Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter May 21, 2014

A Single Parent’s Labor Supply: Evaluating Different Child Care Fees within an Intertemporal Framework

  • Junichi Minagawa EMAIL logo and Thorsten Upmann

Abstract

In this paper, we present a model of a one-parent, one-child household where parental decisions on labor supply, leisure, and the demand for parental and public child care are simultaneously endogenized and intertemporally determined. We characterize the path of the optimal decisions and investigate the effects of various public child care fees and of the quality of public child care services on the parent’s time allocation and the child’s performance level. Our results show that different public child care policies may induce substantially diverging effects and reveal that each policy frequently faces a trade off between an encouragement of labor supply and an enhancement of the child’s performance. In addition, we find that, from an efficiency perspective, an income-based fee levied on public child care services is dominated by both a flat fee and a use-based fee system.

JEL Classification: D91; H24; H42; J13; J22

Acknowledgments

Valuable suggestions by an anonymous referee are gratefully acknowledged. This work has been initiated during a visit of Junichi Minagawa at the University of Bielefeld in 2005/06, which was supported by the Bielefeld Graduate School of Economics and Management and financed by the German Academic Exchange Service (DAAD). The author gratefully acknowledges the support of both institutions. Financial support from the Japan Society for the Promotion of Science is also highly appreciated.

Appendix A

Problem (c.f. Seierstad and Sydsæter (1987, 275, 291, 390)). Let χ(t)(χ1(t),,χn(t))n be an state vector and ν(t)(ν1(t),,νr(t))r be an control vector. Consider the following problem,

[15]maxt0t1ζ0(χ(t),ν(t),t)dt+Ψ(χ(t1)),(t0,t1fixed)

subject to the vector differential equation and the initial condition

[16]χ˙(t)=ζ(χ(t),ν(t),t),χ(t0)=χ0,(χ0fixed),

the terminal conditions

[17]χi(t1)χi1,i=1,,n,(χi1allfixed),

and subject to the constraints

[18]ξk(ν(t),t)0,k=1,,s

for all t[t0,t1].

We assume that ζi(χ,ν,t), ζi(χ,ν,t)/χj, and ζi(χ,ν,t)/νk are continuous with respect to all the n+r+1 variables for i=0,1,,n; j=1,,n; k=1,,r, ξk(ν(t),t) and ξk(ν(t),t)/νj are continuous with respect to all the r+1 variables for j=1,,r; k=1,,s, and Ψ is a C1-function. We call (χ(t),ν(t)) an admissible pair if ν(t) is piecewise continuous, χ(t) is continuous and piecewise continuously differentiable such that [16]–[18] are satisfied.

Let ν(t) be an optimal control and define the two sets It and It+ by

[19]It{k|ξk(ν(t),t)=0,k=1,,s},
[20]It+{k|ξk(ν(t+),t)=0,k=1,,s},

where ν(t) denotes the left-hand limit of ν(t) at t and ν(t+) is the corresponding right-hand limit. Then the constraint qualification is as follows: For every t[t0,t1],

(C1) If It, the matrix {ξk(ν(t),t)/νi},kIt,i=1,,r has a rank equal to the number of elements in It.

(C2) If It+, the matrix {ξk(ν(t+),t)/νi},kIt+,i=1,,r has a rank equal to the number of elements in It+.

(C3) If t=t0, drop (C1), if t=t1, drop (C2).

Theorem 1 (The maximum principle (c.f. Seierstad and Sydsæter (1987, 276, 291, 396)). Let(χ(t),ν(t))be an admissible pair which solves problem [15]–[18]. Assume that the constraint qualification (C1)–(C3) is satisfied. Then there exist numbersπ0andρ1,,ρn, vector functionsπ(t)(π1(t),,πn(t))andμ(t)(μ1(t),,μs(t)), whereπ(t)is continuous and piecewise continuously differentiable andμ(t)piecewise continuous, such that for allt[t0,t1]:

  1. (π0,ρ1,,ρn)(0,0,,0).

  2. H(χ(t),ν(t),π(t),t)H(χ(t),ν,π(t),t)for allνsuch thatξk(ν,t)>0,k=1,,swhereH(χ,ν,π,t):=π0ζ0(χ,ν,t)+i=1nπiζi(χ,ν,t).

  3. j=1r(L/νj)(νjνj(t))0for allν=(ν1,,νr)U˜(t), whereL(χ,ν,π,μ,t):=H+k=1sμkξk(ν,t), L/νjmeansL/νjevaluated at(χ(t),ν(t),π(t),μ(t),t), andU˜(t){ν|j=1r(ξk/νj)(νjνj(t))0forallk,ξk(ν(t),t)=0}.

  4. μk(t)0(=0ifξk(ν(t),t)>0),k=1,,s.

  5. π˙(t)=L/χi,i=1,,nexcept at points of discontinuity ofν(t), whereL/χimeansL/χievaluated at(χ(t),ν(t),π(t),μ(t),t).

