher salaries, will also find it convenient to work. Having the expectation that they
will certainly work, they decide to invest in education by comparing their consumption
possibilities in case they work as skilled or as unskilled. The decision is taken under
the expectation that, with probability  the cost of the kid is equal to  and with
probability (1 − ) it is equal to  . Remembering that unskilled women work for two
periods, a woman invests in education if her consumption as skilled is larger than the one
as unskilled, i.e:
  − (1−)−(1−) (1−) ≥  (1+)+ − (1−)−(1−) (1−) (11)
which delivers the following threshold level of ability such that all women with an ability
type larger than this invest in education:
 ≥

(2 + )

(12)
Since both skilled and unskilled women will work, the investment in education depends
only on the wage premium, i.e. the ratio between the unskilled and the skilled wage. This
case is similar to what happens to men: since, as we argued before, men always work, their
education decision depends only on the wage premium, i.e. men with a level of ability such
that  ≥


(2

+ ) invest in education and the others remain unskilled. Interestingly,
if the wage premium for female is larger than the one for males, women have stronger
incentives to educate than men.
2.2.2
Case 2: Unskilled women do not work
If the wage of the unskilled workers is below the threshold  +  (1 − )(1 − ), none of
them will find it convenient to continue working after the child’s birth. For the skilled,
different possibilities arise depending on their level of ability.
• Case 2a
11
All skilled women with a level of ability such that
 
 +  (1 − )(1 − )

(13)
find it convenient to work. Given this expectation, they will decide to invest in education
if their consumption possibilities in the case they acquire education are larger than if they
do not, i.e. if the following condition is satisfied:
¤
£
¤
£
   −  (1 − ) + (1 − )   −  (1 − )
£
¤
£
¤
≥  (1 + ) +   −  (1 − ) + (1 − )  −  (1 − )
i.e. if their level of ability is above the following threshold:
 ≥
 (1 + ) +  + (1 − )(1 − )

(14)
where  =  + (1 − ) .
Interestingly, in this case a woman invests in education on the basis of the expected
value of the cost of the kid.
• Case 2b
All skilled women with a level of ability such that
 +  (1 − )(1 − )
 +  (1 − )(1 − )

≤

≤


(15)
find it convenient to work if they have a low-cost kid  and not to work if they have a
high-cost kid  . Knowing this, they will decide to invest in education if the following
condition is satisfied:
¤
£
¤
£
  −  (1 − ) + (1 − )   −  (1 − )
£
¤
£
¤
≥  (1 + ) +   −  (1 − ) + (1 − )  −  (1 − ) 
(16)
which delivers the following threshold level of ability:
 ≥
 (1 + )  +  (1 − )(1 − )
+
 (1 − )

12
(17)
This means that all women with ability levels below this threshold remain unskilled.
Notice that not all skilled women work: women who have invested in education under
the imperfect information about the cost of their kid may find it convenient not to work if
this cost turns out to be high. Their human capital is thus wasted or unused, in the sense
that it does not translate into output on the market.5
• Case 2c
All skilled women with a level of ability such that
 
 +  (1 − )(1 − )

(18)
find it convenient not to work. Under the expectations of not working it is immediate to
verify that no woman invests in education.
To summarize, in Case 2 we have found that: low ability women (the ones with  
 (1+)
+ (1−)(1−)

+
)


 (1−)

≤
abilities ( + (1−)(1−)

remain unskilled and do not work; for intermediate levels of
 ≤
+ (1−)(1−)
)

women invest in education, and a fraction
(1 − ) of them, who turns out to have a low-cost kid, works, while a fraction  who
have a high-cost kid, does not work; the high ability types ( 
+ (1−)(1−)
)

invest in
education and work.
2.2.3
Case 3: Women with a low-cost kid work
If the wage of the unskilled workers is such that  +  (1 − )(1 − )     +  (1 −
)(1 − ) the unskilled women with a low-cost kid find it convenient to continue working
after the child’s birth, while the unskilled women with a high-cost kid decide to drop out.
Since unskilled women with a low-cost kid work, we are sure that also skilled women with
5
We are not explicitly accounting for the benefits that human capital investment provides when it is
not employed on the market. This is not to say that human capital investments by non-working mothers
do not have positive effects on the economy (for instance through private benefits to children or through
externalities to the society as a whole). However, as long as working on the market does not completely
crowd out these effects, there are further benefits (or lower losses) to be reaped when educated women
work on the market. On the effects of maternal employment on children’s human capital see Bernal (2008).
13
a low-cost kid do indeed work. The skilled with a high-cost kid instead may be of two
types, depending on their level of ability. We analyze them separately.
• Case 3a
All skilled women with a high-cost kid and a level of ability such that
 
 +  (1 − )(1 − )

