# **B Computational Aspects**

In this appendix we describe our model’s quantitative version and equilibrium computation. For simplicity, the description below focuses on two-parent households but all features are analogous for single-mother households.

## **B.1 Quantitative Model**

We present the quantitative version of the aggregate model and discuss, as needed, modifications for the location-specific model.

Our sample includes 750 households. We assume each one represents a household type, thereby yielding _J_ = 750 types. In our empirical application for the aggregate model, a household type is a combination of parental wage offers, preferences, initial child development, relative care availability, and parental education. We construct types as follows. First, for each type (household in the sample) we take from the data the initial child development, father’s education, and mother’s education. Second, for each type and a given parameter vector, we generate parental wage offers and preference parameters. We draw the vector ( _vc, vq, vf_ ) from a joint normal distribution to create the vector of preference parameters, ( _αc, αq, αf , αm_ ), as defined in equations (18)-(21). Similarly, we draw a pair of wage offers, ( _wf , wm_ ) from a joint lognormal distribution with the mean vector and variance-covariance matrix defined in (22) and (23), respectively. We set the non-labor income as follows: _Ij_ = 0 if _j_ is a two-parent household and _Ij_ = 41 if _j_ is a single-mother household (see Section C.2.3). For location-specific models, we use the location-specific wage offer distributions and scale the non-labor income transfer using the “Wage” factors from Table D.1. Given the calibrated proportions of households with available relative care by location (Table D.2), in each location we randomly assign relative care to the proportion of households indicated in this table.

On the child care supply side, an important issue is the number of potential entrants, _S_<sup> ̄</sup> , as it relates to the total measure of households in the economy,<sup>�</sup><sup>_J_</sup> _j_ =1<sup>_Mj_.Ahigher</sup><sup>_S_ ̄gives</sup> more heterogeneity among potential entrants yet also raises the computational cost of solving the two-stage supply side game. In the aggregate model, we strike a balance by setting _S_<sup> ̄</sup> = 7 and _Mj_ = _M_ = 1 _._ 3 for all _j_ = 1 _, ..., J_ . In the location-specific model, we set _M_ to the following values in locations 1 through 4: 2.93, 3.24, 3.98 and 3.17, respectively, to reflect the fact that locations in size.

̄ Since potential entrant _s_ is defined by the pair ( _c_<sup>_L_</sup> _s_ ̄<sup>_, cH_</sup> _s_ ̄<sup>),wedraw</sup><sup>_εs_ ̄fromastandard</sup> normal distribution and define the marginal costs for providing low and high quality as _c_<sup>_L_</sup> _s_ ̄<sup>=exp(</sup><sup>_μL_</sup> _c_<sup>+</sup><sup>_σ_</sup> _c_<sup>_Lεs_ ̄)and</sup><sup>_cH_</sup> _s_ ̄<sup>=exp(</sup><sup>_μH_</sup> _c_<sup>+</sup><sup>_σ_</sup> _c_<sup>_Hεs_ ̄),respectively.Asaresultofthisprocedure,</sup> the ordering of potential entrants based on marginal costs is the same for low- and highquality provision. In other words, potential entrants who are more efficient in the provision of high-quality services are also more efficient for low-quality services, a feature that simplifies the equilibrium computation below.

In the aggregate model, we set the common pair of daily fixed costs for low- and highquality provision to _F_<sup>_L_</sup> = 128 _._ 90 and _F_<sup>_H_</sup> = 322 _._ 25, respectively, based on industry cost reports (see Mitchell and Stoney (2012) and footnote 23 for more details). In the location61