  6. π0=0orπ0=1.

  7. πi(t1)=π0(Ψ(χ(t1))/χi)+k=1nρkwhereρk0(=0ifχi(t1)>0), i=1,,n.

Theorem 2 (Filippov-Cesari (c.f. Seierstad and Sydsæter (1987, 285, 400)). Consider the problem [15]–[18]. Assume that
  1. There exists an admissible pair(χ(t),ν(t)).

  2. The setN(χ,t){(ζ0(χ,ν,t)+δ,ζ(χ,ν,t))|δ0,ξ(ν,t)0}is convex for allχand allt[t0,t1].

  3. There exists a numberκsuch that||χ(t)||κfor all admissible pairs(χ(t),ν(t)), and allt[t0,t1].

  4. There exists an open ballB(0,κ1){θr|||θ||<κ1}which, for allχwith||χ||κand allt[t0,t1], contains the setU(χ,t){ν|ξ(ν,t)0}.

Then there exists an optimal pair(χ(t),ν(t))(whereν(t)is measurable).

Proof of Proposition 1. First, we argue that an optimal solution must lie in the interior of its admissible domain, viz. in the interior of U{(d,h)|1d0,dh,h(βd+γ1d>0)/ωα}. The properties of the utility functions u and v, formulated in (i) and (ii), namely u(x) as x0+, vl(l,c) as l0+, and vc(l,c) as c0+, exclude x=0, l=0, and c=0 being optimal. It follows that d(t)=1 cannot be optimal since d(t)=1c(t)=0; and d(t)=0 is ruled out since d(t)=0c(t)=1l(t)=0; also, d(t)=h(t) is not optimal since d(t)=h(t)l(t)=0; finally, h(t)=(βd(t)+γ)/ωα cannot be optimal since h(t)=(βd(t)+γ)/ωαx(t)=0. Hence an optimal control (d(t),h(t)) belongs to the interior of U, that is, to U{(d,h)|1>d>0,d>h,h>(βd+γ)/ωα}.

Since d(t)=0 cannot be optimal, we simply drop the indicator function in the fee function, ϕ. Economically, this implies that the parent were to pay the fixed fees αM(t) and γ irrespective of whether or not public child care is used at all. The only effect of this modification is that it makes the sub-optimal choice d(t)=0 even more unattractive, and thus does not affect the optimal control, whatever it looks like.

With this pre-requisite, we now consider Theorem 1 (necessary conditions for optimality) for the control problem [6]–[11]. Taking account of the form [18], we write ξ1:=1d, ξ2:=d, ξ3:=dh, and ξ4:=h(βd+γ)/ωα. Since (d(t),h(t))U, it follows from (IV) that μi=0,i=1,2,3,4. Then, define the “current value” Hamiltonian by

[21]H(b,d,h,π):=π0u(ωαhβdγ)+v(dh,1d)+w(b)+πf(1d,qd)g(b).

We first show that π0=1. By b0>0, (iv), and (v), we have b(t)>0 for all t[0,T]. This implies, from (VII), ρ1=0. Therefore, by (I) and (VI), π0=1. Then (III) implies

[22]Hd=βu+vlvc+π(fc+qfy)g=0,
[23]Hh=ωαuvl=0.

This condition is sufficient for (II) if H(b,d,h,π)=π0uˆ(d,h)+w(b)+πfˆ(d,q)g(b) is concave in (d,h), where u^(d,h):=u(ωαhβdγ)+v(dh,1d) and fˆ(d,q):=f(1d,qd). We show that H(b,d,h,π) is strictly concave in (d,h). First, we obtain uˆdd=β2u′′+vll2vlc+vcc, uˆdh=ωαβu′′vll+vlc, and uˆhh=ωα2u′′+vll. By Assumptions (i) and (ii), we have uˆdd<0, uˆhh<0, and uˆdduˆhh(uˆdh)2=((ωαβ)2vll+ωα(ωαvcc2vlc(ωαβ)))u′′+vllvcc(vlc)2>0, which shows that uˆ is strictly concave. Second, we obtain fˆdd=fcc2qfcy+q2fyy. By Assumption (iv), fccqfcy0 and qfyyfcy0, and implying fˆdd0, and thus fˆ is concave in d. Last, with the above results, π0=1, and g(b)>0, it remains to be shown that π0.30 We claim that π(t)>0 for all t[0,T]. Suppose that π(t1)0 for some t1. It follows from (VII) with b(T)>0, π0=1, and (iii), that π(T)>0. Then, by continuity, there exists some t2 such that π(t2)=0 and π˙(t2)0. On the other hand, condition (V) can be written as

[24]π˙=(rf(1d,qd)g(b))πw(b).

This equation yields π˙(t2)=w(b(t2))<0, which is contradiction. Therefore π(t)>0 for all t, and we conclude that H(b,d,h,π) is strictly concave in (d,h).

