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A Structural Approach to Assessing Retention Policies in Public Schools

Celia P. Vera

Abstract

One out of five entering public school teachers leave the field within the first 4 years. Despite that the presence of a newborn child is the single most important determinant of exits of female teachers, retention policy recommendations rely on models that take children as predetermined. This article formulates and estimates a structural dynamic model that explicitly addresses the interdependence between fertility and labor force participation choices. The model with unobserved heterogeneity in preferences for children fits the data and produces reasonable forecasts of labor force attachment to the teaching sector. Structural estimates of the model are used to predict the effects that wage increases and reductions in the cost of childcare would have on female teachers’ employment and fertility choices. The estimates unpack important features of the interdependence of fertility and labor supply and contradict previous studies that did not consider the endogeneity between these two choices.

JEL Classification: J13; J44; J45; C61

Appendix

A Model Description

A.1 Dynamic Choice

I now describe in more detail the dynamic choices that individuals make. Section 2 presents the generic Bellman equation:

Vjt(Ωt)=Ujt(Ωt)+δEtVt+1(Ωt+1|Ωt,djt=1),t<A0+TVj,A0+T(ΩA0+T)=Uj,A0+T(ΩA0+T),

where j denotes the joint option of employment and fertility and Ωt denotes the state space defined as Ωt=(Kt1,A0,jt1,ω1I(et1=1),ω2I(nt1=1)).

As mentioned in Section 2, there are six possible alternatives. The value functions of each of these alternatives are[34]:

(9)VT,NB(Ωt)=UtT,NB+δEtVt+1(Ωt+1),
(10)VT,B(Ωt)=UtT,B+δEtVt+1(Ωt+1P),
(11)VNT,NB(Ωt)=UtNT,NB+δEtVt+1(Ωt+1),
(12)VNT,B(Ωt)=UtNT,B+δEtVt+1(Ωt+1P),
(13)VH,NB(Ωt)=UtH,NB+δEtVt+1(Ωt+1),
(14)VH,B(Ωt)=UtH,B+δEtVt+1(Ωt+1P).

The individual maximizes these conditional value functions in sequence. I denote these conditional value functions by indexing them with B for birth and NB if the woman does not give birth. I also index them with T for teaching, NT for nonteaching and H for out of the workforce. The subscript P indicates that the woman gives birth in t, so the number of children in the future state space is increased by one.

At the beginning of a period, women take as given their age, occupation, number of children, and their labor supply in the previous period. Women then decide to conceive a child or not. Women next decide how much to consume. Once fertility and consumption choices have been made, individuals observe shocks to labor supply, which consist of job offer arrivals. These shocks determine the labor status at the beginning of the next period.

I present below the employment-specific value functions. In all cases, the tilde ()˜ in the future state space (Ω˜t+1PorΩ˜t+1) describes the future state space when the individual accepts the job offer from the alternative sector.

A.2 Value of Teaching

I start with the value of teaching and conceiving a child. A woman working in a teaching job receives a job offer from the nonteaching sector. If she accepts it, she switches to the nonteaching sector. If she rejects it, she can either keep her current job for the next period or she can drop out of the workforce. The value is written as:

VT,B(Ωt)=UtT,B+δE1max.

The first term consists of the current utility of consumption, leisure, and children, as described in eq. 2. The second term is the future flow of utility, defined as:

E1max=Etmax[VT(Ωt+1P),VNT(Ω˜t+1P),VH(Ωt+1P)].

The woman compares the future utility flows of keeping her current teaching job, accepting the job offer from the nonteaching sector, and dropping out of the workforce, and chooses the sector that provides the highest utility. The employment decision is made conditional on having an additional child. That is, the number of children in the future state space is increased by one.

The value of teaching and not giving birth is defined as:

VT,NB(Ωt)=UtT,NB+δE2max,

where

E2max=Etmax[VT(Ωt+1),VNT(Ω˜t+1),VH(Ωt+1))].

Since there is no birth in period t, the individual starts the next period with an updated state space Ωt+1, where all the state variables but the number of children have been updated.

A.3 Value of Nonteaching

When working in the nonteaching sector, a woman receives a job offer from the teaching sector. If she accepts the job offer, she becomes employed as a teacher. If she rejects it, she can either keep her current job or drop out of the workforce. The value of being in nonteaching and giving birth is:

VNT,B(Ωt)=UtNT,B+δE3max.

The term E3max is defined as:

E3max=Etmax[VT(Ω˜t+1P),VNT(Ωt+1P),VH(Ωt+1P)].

The woman compares the future utility flows of keeping her current nonteaching job, accepting the job offer from the teaching sector, and dropping out of the workforce, then chooses the sector with the highest utility. The employment decision is made conditional on a future state space where the number of children is increased by one.

The value of nonteaching and not giving birth is:

VNT,NB(Ωt)=UtNT,NB+δE4max,

where the term E4max is defined as:

E4max=Etmax[VT(Ω˜t+1),VNT(Ωt+1),VH(Ωt+1)].

Since the woman chooses not to give birth in t, the future state space Ωt+1 is updated but the number of children remains the same.

A.4 Value of Being Out of the Workforce

When a woman is out of the workforce, she receives a job offer from the teaching sector with probability ρ and a job offer from the nonteaching sector with probability 1ρ. The value of being out of work and giving birth is modeled as:

VH,B(Ωt)=UtH,B+δ[ρE5max+(1ρ)E6max],

where

E5max=Etmax[VT(Ω˜t+1P),VH(Ωt+1P)],

and

E6max=Etmax[VNT(Ω˜t+1P),VH(Ωt+1P)].

The woman compares the utility flows of remaining out of the workforce and accepting the job offer from the corresponding sector. As in VT,BandVNT,B, the employment decision is made conditional on a future state space where the number of children is increased by one.

The value of being out of the workforce and not giving birth is modeled as :

VH,NB(Ωt)=UtH,NB+δ[ρE7max+(1ρ)E8max],

where

E7max=Etmax[VT(Ω˜t+1),VH(Ωt+1)],

and

E8max=Etmax[VNT(Ω˜t+1),VH(Ωt+1)].

Since there is no birth occurs in t, all variables but the number of children in the future state space are updated.

A.5 Fertility Decision

The decision of whether to give birth or not is taken as:

VT(Ωt)=max[VT,B(Ωt),VT,NB(Ωt)],VNT(Ωt)=max[VNT,B(Ωt),VNT,NB(Ωt)],VH(Ωt)=max[VH,B(Ωt),VH,NB(Ωt)].

The decision to give birth, noted by kt in Section 2, is the arg max of the expressions above.

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Published Online: 2019-04-20

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