(19)
find it convenient to work. Given this expectation, they decide to invest in education if:
£
¤
£
¤
   −  (1 − ) + (1 − )   −  (1 − )
£
¤
£
¤
≥  (1 + ) +   −  (1 − ) + (1 − )  −  (1 − )
(20)
which requires a level of ability above the following threshold:
 ≥
 [1 +  + (1 − )]
 +  (1 − )(1 − )
+




(21)
• Case 3b: All skilled women with a high-cost kid and a level of ability such that
 ≤
 +  (1 − )(1 − )

(22)
find it convenient not to enter the labor market. Their investment in education depends
on the following comparison:
¤
£
¤
£
  −  (1 − ) + (1 − )   −  (1 − )
£
¤
£
¤
≥  (1 + ) +   −  (1 − ) + (1 − )  −  (1 − )
(23)
which delivers the following threshold of ability:
 ≥
 [1 +  + (1 − )]
 (1 − )
This last case is particularly interesting: women whose abilities are such that
 ≤
+ (1−)(1−)

(24)
 [1++(1−)]

 (1−)
≤
have invested in education but do not find it convenient to work. This
is again a situation where human capital is unused.
14
Remark 1 The higher the share of child care time under her responsibility (1 − ), the
higher the level of ability such that a woman find it convenient to invest in education.
Moreover, the higher the value of home production (), the higher the level of ability to
make the educational choice a convenient one.
Proof: When  is smaller, the threshold levels of ability related to the education
decision at Eq. 14, Eq. 17 and Eq.21 are smaller. Moreover 1 −  (1 − )  0 for any 
guarantees that the same threshold levels are increasing in .
The remark clarifies the role of the division of child care duties within the couple,
captured by the parameter . A more balanced division (a lower ) helps not only female
participation to the labor market (see Eq.8 and Eq. 10), but also female investments in
education, with respect to the case where a woman is the main responsible of the kid’s
care. Thus, the division of child-care duties within the couple is crucial to understand the
level of female education and employment. Similarly, a female comparative advantage in
home production (higher ) reduces female education and employment.
3
The role of imperfect information
Imperfect information plays a crucial role in the results described. It is intuitive to understand that in our set-up, if there is perfect information on the ability type and kid’s
cost at the stage of the education decision, all those who invest in education will certainly
work and there is no unused human capital.
Consider a woman whose kid costs  with  =   If, drawing upon Eq. 10,  ≥
 + (1−)(1−), then participating is always preferable to staying at home. In this case,
all women with abilities such that  ≥



(2+) will invest in education and the others will
remain unskilled (irrespective of the  ). When    +  (1 − )(1 − ), the unskilled
will not find it convenient to work. As to skilled women, they will find it convenient
to participate only if their ability level is such that  ≥
+ (1−)(1−)
.

If we turn to
the education decision, we find that only those whose ability satisfies  ≥
 (1−)(1−)

 (1+)+


+
will decide to invest in human capital. Given that the education threshold is
15
higher than the participation one, all women whose ability is  
 (1+)+


 (1−)(1−)
+

are unskilled and will stop participating at the child’s birth.
A comparison between the perfect and imperfect information framework delivers interesting results. Having the perfect information context as a benchmark, we can study the
implications of the presence of imperfect information for the education and participation
female decisions and, in the aggregate, for the level of human capital and output of the
economy. On the one side, in the perfect information scenario there is no unused human
capital as only those whose ability type and kid’s type will be such that they can work as
skilled will actually decide to invest in education. On the other side, it is not clear whether
the aggregate level of output is larger or smaller than what emerges in a context of imperfect information. In our first scenario (Case 1) imperfect information does not play
any role, since all women work in both contexts and the cut-off level of ability such that
women find it convenient to invest is the same in the perfect and imperfect information
set-up. A similar result is obtained when no women work (Case 2c), and thus there are
no incentives to invest in education. In the other scenarios instead, imperfect information
plays an important role, as described in the following proposition.
Proposition 2 Imperfect information enlarges the set of women’ ability types that find it
convenient to invest in education and turn out to have a high-cost kid, while it restricts
the one of those who become skilled and turn out to have a low-cost kid. As a consequence,
a context of imperfect information may be associated with a different level of output than
the perfect information one.
The general result stated above has to be proved and studied separately in our scenarios. The implications on aggregate human capital and output are different in each of
them.
If unskilled women have no incentives to work, while skilled women do (Case 2a), in a
context of imperfect information the decision of investing in education is taken under the
expectation that the cost of the kid will be on average equal to , rather than the true
16
cost  and  . Thus, women with ability types such that
 (1 + ) +  +  (1 − )(1 − )
 (1 + ) +  + (1 − )(1 − )