Next, we show that the necessary conditions yield a unique candidate for (b(t),d(t),h(t)). To this end, define z(t)g(b(t))π(t), and Hz(b,d,h,z)H(b,d,h,z/g(b)). Then z(t)=g(b(t))π(t)>0. By z˙=g(b)b˙π+g(b)π˙ and eqs [7] and [24], we obtain z˙=rzw(b)g(b). Then, by the assumption in Proposition 1, we have z˙=rzC. On the other hand, multiplying both sides of the condition π(T)w(b(T))=0 by g(b(T)) we obtain z(T)w(b(T))g(b(T))=z(T)C=0. The solution to z˙=rzC with z(T)=C is given by

[25]z˜(t)=C((r1)er(Tt)+1)/r.

Recall that H(b,d,h,π), and thus Hz(b,d,h,z) is strictly concave in (d,h), and note that [22] and [23] can be written as

[26]Hdz=βu+vlvc+(fc+qfy)z=0,
[27]Hhz=ωαuvl=0.

Hence, for any given z, [26] and [27] has a unique solution (d,h). We thus write it as a function of z, that is, (d,h)=(dz(z),hz(z)), and, from the implicit function theorem, it is continuous and differentiable at every z. Substituting z=z˜(t) into the solution, we get a control (d(t),h(t))=(dz(z˜(t)),hz(z˜(t))). Then, by concavity of g and the standard existence theorem, the solution to the initial value problem [7] and [9] with the control d=d exists, and is denoted by b(t); and, by continuous differentiability of g and the usual uniqueness theorem, b(t)is unique.31 We can now conclude that the necessary conditions provide a unique candidate for (b(t),d(t),h(t)).

In order to show that this candidate is indeed optimal, we apply Theorem 2. We verify that the conditions (a)–(d) are satisfied: Set (d(t),h(t))(dˉ,hˉ)U. The solution of the initial value problem [7] and [9] with d=dˉ is denoted by bˉ(t). The existence of bˉ(t) is established by concavity of g and the standard existence theorem for ordinary differential equations. From Assumptions (iv) and (v) together with bˉ(t)>0 it follows that an admissible triple (bˉ(t),dˉ,hˉ) exists. As for the condition (b), we try an alternative route.32 Let η(χ,ν,t)f(1d,qd)g(b) and λ(t)0. Then, η and λ are continuous. Thus (b1) is satisfied. Let ζ0(χ,ν,t)(u(ωαhβdγ)+v(dh,1d)+w(b))ert and Ψ(χ)w(b(T))erT. Then, ζ0, Ψ, and η are non-decreasing in χb. Moreover, ζ0, ξk, and η are concave in ν(d,h). Thus (b2) is satisfied. The condition (c) follows from the concavity of g and the standard existence theorem for ordinary differential equations. It is clear that U(b,t) is bounded. We therefore conclude that, by Theorem 2, an optimal triple (b(t),d(t),h(t)) exists.33 Hence, the unique candidate produced by the necessary conditions is optimal.⃞

Proof of Corollary 1. In this case, with w=0, C must equal zero, implying z˜0. See eq. [25].⃞

Appendix B

Proof of Proposition 2. We first consider the optimal control (d(t),h(t)). To prove part (A), recall that d(t)=dz(z˜(t)) and h(t)=hz(z˜(t)), and then d˙*=dz(z˜*)z˜˙* and h˙*=hz(z˜*)z˜˙*. First, by eq. [25], z˜˙*=C(r1)er(Tt). Second, applying the implicit function theorem to eqs [26] and [27], we obtain

[28][dzhz]=(J*)1[Hdzz*Hhzz*]=(J*)1[(fcqfy)0],

where J1 is the inverse of the Jacobian matrix of (Hdz,Hhz) evaluated along the optimal path, whose elements are all negative since Hddz*=u^dd+z*f^dd<0, Hhhz*=u^hh<0, and |J*|=Hddz*Hhhz*(Hdhz*)2>0 by strict concavity, and Hdhz*=u^dh=ωαβuvll+vlc>0 by Assumption (ii). We next consider the optimal control (c(t),l(t)). The result for c(t) is immediate from the above result together with c(t)=1d(t) for all t. For the result for l(t) we use l(t)=d(t)h(t) for all t implying l˙*(t)=d˙*(t)h˙*(t)=(dz(z˜*)hz(z˜*))z˜˙*. Substituting from eq. [28] and using ωα>β and vlc=0 yields the result.⃞

Proof of Corollary 2. This can easily be seen by differentiating fcqfy with respect to time, yielding (fccqfcy)c˙(qfyyfcy)qd˙. By Assumption (iv) both bracket terms are non-positive, and by Proposition 2 the derivatives c˙ and d˙have opposite signs.⃞

