We focus on the effect of mother’s education and ignore the effect of father’s given previous findings on the asymmetric roles of mothers and fathers in child-rearing and limitations of our estimation sample.<sup>17</sup> In single-mother households, we allow mother’s productivity in child development production to depend on her education differently from two-parent households, leading to: where _{γ_ 0<sup>_o, γ_</sup> _m_<sup>_o}_arethesingle-mothershifters.</sup> Fourth, we assume a parametric distribution for center-based variable costs. Recall from (7) that potential child care provider _s_ has variable costs _c_<sup>_L_</sup> _s_<sup>and</sup><sup>_cH_</sup> _s_<sup>to provide low- and high-</sup> quality care, respectively. We assume _c_<sup>_L_</sup> _s_<sup>and</sup><sup>_cH_</sup> _s_<sup>are drawn from lognormal distributions with</sup> parameters _{μ_<sup>_L_</sup> _c_<sup>_, σ_</sup> _c_<sup>_L}_and</sup><sup>_{μH_</sup> _c_<sup>_, σ_</sup> _c_<sup>_H}_,respectively.Forcomputationalreasonsweassumethat</sup> _c_<sup>_L_</sup> _s_<sup>and</sup><sup>_cH_</sup> _s_<sup>arepositivelycorrelatedforagivenentrantbutareindependentacrossentrants</sup> (see Appendix B.1 for further details.) As a result, potential entrants that are more efficient in the provision of high-quality care are also more efficient in the provision of low-quality care–for instance, because greater managerial ability leads to greater efficiency regardless of service quality. Finally, we assume that fixed costs depend on quality, leading to _{F_<sup>_L_</sup> _, F_<sup>_H_</sup> _}_ for low- and high-quality provision respectively, but are homogeneous across firms.

In addition, we assume that center-based providers face a capacity constraint or maximum allowable enrollment, _N_<sup> ̄</sup> . To enroll additional children, they can increase capacity by paying the (quasi) fixed cost as many times as needed: where _⌈·⌉_ is the ceiling operator. We set _N_<sup> ̄</sup> = 40. For instance, in order to serve 45 children, a firm must incur the fixed cost twice; for 92 children, three times, and so forth. The firm can therefore run multiple centers, each with a 40-children maximum capacity.<sup>18</sup> Fifth, we set values for non-labor income _I_ (see equation (2)). We focus on income from transfers since income from assets is mostly negligible. Households in the data report whether they receive a transfer but not the amount. Given this limitation and the complexity of actual transfer rules,<sup>19</sup> we assume that two-parent households receive no transfers while single mothers receive a positive, homogeneous transfer amount, calibrated to match the 2003 poverty line.<sup>20</sup> We also assume that non-college single mothers receive a 35% discount at child care centers, since in the sample they pay about 65% percent of center-reported prices.

> 17Correlation studies indicate that “the productivity of women in home production appears to increase as their education increases” (Schultz (1993), p 53) while fathers’ education seems substantially less important. Admittedly, the conclusion is more nuanced among the several studies that use arguably exogenous variation on maternal education to look at its effect on children in the short and long run (for a survey, see Black and Devereux (2010)). An example is Carneiro et al. (2013)’s finding that maternal education has a positive effect on children when they are 7 to 8 years old but a negative one when they are younger than 24 months old. In addition, the reduced form relationship between mother’s education and children outcomes estimated by these studies involves a variety of mechanisms such as labor market participation and assortative mating in the marriage market. At least the second one is clearly not independent of father’s education.

> 18In our sample, this is the maximum observed enrollment.

> 19For a survey on children-related transfers, see Fang and Keane (2004); for one focused on single-mother households, see Hoynes and Schanzenbach (2018).

> 20See Appendix C.2.3 for additional details. Transfers are scaled by location.

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