(25)
will not find it convenient to invest in education in the presence of imperfect information,
while they would find it profitable in the case they knew from the beginning that their
kids were of a low-cost type. Similarly, women whose ability types are such that
 (1 + ) +  +  (1 − )(1 − )
 (1 + ) +  + (1 − )(1 − )






(26)
find it convenient to invest in education in the presence of imperfect information, while
they would not find it convenient it if they knew from the beginning that they have a
high-cost kid. Remember that in this case all skilled women work. Thus, the presence
of imperfect information is associated with a loss in terms of human capital and output
due to women with low-cost kids and a gain produced by individuals with high-cost kids.
It is easy to check that the total number of skilled women is the same in case of perfect
and imperfect information.6 However, the skill composition is different, since under imperfect information we gain skilled women with higher level of abilities than those that
we lose. Imperfect information increases the risk sharing opportunities (see Eckwert and
Zilcha, 2009 for a similar implication) and may thus lead to a better allocation of talents:
high-talented individuals become skilled and low-talented individuals remain unskilled,
irrespectively of the cost of the kid. This implies that imperfect information may increase
the level of human capital and output of the economy.
If skilled women work only if they have a low-cost kid (Case 2b), the increase of the
number of educated women with high-cost kids in a context of imperfect information does
not traslate into an increase of output. Thus imperfect information creates unused human
capital. Moreover, by reducing the number of skilled women with low-cost kids who would
work in a context of perfect information,7 imperfect information reduces the level of output
in the economy.
6
Fewer skilled women with low-cost kids are exactly compensated by more skilled women with high-cost

 
 (1+)++(1−)(1−)
 (1+)++ (1−)(1−)
− 
kids, since (1 − ) 





 
 (1+)++ (1−)(1−)
 (1+)++(1−)(1−)
= 
− 




7
This result derives from the following comparison: in a context of perfect information, women with low-
17
Moving to our third scenario, when the workers are all skilled women and the unskilled
with low-cost kids (Case 3a), the incentives to invest in education for women with highcost kids associated with the imperfect information context8 induce a gain of output. The
reduction in the share of skilled women with low-cost kids9 implies, instead, a loss of
output. To be more precise, the gain is equal to the difference in the threshold of abilities
for the educational choice of women who turn out to have high-cost kids corresponding to
the case of perfect and imperfect information, i.e.
 (1 + ) +  +  (1 − )(1 − )  (1 + ) +  (1 − ) +  +  (1 − )(1 − )
−


(27)
which can be rewritten as
∙
 +  (1 − )(1 − ) − 
(1 − )

¸
(28)
while, similarly, the loss, which is equal to the difference in the threshold of abilities for
the educational choice of women who turn out to have low-cost kids corresponding to the
case of imperfect and perfect information, can be written as
¸
∙
 +  (1 − )(1 − ) − 


Thus, if  
1
2
(29)
there is a positive gain of the number of skilled individuals in the
case of imperfect information. Remembering also that the larger number of individuals
who invest in education thanks to imperfect information are high-ability types, imperfect
information increases the level of output in the economy also in this case.


(1+)++ (1−)(1−)
, while in a context of imperfect


+ (1−)(1−)
. Since the second threshold is larger than the


cost kids invest in education if  ≥
information
this happens if  ≥
first one we



(1+)
 (1−)

+
have less educated women with low-cost kids in a context of imperfect information. .
8
In a context of perfect information women with low-cost kids invest in education if  ≥


(1+)++ (1−)(1−)
, while


 (2+−)++ (1−)(1−)

 It


in the context of imperfect information the threshold is:

≥
is easy to check that the second threshold is smaller than the first one,
thus for a larger set of abilities women with high-cost kid invest in education in the presence of imperfect
information.
9
The threshold of abilities such that women with ability higher than this become educated is larger
in case of imperfect information (
 (2+)
(  ).

 (2+−)++ (1−)(1−)

)


18
than in the context of perfect information
Finally, if women with low-cost kids work, either as skilled or unskilled, while women
with high-cost kids never work (case 3b), the increased number of skilled women who turns
out to have high-cost kids, due to imperfect information, does not translate into output.
Moreover, since the set of women who become skilled and turn out to have a low-cost kid
is restricted, although women with a low-cost kid work also as unskilled, they obtain a
lower salary. Thus, in this case imperfect information reduces total output in the economy.
4
The role of government: A tax-transfer scheme
In this section we study the introduction of a tax-transfer scheme targeted to working
women and how it affects the education and participation decision. For a working woman,
the total cost of buying child care on the market is reduced from  to  (1 − ) with
 ∈ (0 1). The subsidy  to working women is financed by a proportional tax  on wages
paid by all workers, men and women. We assume that the scheme is balanced, that is
⎤
⎡
R
R
 (∗ ) +  ∗  () + (1 − ) ∗  ()+