Appendix C

Proof of Proposition 3. We first consider the optimal control (d(t),h(t)). Let J be given as in the proof of Proposition 2, and let Jα*:=J*|β=0,γ=0, Jβ*:=J*|α=0,γ=0, and Jγ*:=J*|α=0,β=0. The results follow from an application of the implicit function theorem to eqs [26] and [27]:

[29][dα*hα*]β=0,γ=0=(Jα*)1[Hdαz*Hhαz*]β=0,γ=0=(Jα*)1[0ωωαh*uωu],
[30][dβ*hβ*]α=0,γ=0=(Jβ*)1[Hdβz*Hhβz*]α=0,γ=0=(Jβ*)1[d*βuuωd*u],
[31][dγ*hγ*]α=0,β=0=(Jγ*)1[Hdγz*Hhγz*]α=0,β=0=(Jγ*)1[0ωu],
[32][dq*hq*]=(J*)1[Hdqz*Hhqz*]α=0,β=0=(J*)1[z*f^dq0],

where we treated α,β and γ as additional arguments of all functions.

We next consider the optimal control (c(t),l(t)). The proof follows immediately from the above result together with the relations c(t)=1d(t) and l(t)=d(t)h(t) for all t.⃞

Appendix D

Proof of Proposition 4. We show only Part (A). The optimal path of the state variable, b(t), is given implicitly as a solution of the initial value problem: b˙(t)=f(1d(t,α),qd(t,α))g(b(t)),b(0)=b0>0,t[0,T]. Since b(t) depends on α, we subsequently write b(t,α)to make this dependence explicit. We define fˉ(t,α):=f(1d(t,α),qd(t,α)). Let α0(0,1) and κ>0 be fixed constants. Then,

fˉ(s,α0+κ)g(b(s))fˉ(s,α0)g(b(s)),s[0,t]b(t,α0+κ)b(t,α0).

Here, g(b(s))>0 for all s due to Assumptions (iv) and (v) with b0>0. It thus follows that

(fˉ(s,α0+κ)fˉ(s,α0))/κ0,s[0,t](b(t,α0+κ)b(t,α0))/κ0.

Taking the limit as κ0, we have

fˉα(s,α0)0,s[0,t]bα(t,α0)0.

Here, f¯α(s,α0)(fc(1d*(s,α0),qd*(s,α0))qfy(1d*(s,α0),qd*(s,α0)))dα*(s,α0) and α0 can be taken arbitrarily. Thus,

(fcqfy)dα0,s[0,t]bα0.

Finally, together with Corollary 2 and Proposition 3(A), Part (A) follows.⃞

Appendix E

Proof of Proposition 5. We first prove Part (A). Solving eqs [26] and [27] under the assumption fcqfy=0, we have the constant optimal controls d=(ω+ω1/τ)/(ω+ω1/τ+ω1/τ(ωβ)11/τ) and h=(β+ω1/τ)/(ω+ω1/τ+ω1/τ(ωβ)11/τ). Then it is straightforward to show that dβ<0, and hβ0(ω1/τ+(ωβ)1/τ)(ω+ω1/τ)τω1/τ(ω1/τ+β)0. Finally, lβ<0 and cβ>0 are easily verified from the relations l(t)=d(t)h(t) and c(t)=1d(t) for all t, respectively. Next, by using the similar argument in the proof of Proposition 4, it can be shown that if, for all t: (fcqfy)dβ0, then bβ0. Then, Part (B) is clear from the assumption fcqfy=0 for all t.⃞

Proof of Corollary 3. This may be seen as follows. Due to the linearity of f on c and y, we have fcc=fcy=fyy=0. Then, the Jacobian matrix Jβ* for the present case is the same for the previous case with fcqfy=0 in Proposition 5(A), and thus the same comparative static results apply (see eq. [30]); we have dβ<0. Therefore, combined with the result in the proof of Proposition 5(B), we conclude that if, for all t: fcqfy0, then bβ*0.⃞

Appendix F

Proof of Proposition 6. The process establishing the existence of a unique optimal triple is similar to that in the proof of Proposition 1 and is therefore omitted. The only difference is that the restriction dh0 can be active. First, let ϕ:=ϕα. The optimal control, l=dh, is given by l=max{lα(α,τ,ω,p),0} where lα:=(1p(1+((1α)ω)1+1/τ))/(2+((1α)ω)1+1/τ). Then, it is immediate that lpα<0. Moreover, it follows that lα0α1(1+1/p)τ/(1τ)/ω=:α˜(p), and then α˜>0, α˜(1/(1+ω1+1/τ))=0, and α˜1 as p1. Second, let ϕ:=ϕβ. The optimal control is given by l=max{lβ(β,τ,ω,p),0} where lβ:=(ωβpω1/τ(1+(ωβ)11/τ))/(ω+ω1/τ+ω1/τ(ωβ)11/τ). Then, it is clear that lpβ<0. Moreover, it follows that lβ0ββ˜(p) such that ωβ˜(p)pω1/τ(1+(ωβ˜(p))11/τ)=0, and then β˜<0, β˜ω as p0+, and β˜(1/(1+ω1+1/τ))=0. Last, let ϕ:=ϕγ. The optimal control is given by l=max{lγ(γ,τ,ω,p),0} where lγ:=(ωγp(ω+ω1/τ))/(2ω+ω1/τ). Then, it is obvious that lpγ<0. Moreover, it follows that lγ0γωp(ω+ω1/τ)=:γ˜(p), and then γ˜<0, γ˜ω as p0+, and γ˜(1/(1+ω1+1/τ))=0.⃞

Appendix G

Proof of Proposition 7. The argument of the proof is similar to that of the proof of Proposition 6. In the present case, only the restriction h0 can be active. First, let ϕ:=ϕα. The optimal control is given by h=max{hα(α,τ,ω,m),0} where hα:=((1α)1+1/τω1/τ2m)/(ω(2+((1α)ω)1+1/τ)). Then, it is immediate that hmα<0. Moreover, it follows that hα0α1(2m)τ/(1τ)/ω1/(1τ)=:αˆ(m), and then αˆ<0, αˆ(ω1/τ/2)=0, and αˆ1 as m0+. Second, let ϕ:=ϕβ. The optimal control is given by h=max{hβ(β,τ,ω,m),0} where hβ:=((β+ω1/τ)(ωβ)1/τ(ω1/τ+(ωβ)1/τ)m)/((ω+ω1/τ+ω1/τ(ωβ)11/τ)(ωβ)1/τ). Then, it is clear that hmβ<0. Moreover, it follows that hβ0ββˆ(m) such that (βˆ(m)+ω1/τ)(ωβˆ(m))1/τ(ω1/τ+(ωβˆ(m))1/τ)m=0 and βˆ(m)is close to ω, and then βˆ<0, βˆω as m0+. Last, let ϕ:=ϕγ. The optimal control is given by h=max{hγ(γ,τ,ω,m),0} where hγ:=(ω1/τ+2γ2m)/(2ω+ω1/τ). Then, it is obvious that hmγ<0. Moreover, it follows that hγ0γmω1/τ/2=:γˆ(m), and then γˆ>0, γˆω1/τ/2 as m0+, and γ^(ω1/τ/2)=0.⃞

References

Aakvik, A., K. G.Salvanes, and K.Vaage. 2005. “Educational Attainment and Family Background.” German Economic Review6:37794.10.1111/j.1468-0475.2005.00138.xSearch in Google Scholar

Aiyagari, S. R., J.Greenwood, and A.Seshadri. 2002. “Efficient Investment in Children.” Journal of Economic Theory102:290321.10.1006/jeth.2001.2852Search in Google Scholar

Attanasio, O., H.Low, and V.Sánchez-Marcos. 2008. “Explaining Changes in Female Labor Supply in a Life-Cycle Model.” American Economic Review98:151752.10.1257/aer.98.4.1517Search in Google Scholar

Averett, S. L., L. A.Gennetian, and H. E.Peters. 2005. “Paternal Child Care and Children’s Development.” Journal of Population Economics18:391414.10.1007/s00148-004-0203-4Search in Google Scholar

Averett, S. L., H. E.Peters, and D. M.Waldman. 1997. “Tax Credits, Labor Supply, and Child Care.” Review of Economics and Statistics79:12535.10.1162/003465397556467Search in Google Scholar

Baum, C. L. 2002. “A Dynamic Analysis of the Effect of Child Care Costs on the Work Decisions of Low-Income Mothers With Infants.” Demography39:13964.10.1353/dem.2002.0002Search in Google Scholar

Becker, G. S., and H. G.Lewis. 1973. “On the Interaction between the Quantity and Quality of Children.” Journal of Political Economy81:S27988.10.1086/260166Search in Google Scholar

Becker, G. S., K. M.Murphy, and R.Tamura. 1990. “Human Capital, Fertility, and Economic Growth.” Journal of Political Economy98:S1237.Search in Google Scholar

Becker, G. S., and N.Tomes. 1976. “Child Endowments and the Quantity and Quality of Children.” Journal of Political Economy84:S14362.10.1086/260536Search in Google Scholar

Bergstrom, T., and S.Blomquist. 1996. “The Political Economy of Subsidized Day Care.” European Journal of Political Economy12:44357.10.1016/S0176-2680(96)00009-2Search in Google Scholar

Bernal, R. 2008. “The Effect of Maternal Employment and Child Care on Children’s Cognitive Development.” International Economic Review49:1173209.10.1111/j.1468-2354.2008.00510.xSearch in Google Scholar

Bernal, R., and A.Fruttero. 2008. “Parental Leave Policies, Intra-Household Time Allocations and Children’s Human Capital.” Journal of Population Economics21:779825.10.1007/s00148-007-0141-zSearch in Google Scholar