⎦
⎣
(30)
R
 (∗ )
)
+

 ∗  () + (1 − ) (∗




#
"
Z 
Z 



∗

∗
() + (1 − )
() +  ( ) + (1 − ) ( )
=  
∗

∗

where ∗
 ( =  ) is the threshold level of ability which identifies the participation
decision for skilled women with kid of cost-type  (which takes a different value in each
specific scenario, see sections from 2.2.1 to 2.2.3), and ∗ is the threshold level of ability
such that men with abilities above this threshold invest in education10 . The budget constraint above is a general formula encompassing all the scenarios identified. Depending
on the participation decision arising in each of the different scenarios some of the terms
disappear as women who do not work do not pay the tax and do not receive the subsidy.
Considering now the participation decision in the presence of the tax-transfer scheme.
A skilled woman with ability  and with a kid of cost  ( =  ) works if:
  (1 −  ) −  (1 − )(1 − ) ≥  −  (1 − )
10
This threshold is given by ∗ =


(2+− )
See

1−

below Eq. 35.
19
(31)
which delivers the following threshold level of ability:
 ≥
 +  (1 − )(1 −  − )
 (1 −  )
(32)
Assume that 1 −  −   0: as in the case where the tax transfer scheme is absent,
it still holds that a high cost kid requires a higher level of ability for a woman to find it
convenient to work.
Similarly, the unskilled continue working in the second period if the following condition
is satisfied:
 (1 −  ) −  (1 − )(1 − ) ≥  −  (1 − )
(33)
i.e. if the wage of the unskilled is larger than the following threshold:
 ≥
 +  (1 − )(1 − )

1−
(34)
Again, this condition is more easily satisfied for women with a low-cost kid than for
those with a high-cost one.
As in the case with  =  = 0, there are three possible scenarios, each of which delivers
different thresholds of abilities for the education decision. The payoffs and the threshold
level of ability for the education decision in each scenario are summarized in Figure 4. We
here comment only some major findings.
Similarly to what we have obtained in Section 2, in the first scenario (Case 1) all
women work and their education decision depends on the wage premium and on the tax
rate, besides the interest rate. More precisely, all women with ability above the following
threshold invest in education:
 ≥
 (2 +  −  )

1−
(35)
Comparing ( 35) and (12) we find that when all women work, the presence of the taxtransfer scheme increases the threshold level of ability associated with the education decision, i.e. it reduces the set of ability types that become skilled. Notice also that in
this case a marginal increase of the tax rate reduces the share of skilled women. This is
because incentives to education goes hand in hand with incentives to work. As in this
case all women are already working, the distortionary effects of the tax dominate and a
20
further use of this policy has no efficacy, since there is no need to use a higher tax rate to
increase female participation.
In the second scenario, the results that we have obtained when  =  = 0 are confirmed.
In particular, in case 2b unused human capital emerges due to the presence of imperfect
information: women who have invested in education under the imperfect information
about the cost of their kid may find it convenient not to work if this cost turns out to be
high. This happens also in the third scenario, case 3b.
What is the role of this policy? First, we show the implications of this policy in terms
of female employment and education. In particular, what are the effects of this policy on
the presence of unused human capital and on the overall level of output? The following
propositions summarize the results.
Proposition 3 If  (1 − ) [ (1 − ) − ] +   0 and  (1 − ) −    0, the
tax-transfer scheme makes it more likely that a scenario where women always work independently of their ability and kid’s cost type is realized.
Proof: See the appendix
The result stated above indicates that the economy is more likely to replicate the
scenario described in Case 1 when a tax-transfer scheme is present. The proposition
confirms that we have identified a policy which increases female participation, i.e. it
increases the likelihood that a woman always works. Moreover, it is intuitive and easy to
check that, when all women have the expectation that they will always work irrespective
of the idiosyncratic ability and cost type, incentives to invest in education are maximised,
that is, the threshold level of ability such that women find it convenient to invest in
education is the lowest. As a consequence, this policy also delivers the highest number
of skilled women, thus having positive repercussions on the level of human capital in the
economy. The total amount of output produced in the economy when all women work is
also maximum.
Propositions 2 suggests that the tax-transfer scheme makes the realization of the first
scenario more likely. The following corollary emphasizes one of the consequences of this
result, i.e. that the policy reduces the presence of unused human capital (Case 2b and
21
3b).
Corollary 4 The tax-transfer scheme may reduce the emergence of unused human capital.
Proof: See appendix
Focusing on cases 2b and 3b where unused human capital emerges in the presence of
imperfect information, the proposition suggests that while imperfect information cannot be
completely eliminated, its effect of generating unused human capital can be attenuated by
the government intervention. The tax-transfer scheme can be interpreted as an insurance
device against the risk of having high-cost kids. In the presence of the government policy,
the difference between having a high-cost kid and a low-cost kid is reduced and so is
the possibility that women who have invested in education do not work after the child
birth. We may also argue that the absence of such a policy can be interpreted as the real
responsible for non participation. Notice however that this potential gain in the output
of skilled women with a high-cost kid associated to the policy does not necessarily imply
that the total amount of output is larger when the government intervenes. This is because
the tax-transfer scheme may enlarge the loss of output due to a reduction in the number
of skilled working women with a low-cost kid associated with the imperfect information
with respect to the perfect information benchmark11 . In other words, while for skilled
women with high-cost kid the policy works as an insurance, for skilled women with a lowcost kid it magnifies the impact of imperfect information as a deviation from the perfect
information benchmark.
The next proposition summarizes the role of the policy in the other cases.
11
In case 2b this loss is given by the difference in the threshold level of ability characterizing the
education decision of women with a low-cost kid in the case of imperfect and perfect information:

 
 (1+)
+ (1−)(1−)
−
 (1−) +




 

 (1+)
 (1+) 
=  1−
+ + (1−)(1−)
, which, in presence of the tax transfer scheme enlarges to



 (1+)


 (1− ) 1− 



The same emerges in case3b, where the difference has to be interpreted as due to an increase
of the number of women with a low-cost kid working as unskilled rather than as skilled in case of imperfect
information.
22
Proposition 5 When imperfect information does not create unused human capital, if
 (1 − ) [ (1 − ) − ] +   0 and    (1 − ) the tax-transfer scheme reduces the
consequences of the imperfect information, by inducing an equilibrium closer to the perfect
information benchmark.
Proof: See appendix.
The proposition shows that in case 2a and 3a the role of government is to bring the
imperfect information equilibrium closer to the perfect information one. Again, the consequences on the level output are not unambiguous. The tax-transfer scheme, by attenuating
the impact of imperfect information, may limit the benefits of imperfect information, in
terms of output, arising for instance when skilled women work and unskilled do not work
(case 2a), while it may also limit the loss of output arising when women with a high-cost
kid do not work and those with a low-cost kid work (case 3a).
Finally, we may ask whether this policy is welfare improving. Given that in our model
utility is defined by the consumption possibilities, it is immediate to show that the taxtransfer scheme may increase the utility of the women who work and of the fathers of their
kids. The utility of women who do not work is not affected by the presence of the tax
scheme, while the utility of men who have a kid with a non working woman decreases.
Thus, the scheme may increase the overall welfare if the gains obtained by the working
women and the fathers of their kid is large enough to overcome the loss of the men who
have a kid with non working women.
5
Concluding Remarks
In a context of imperfect information on the time required to provide child care to their
kid, women make education decisions. Imperfect information may induce a waste of talent
since women who have invested in education but turn out to have high-cost kids may find
it convenient not to work. However, imperfect information may also be associated with
a higher level of output if, in the presence of uncertainty, high ability working women
who turn out to have high-cost kids are induced to acquire education, an investment they
would not have performed had they known from start the high cost of their kid.
23
We have studied the introduction of a tax-transfer policy and shown that its presence
makes it more likely that women work irrespective of their idiosyncratic ability and cost
type, strengthening the incentives to invest in education. In particular, the government
intervention, by reducing the costs of buying child care on the market, diminishes the
chances of observing a waste of talent of skilled women who turn out to have a high-cost
kid and it induces them to work and realize on the job the return of their educational
investment.
Our model can be extended in several directions. First, in a political economy context,
we may ask whether the policy that we have introduced is politically feasible. In an
overlapping generation framework, individuals, men and women, live three periods: in the
first they are young and they do not take any decision; in the second they make their
education investment or work as unskilled; at the beginning of the third period they have
a kid and they make their participation decision. Each time two persons are alive, a man
and a woman and two kids are generated, so that the population growth rate is zero. The
education decision is done under imperfect information and under expectations on the
level of taxation which will be in place in the next period. We will look at fixed point
equilibrium such that the expected and the actual level of tax are equal. Two candidates
for the equilibrium are: a zero tax rate and a positive tax rate. If individuals expect
the tax rate to be zero, it is more likely that women will not work, thus decreasing their
incentive to invest in education. Non-working women and the father of their kids will
obviously vote for a zero level of the tax rate, thus reinforcing their expectation. On the
contrary, if individuals expect the tax-transfer scheme to be in place, women are induced
to work and to invest in education, and they will vote for a positive tax rate, together
with the father of their kids.
Second, as suggested by the figures shown in the introduction, cross-country differences
in child-care policies may be associated with differences in female educational and employment outcomes. An empirical analysis founded on microdata would be particularly useful
to assess the relationship between the presence of institutions and the female education
and participation decisions.
Finally, in terms of policy implications, our analysis suggests that promoting an institu24
tional environment in favor of women education and employment may generate economic
benefits. The Lisbon strategy which identifies target values for European countries on
female employment and on the institutional measures which may favor it (child care services, parental leaves etc.) goes in this direction. In particular, we have stressed that
incentives to employment may also reinforce - through their effect on expectations - educational investments. Focusing on them may represent a good strategy to increase the
human capital and output of any country, a priority especially in countries where human
capital is scarse.
6
Appendix
6.1
Proof of proposition 2
We start from case 2c where a woman never works. If she is skilled, she does not work
for  ≤
+ (1−)(1−)