Bernal, R., and M. P.Keane. 2010. “Quasi-Structural Estimation of a Model of Childcare Choices and Child Cognitive Ability Production.” Journal of Econometrics156:16489.10.1016/j.jeconom.2009.09.015Search in Google Scholar

Blau, D. M. 1999. “The Effect of Child Care Characteristics on Child Development.” Journal of Human Resources34:786822.10.2307/146417Search in Google Scholar

Blau, D. M., and A. P.Hagy. 1998. “The Demand for Quality in Child Care.” Journal of Political Economy106:10446.10.1086/250004Search in Google Scholar

Blau, D. M., and P. K.Robins. 1988. “Child-Care Costs and Family Labor Supply.” Review of Economics and Statistics70:37481.10.2307/1926774Search in Google Scholar

Brink, A., and K.Nordblom. “Child-Care Quality and Fee Structure: Effects On Labor Supply and Leisure Composition.” Working Paper No. 157, Göteborg University, 2005.Search in Google Scholar

Cardia, E., and S.Ng. 2003. “Intergenerational Time Transfers and Childcare.” Review of Economic Dynamics6:43154.10.1016/S1094-2025(03)00009-7Search in Google Scholar

Chakrabarti, R. 1999. “Endogenous Fertility and Growth in a Model with Old Age Support.” Economic Theory13:393416.10.1007/s001990050261Search in Google Scholar

Connelly, R. 1992. “The Effect of Child Care Costs on Married Women’s Labor Force Participation.” Review of Economics and Statistics74:8390.10.2307/2109545Search in Google Scholar

Cunha, F., J. J.Heckman, and S. M.Schennach. 2010. “Estimating the Technology of Cognitive and Noncognitive Skill Formation.” Econometrica78:883931.10.3982/ECTA6551Search in Google Scholar

Desai, S., P. L.Chase-Lansdale, and R. T.Michael. 1989. “Mother or Market? Effects of Maternal Employment on the Intellectual Ability of 4-Year-Old Children.” Demography26:54561.10.2307/2061257Search in Google Scholar

Heckman, J. J. 1974. “Effects of Child-Care Programs on Women’s Work Effort.” Journal of Political Economy82:S13663.10.1086/260297Search in Google Scholar

Heckman, J. J. 2006. “Skill Formation and the Economics of Investing in Disadvantaged Children.” Science312:19002.10.1126/science.1128898Search in Google Scholar

Hotz, V. J., and R. A.Miller. 1988. “An Empirical Analysis of Life Cycle Fertility and Female Labor Supply.” Econometrica56:91118.10.2307/1911843Search in Google Scholar

James-Burdumy, S. 2005. “The Effect of Maternal Labor Force Participation on Child Development.” Journal of Labor Economics23:177211.10.1086/425437Search in Google Scholar

Kimmel, J., and R.Connelly. 2007. “Mothers’ Time Choices: Caregiving, Leisure, Home Production, and Paid Work.” Journal of Human Resources42:64381.10.3368/jhr.XLII.3.643Search in Google Scholar

Lamb, M. E. 1996. “Effects of Nonparental Child Care on Child Development: An Update.” Canadian Journal of Psychiatry41:33042.10.1177/070674379604100603Search in Google Scholar

Lundholm, M., and H.Ohlsson. 1998. “Wages, Taxes and Publicly Provided Day Care.” Journal of Population Economics11:185204.10.1007/s001480050064Search in Google Scholar

Lundholm, M., and H.Ohlsson. 2002. “Who Takes Care of the Children? The Quantity-Quality Model Revisited.” Journal of Population Economics15:45561.10.1007/s001480100071Search in Google Scholar

Michalopoulos, C., P. K.Robins, and I.Garfinkel. 1992. “A Structural Model of Labor Supply and Child Care Demand.” Journal of Human Resources27:166203.10.2307/145916Search in Google Scholar

Michel, P., and B.Wigniolle. 2007. “On Efficient Child Making.” Economic Theory31:30726.10.1007/s00199-006-0099-1Search in Google Scholar

Raut, L. K., and T. N.Srinivasan. 1994. “Dynamics of Endogenous Growth.” Economic Theory4:77790.10.1007/BF01212030Search in Google Scholar

Ribar, D. C. 1992. “Child Care and the Labor Supply of Married Woman: Reduced Form Evidence.” Journal of Human Resources27:13465.10.2307/145915Search in Google Scholar

Ruhm, C. J. 2004. “Parental Employment and Child Cognitive Development.” Journal of Human Resources39:15592.10.2307/3559009Search in Google Scholar

Seierstad, A., and K.Sydsæter. 1987. Optimal Control Theory with Economic Applications. Amsterdam: North-Holland.Search in Google Scholar

Tamura, R. 1994. “Fertility, Human Capital and the Wealth of Families.” Economic Theory4:593603.10.1007/BF01213626Search in Google Scholar

Tanaka, R. 2008. “The Gender-Asymmetric Effect of Working Mothers on Children’s Education: Evidence From Japan.” Journal of the Japanese and International Economies22:586604.10.1016/j.jjie.2008.05.003Search in Google Scholar

Waldfogel, J., W.Han, and J.Brooks-Gunn. 2002. “The Effects of Early Maternal Employment on Child Cognitive Development.” Demography39:36992.10.1353/dem.2002.0021Search in Google Scholar

Xie, D. 1991. “Increasing Returns and Increasing Rates of Growth.” Journal of political economy99:42935.10.1086/261759Search in Google Scholar

Xie, D. 1997. “On Time Inconsistency: A Technical Issue in Stackelberg Differential Games.” Journal of Economic Theory76:41230.10.1006/jeth.1997.2308Search in Google Scholar

  1. 1

    See, for example, Blau and Robins (1988), Connelly (1992), Ribar (1992), Averett, Peters, and Waldman (1997), and Blau and Hagy (1998). Only a few authors conclude differently. For example, Michalopoulos, Robins, and Garfinkel (1992) and Baum (2002) find the cost effect to be close to zero or positive, respectively.

  2. 2

    Moreover, other empirical studies suggest that the effect varies with the child’s age (Waldfogel, Han, and Brooks-Gunn (2002) and James-Burdumy (2005)) and gender (Tanaka (2008)).

  3. 3

    Further related literature includes, e.g., Becker, Murphy, and Tamura (1990), Tamura (1994), Raut and Srinivasan (1994), Chakrabarti (1999), and Michel and Wigniolle (2007), who study the long-run relationship between fertility and growth, explicitly considering child care cost.

  4. 4

    Results in the static model of Brink and Nordblom (2005) can be interpreted as showing a similar trade off when it is simply assumed that public child care is harmful for children. On the other hand, the trade off presented in this paper is based on the proposition that the productivity of parental child care exceeds that of public child care, which follows from the implied consequences to be consistent with common real-world intertemporal patterns of labor supply and child care demand. That is, we do not assume from the outset that public child care is less productive. Also, Bernal (2008) experimentally finds that child care subsidy is detrimental for children’s cognitive development yet increases parents’ expected lifetime utility. The subsidy corresponds to a policy of reducing use-based fee in our theoretical model (see below).

  5. 5

    In order to make the model more tractable, we introduce the assumption of additive separability except for the utility from leisure with and without the child. The exception is made because it may be less plausible that the value of one additional hour of leisure time which the parent is going to spend with the child this afternoon is independent of the time which the parent has spent alone this morning.

  6. 6

    The subutility function w captures the altruistic behavior of the parent. In some instances, however, the assumption of an altruistic parent may be questioned. But the case of a purely egoistic parent is included in our analysis as well and can be accomplished by setting w0.

  7. 7

    We use the term “public” child care in a broad sense. It means all types of nonparental child care on which the government may levy a fee.

  8. 8

    An interpretation of the parameter q is that it reflects the number of supervisors per child: for example, when one care provider supervises 10 children for say an hour, the actual time for which each of the children is adequately supervised by the care provider would be less than an hour. In this context, given the quality of the child care provided by an average parent, the quality of public child care, q, can become much smaller than 1 as the number of supervisors per child decreases, but cannot be much larger than 1 because of congestion effects.

  9. 9

    A justification for this formulation is provided by Bernal (2008), who finds that negative effects of nonparental child care on children’s ability are stronger for children with high ability endowments.

  10. 10

    A more general form of the technology includes goods inputs in addition to time inputs. We disregard such monetary investments. This assumption seems to be not problematic since the effect of income on the child’s performance is found to be much less important than that of parent’s education level (see Aakvik, Salvanes, and Vaage (2005) and Bernal and Keane (2010)).

  11. 11

    As we shall see below, we focus on the early childhood period, so that for this moderate period of time we reasonably assume a constant wage rate.

  12. 12

    For example, in Germany and Japan kindergarten fees are, in principle, based on the parent’s income, the child’s age, and the number of siblings. Moreover, additional fees are levied on extended child care hours, usually beyond the standard time of 4 or 5 hours.

  13. 13

    This should be understood as a special case of a more general formulation for the future welfare of the child.

  14. 14

    As shown in this proposition, the problem generates a unique interior solution. We first focus on the interior solution since our primary objective is to obtain standard comparative static results for the baseline scenario with respect to the effects of policy parameters on core variables. Then, later, in Section 8, we complement the analysis by allowing for corner solutions to suggest possible generalizations of the main results. We believe that this procedure is didactically more suitable than starting with a fully fledged version of an already sophisticated problem from the outset.

  15. 15

    This type of assumption, together with a pair of functions satisfying it, which we also apply in Section 9, have been presented by Xie (1991,1997), though in a different setting and context.