or, in presence of the tax scheme,  ≤
+ (1−)(1−−)
.
 (1− )
If
 (1 − ) [ (1 − ) − ] +   0, it is  (1 − ) [ (1 − ) − ] +   0 which guarantees
that
+ (1−)(1−−)
 (1− )

+ (1−)(1−)
,

i.e. the tax-transfer scheme makes the realization
of this case more difficult. In particular, it increases the likelihood that a skilled woman
with a low-cost kid works. This corresponds to case 2b.
Now, in case 2b a woman invests in education if  ≥
 (1+)

 (1−)
+
+ (1−)(1−)


+ (1−)(1−)

 (1+)

 (1−)
 (1−)(1−)
+ +

. Since
a skilled woman with a low-cost kid always
works. This is not necessarily true for a skilled woman with a high-cost kid. In fact, if
¡
¢
 (1+)



+ + (1−)(1−)
 + (1−)(1−)
, i.e.  (1+)   −  (1−)(1−)(1−),
 (1−)





a skilled woman with a high cost kid does not work. It is easy to find that, when the tax¢
¡
transfer scheme is introduced, this condition becomes:  (1 + )   −  (1 − )(1 −
 − )(1 − ), which is more difficult to be satisfied for a given level of  . Thus, the
presence of the tax-transfer scheme increases the likelihood that a skilled woman even with
a high-cost kid works. This corresponds to case 2a.
In case 2a an unskilled woman never works, since    +  (1 − )(1 − ). In
presence of the tax-transfer scheme, this condition becomes  
+ (1−)(1−−)
1−
which,
for  (1 − ) [ (1 − ) − ] +   0 is more difficult to be satisfied than the first one.
25
Thus, the policy increases the likelihood that even an unskilled woman works. When a
woman always works we are in case 1.
Now we start from case 3b, where a woman works only if she has a low-cost kid. She
invests in education if  ≥
 [1++(1−)]

.
 (1−)
If  [1 +  + (1 − )]−(1−)− (1−)(1−
)(1 − )  0 a skilled woman who turns out to have a high-cost kid does not work. In
presence of the tax scheme, this condition becomes  [1 +  + (1 − )]− (1−) −(1−
)− (1−)(1−)(1−)+ (1−)(1−)  0If  (1−)−   0 the tax-transfer
scheme makes it more difficult the realization of this case, by increasing the likelihood that
even a skilled woman with high-cost kid works. This corresponds to case 3a. In case 3a a
woman does not work only if she is unskilled with a high-cost kid:   + (1−)(1−)
In presence of the tax scheme she does not work if  
 (1 − ) [ (1 − ) − ] +   0 it is
+ (1−)(1−−)
1−
+ (1−)(1−−)
.
1−
Since for
  +  (1 − )(1 − ) the
policy makes it more likely that even an unskilled woman with a high-cost kid works, i.e.
increases the possibility that a woman always works (case 1).
6.2
Proof of corollary 3
Unused human capital emerges in case 2b and 3b.
Starting from case 2b, we may have a non working skilled woman with a high-cost
kid if the threshold level of ability which defines the education decision is lower than
¡
¢
the one that defines the participation decision, i.e. that  (1 + )   −  (1 −
)(1 −  − )(1 − ). In presence of the tax transfer scheme this condition becomes
¡
¢
 (1 + )   −  (1 − )(1 −  − )(1 − ), which is more difficult to be satisfied for
a given level of  . Similarly, in case 3b the same condition reads  [1 +  + (1 − )] −
(1−)− (1−)(1−)(1−)  0, which, in presence of the tax-transfer scheme becomes
 [1 +  + (1 − )]− (1−) −(1−)− (1−)(1−)(1−)+ (1−)(1−)  0,
which, again, is more difficult to be satisfied.
6.3
Proof of proposition 4
Case 1 and 2c are not interesting, since imperfect information has no consequences.
26
Let first analyze case 2a. Imperfect information is associated with a loss in terms
of human capital and output due to individuals with low-cost kids that do not invest in
education and a gain related to individuals with high-cost kids who would indeed invest.
If    (1 − ), both this loss and the gain are smaller in presence of the tax scheme.
In fact, the loss related to women with a low-cost kid is equal to the difference in the
threshold levels of abilities that define their education decision in case of imperfect and