  16. 16

    See, for example, Bernal (2008) who finds that mothers get diminishing marginal utility from child ability and that mothers of low ability children spend more time with their children.

  17. 17

    As previously noted, this case is supported by the finding of negative effects of nonparental child care on child development as reported by Ruhm (2004) and Bernal (2008). Negligible effects, found by Desai, Chase-Lansdale, and Michael (1989), Blau (1999), and Averett, Gennetian, and Peters (2005), may be expressed by fcqfy0 in our model; we consider this case in later sections.

  18. 18

    Hotz and Miller (1988) and Kimmel and Connelly (2007) find that mothers’ caregiving time decreases with age of the child.

  19. 19

    In contrast to our analysis, Brink and Nordblom (2005) obtained clear-cut comparative static effects. This is, however, a simple consequence that they assumed that in each case the substitution effect dominates the corresponding income effect.

  20. 20

    In fact an estimation of Bernal (2008), for example, indicates an elasticity less than unity.

  21. 21

    As we infer from the relationship fˆdq=fy+yfyydfcy there are quite a few conditions that guarantee fˆdq>0. For example, if (a) f is separable, fcy=0, and ‘not too concave’ in y, in the sense that fy+yfyy>0 or if (b) parental and public child care are (weak) substitutes fcy0 and f is linear in y, we obtain fˆdq>0. In our subsequent numerical example, we use f(c,y):=c+y which implies fˆdq=1.

  22. 22

    This assumption is a joint condition on endogenous variables, but we can easily construct a situation where the assumption applies. The simplest way to achieve this is to specify f as a linear function f(c,y):=f˜(qc+y),f˜>0. In this case, we have fcqfy=0 for all c and y. Another way with a linear function will be found in Section 9. Moreover, we may specify a non-linear function f and then find the situation above, since with the assumption the unique optimal path is constant over time.

  23. 23

    We set Δ(β):=(ω1/τ+(ωβ)1/τ)(ω+ω1/τ)τω1/τ(ω1/τ+β). Then, we have Δ<0, Δ(0)=ω2/τ(2(ω11/τ+1)τ1) which is positive, at least, for τ>1/2, and Δ(ω+ω1/τ)ω1/τ(1τ)<0 as βω.

  24. 24

    Formally, βˉ is defined as the unique root of Δ (c.f. footnote 23).

  25. 25

    We already suggested a linear technology. See footnote 21.

  26. 26

    Alternatively, in the context of a two-parent household, we may interpret m as the other parent’s income.

  27. 27

    It is important to note that public revenue is endogenously determined implying that generally the same revenue is collected by different values of the respective fee. In other words, the same effective fee paid per hour of public child care generally varies over the three fee systems even if the collected public revenue is the same.

  28. 28

    Let us note that when g(b):=Abσ, eq. [7] with d=d is a Bernoulli-type equation: b˙=f(1d,qd)Abσ, whose solution for b(0)=b0 is b*(t)={(b0)1σ+A(1σ)0tf(1d*(s),qd*(s))ds}1/(1σ). Given this, in order to obtain Figure 4, we set A=1,b0=1,q=1r=1/10,T=10,σ=1/2,τ=1/2, and ω=1.

  29. 29

    Figure 4 is generated as a parametric plot, displaying for varying values of the respective fee parameter, public revenue and welfare, respectively public revenue per hour of public child care and welfare.

  30. 30

    A nonnegative combination of concave functions is also concave.

  31. 31

    This can be verified in a similar way as in Seierstad and Sydsæter (1987, 138, Example 10 with Theorems A.4 and A.5 in Appendix A). As for existence, since g is concave and since f(1d,qd)0, we have f(1d,qd)[g(b)g(b0)]f(1d,qd)g(b0)(bb0). Thus [A.10] in Theorem A.4 is satisfied. Moreover, since f(1d,qd)g(b)0 implying b˙0, we have b(t)b0 for all t. Thus [A.11] in Theorem A.4 is also satisfied. As for uniqueness, since g is continuously differentiable, [A.12] in Theorem A.5 is satisfied.

  32. 32

    The condition (b) can be replaced by the following two conditions: (b1) ζ(χ,ν,t)=η(χ,ν,t)+λ(t)χ with η and λ as continuous functions. (b2) ζ0, Ψ, and η are non-decreasing in χ, and ζ0, ξk, and η are concave in ν (c.f. Seierstad and Sydsæter (1987, 401)).

  33. 33

    We assume here that the optimal control is piecewise continuous. Regarding the risks of this assumption and the necessary changes if we allow the control only to be measurable, see Seierstad and Sydsæter (1987, 132–133, 276, 285).

Published Online: 2014-5-21
Published in Print: 2014-1-1

©2014 by De Gruyter

Downloaded on 28.3.2024 from https://www.degruyter.com/document/doi/10.1515/bejte-2012-0026/html
Scroll to top button