(1−)(1−)(− )
−  (1+)++(1−)(1−)
=
perfect information:  (1+)++(1−)(1−)




(1−)(1−−)(− )
which, in presence of the tax scheme, becomes
i.e., if    (1 − )
 (1− )

it is smaller. Similarly, the gain of imperfect information related to skilled women with
(1−)(1−)( −)
which, in presence of the tax scheme and
a high-cost kid depends on


(1−)(1−−)( −)
for    (1 − ) reduces to
The loss related to women with a low (1− )

cost kid is equal to the difference in the threshold levels of abilities that define their

−
education decision in case of imperfect and perfect information:  (1+)++(1−)(1−)




(1−)(1−)(− )
 (1+)++(1−)(1−)
=
, which, in presence of the tax scheme becomes



(1−)(1−−)(− )
i.e., if    (1 − ) it is reduced. Similarly, the gain of imperfect
 (1− )
(1−)(1−)( −)
information related to skilled women with a high-cost kid depends on

(1−)(1−−)( −)
which, in presence of the tax scheme and for    (1 − ) reduces to

 (1− )

Moving to case 3a where unskilled women with a low-cost kid are the only ones who
do not work, the presence of imperfect information generates a gain with respect to the
perfect information benchmark in terms of more working skilled women with a high-cost
kid and a loss in terms of less working women with a low-cost kid. If    (1 − ) and
 (1 − ) [ (1 − ) − ] +   0 both the gain and the loss are reduced by the presence
of the tax-transfer scheme. In fact the loss related to women with a low-cost kid is equal
to the difference in the threshold levels of abilities that define their education decision



− (2+)
=
in case of imperfect and perfect information:  (2+−)++(1−)(1−)





 [+(1−)(1−) − ]
, which, in presence of the tax scheme, if  (1 − ) [ (1 − ) − ] +

 (1− )
 [+(1−)(1−−) −
]
Similarly, the gain of imperfect information
  0 reduces to


(1−)[+ (1−)(1−)−
]
related to skilled women with a high-cost kid depends on
which,



(1−)[+ (1−)(1−−)−
]
in presence of the tax scheme and for    (1−) reduces to
Again,

 (1− )

27
the tax transfer scheme leads to an equilibrium closer to the perfect information benchmark.
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[11] Iyigun M. and Walsh R., (2007b), Building the Family Nest: Premarital Investments,
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pages 79-108 National Bureau of Economic Research.
29
Figure 3
Case 1 : wuF > η + ρ H (1 – γ) (1 – η )
work
ρH
π
no work
work
Skilled
1- π
ρL
no work
work
Unskilled
π
work
Educated if
αi wFs - ρL ( 1 – γ)
η – ρL ( 1 – γ) η
wFu– ρH ( 1 – γ)
η - ρH ( 1 – γ) η
wFu – ρL ( 1 – γ)
ρL
no work
1( )
η - ρH ( 1 – γ) η
ρH
no work
1- π
αi wFs – ρH ( 1 – γ)
η – ρL ( 1 – γ) η
αi ≥ wFu (2 + r )
wFs
2a
( )
Educated if
αi ≥ wFu (1+ r)+ η+(1-γ)(1 - η) ρL
wFs
Case 2 : wuF < η + ρ L (1 – γ) (1 – η )
Work if
work
ρH
π
no work
αi wFs – ρH ( 1 – γ)
αi ≥
η+(1-γ) (1 - η) ρH
wFs
η - ρH ( 1 – γ) η
2b ( )
work
Skilled
1- π
ρL
no work
work
Unskilled
π
Educated if
η – ρL ( 1 – γ) η
αi ≥ __wFu (1 + r)____
wFs (1 – π)
wFu – ρH ( 1 – γ)
ρH
+
Work if
no work
work
1- π
αi wFs - ρL ( 1 – γ)
η - ρH ( 1 – γ) η
no work
L
η – ρ ( 1 – γ) η
η + ρL (1- γ) (1 - η)
wFs
η+(1-γ) (1 - η) ρL < αi < η+(1-γ)(1 - η) ρH
wFs
wF s
wFu – ρL ( 1 – γ)
ρL
+
2c (
)
Nobody Educated
Don’t work if
αi
< η+(1-γ)(1- η) ρL
wFs
Case 3 :
η + ρ L (1 – γ) (1– η ) < wuF < η + ρ H (1 – γ) (1 – η )
work
no work
work
Skilled
1- π
ρ
work
Unskilled
η - ρH ( 1 – γ) η
)
αi ≥ wFu (1+ r +( 1 – π))
wFs
αi wFs - ρL ( 1 – γ)
η – ρL ( 1 – γ) η
wFu – ρH ( 1 – γ)
Work if
αi ≥
ρH
no work
+
π η + π (1-γ) (1 - η) ρH
wFs
L
no work
π
3a (
Educated if
ρH
π
αi wFs – ρH ( 1 – γ)
η+(1-γ)(1 - η) ρH
wFs
3b( )
η - ρH ( 1 – γ) η
Educated if
work
1- π
wFu – ρL ( 1 – γ)
ρL
no work
η – ρL ( 1 – γ) η
αi ≥ __wFu (1 + r +( 1 – π))
wFs ( 1 – π)
Don’t work if
αi <
η+(1-γ)(1 - η) ρH
wF s
Figure 4
Case 1 : wuF > η + ρ H (1 – γ) (1 – φ – η )
(1 – τ)
work
ρH
π
no work
work
Skilled
1- π
ρL
no work
work
Unskilled
π
work
Educated if
αi wFs (1- τ) - ρL ( 1 – φ) ( 1 – γ)
η – ρL ( 1 – γ) η
wFu (1- τ) – ρH ( 1 – φ) ( 1 – γ)
η - ρH ( 1 – γ) η
wFu (1- τ) – ρL ( 1 – φ) ( 1 – γ)
ρL
no work
1( )
η - ρH ( 1 – γ) η
ρH
no work
1- π
αi wFs (1- τ) – ρH ( 1 – φ) ( 1 – γ)
η – ρL ( 1 – γ) η
αi ≥ wFu 2 + r - τ
wFs
1-τ
2a
( )
Educated if
αi ≥ wFu (1+ r)+ η+(1-γ)(1 - φ - η) ρL
wFs (1 – τ)
Case 2 : wuF < η + ρ L (1 – γ) (1 – φ – η )
(1 – τ)
Work if
work
ρH
π
no work
work
Skilled
1- π
ρL
no work
work
αi wFs (1- τ) – ρH ( 1 – φ) ( 1 – γ)
αi ≥
η+(1-γ)(1 - φ - η) ρH
wFs (1 – τ)
η - ρH ( 1 – γ) η
i
F
L
α w s (1- τ) - ρ ( 1 – φ) ( 1 – γ)
η – ρL ( 1 – γ) η
wFu (1- τ) – ρH ( 1 – φ) ( 1 – γ)
2b ( )
Educated if
αi ≥ __wFu (1 + r)____
wFs (1 – τ)(1 – π)
+
Work if
+
η + ρL (1- γ) (1 - φ - η)
wFs (1 – τ)
H
Unskilled
π
ρ
no work
work
1- π
H
η - ρ ( 1 – γ) η
wFu (1- τ) – ρL ( 1 – φ) ( 1 – γ)
ρL
no work
η+(1-γ)(1-φ -η) ρL < αi < η+(1-γ)(1-φ- η) ρH
wFs (1 – τ)
wFs (1 – τ)
L
η – ρ ( 1 – γ) η
2c (
)
Nobody Educated
Don’t work if
αi
< η+(1-γ)(1-φ -η) ρL
wFs (1 – τ)
Case 3 :
η + ρ L (1 – γ) (1 – φ – η ) < wuF < η + ρ H (1 – γ) (1 – φ – η )
(1 – τ)
(1 – τ)
work
no work
work
Skilled
1- π
ρL
no work
work
Unskilled
π
3a (
)
Educated if
ρH
π
αi wFs (1- τ) – ρH ( 1 – φ) ( 1 – γ)
η - ρH ( 1 – γ) η
αi ≥ wFu (1+ r +( 1 – π) (1 – τ ))
wFs (1 – τ )
αi wFs (1- τ) - ρL ( 1 – φ) ( 1 – γ)
L
η – ρ ( 1 – γ) η
wFu (1- τ) – ρH ( 1 – φ) ( 1 – γ)
+ π η + π (1-γ) (1 - φ - η) ρH
Work if
αi ≥
ρH
no work
wFs (1 – τ )
η+(1-γ)(1 - φ - η) ρH
wFs (1 – τ)
3b( )
η - ρH ( 1 – γ) η
Educated if
work
1- π
wFu (1- τ) – ρL ( 1 – φ) ( 1 – γ)
ρL
no work
η – ρL ( 1 – γ) η
αi ≥ __wFu (1 + r +( 1 – π)(1 – τ))
wFs (1 – τ) ( 1 – π)
Don’t work if
αi <
η+(1-γ)(1 - φ - η) ρH
wFs (1 – τ)