How to Incorporate Children into Labor Supply Equations? An Equivalence-Scale Approach

The paper joins three literatures.

• Equivalence scales. It borrows the idea that children transform a reference behavior through a few interpretable parameters. In its tractable form, children translate the childless labor supply function twice, in resources and in available time, as Independent-of-the-Base scales translate demand functions (Lewbel, 1989; Blundell & Lewbel, 1991; Blackorby & Donaldson, 1993). Hurd & Pencavel (1981), with Barten scales, is one of the few earlier attempts in labor supply.
• Collective models with children. Parent and child have distinct preferences and share resources, as in Bargain & Donni (2012), Dunbar, Lewbel & Pendakur (2013), Lechene, Pendakur & Wolff (2022) and Penglase (2021). Their proportionality assumption on children’s expenditure is generalized to linearity in non-labor income.
• Childcare and mothers’ labor supply. Structural studies treat childcare as a claim on mothers’ time (Blau & Robins, 1988; Ribar, 1995; Kornstad & Thoresen, 2007) but require childcare-time data and leave out the income effect of children. The paper adds that income effect and needs neither childcare-time data nor child-specific spending.

The novelty is to estimate the total, composite cost of children from labor supply alone. This economy of data is bought with structure, that is fewer variables are observed, so more is assumed.

Picture a single mother as a childless woman with the same tastes and two extra burdens: hours of care she cannot cut, and spending on the child. The hours cost her the wage. Together with the spending, they lower the resources she keeps for herself, which pushes her to work more if leisure is normal. The hours of care also leave fewer hours to sell, which pushes her to work less. The effect of children on hours therefore has no sign a priori, and a coefficient on the number of children cannot tell the two channels apart.

What does tell them apart is how her responses differ from a childless woman’s. Care hours are valued at the wage, so a raise also raises the cost of care and makes her poorer, and she increases her hours by more. Part of each extra dollar of non-labor income goes to the child, so it reduces her hours by less. These slope differences, more than level differences, carry the cost of children.

Example. In the 2019 PSID, at the median, a $1 raise adds 27 hours of work a year for a childless working woman and 40 hours for a working mother; $1,000 of extra resources removes 28 and 25 hours respectively (Table 4).

A single mother chooses market hours $h$⁠, childcare time $t$⁠, purchased care $k$⁠, her consumption $c$ and her children’s consumption $m$⁠. Her preferences are $u_f(h+t,c)+\lambda(n)u_c(m)$⁠. She is altruistic toward a representative child, and market work and childcare enter her utility as perfect substitutes. This last assumption is the keystone of the model. Every hour of care is priced at the wage, and the problem becomes recursive. She first meets the childcare requirement $f(t,n) + k \ge T_c(n)$ at minimum cost, which defines the time cost $e_t(w,n) = \min_{t,k}\,(tw + k)$⁠. She then sets spending on the children, $e_m(y,w,n)$⁠. She finally supplies labor as a childless woman would with fewer resources.

Identification rests on two assumptions.

• Preference stability (Rothbarth–Gronau type). Mothers and childless women share the same trade-off between work and consumption, hence the same function $F$⁠. The paper uses this throughout but does not state it as a formal assumption. It is what makes childless women a valid reference.
• Assumption L. Spending on children is linear in non-labor income: $e_m = a_m(w,n) + b_m(w,n)\,y$⁠, with $0 < b_m < 1$⁠.

$F$ is estimated on childless women. For a mother who works, $\partial h / \partial y = F'_2 \cdot (1 - b_m)$⁠.

• If $F$ is nonlinear in $y$⁠, the first and second income derivatives identify $b_m$⁠. Inverting $F'_2$ then yields the total cost $e = e_t + e_m$ and childcare time $t$ (Proposition).
• If $F$ is linear in $y$⁠, only $b_m$ is identified. The total cost is known only up to an arbitrary function of $(w,n)$⁠.

The total cost is identified, and so are its two economically meaningful parts: the value of the mother’s time, $w\,t$⁠, and the value of goods and services, $e - w\,t$⁠. The primitive pair $(e_t, e_m)$ is not identified. Among non-working women $e_m$ is not identified and the price of time is a reservation wage, so their costs rely on a parametric extension.

Annual hours are quadratic in the wage and in net total expenditures (Brown, Levin & Ulph, 1976), a form that nests Hausman’s (1980) linear specification. Child costs scale with $\delta(n) = \sqrt{n}$⁠. Net total expenditures (household expenditures minus labor income) stand in for non-labor income. They are corrected for endogeneity with a control function built on asset, welfare, transfer and other incomes. A wage equation, with education as the excluded variable, handles endogenous wages and the unobserved wages of non-participants. The likelihood is of the Tobit type: the density of hours for workers, and for non-workers the probability of not participating, integrated over the latent wage.

Working mothers’ time is priced at the wage, non-working mothers’ at the reservation wage solving $h = 0$⁠. That equation has two roots; the reservation wage is the one consistent with Slutsky positivity. For some draws no real root exists, so the price of time and the costs are reported as quartiles over 200 Halton draws per woman.

Quadratic specification translated by children, with control functions and participation (equations 16–18 and equations 22–25). The linear baseline (19) above is the nested special case.

Here $q$ is shorthand for the translated resource argument, not a separately observed quantity. Working women contribute the hours density; non-participants contribute the probability of zero hours integrated over their latent wage. View estimated specification and participation.

The 2019 wave is the first in which the PSID records detailed household expenditures, which makes net total expenditures computable. The sample covers single women aged 20–60. Mothers have at most two children, the youngest aged six or under, the age range in which childcare matters most. The sample has 1,270 women, of whom 987 are childless women and 283 are mothers. Single women fit the model’s single decision-maker, and despite their exposure to poverty they are rarely the subject of structural labor supply analysis.

Table 1 shows why a control variable will not do. Mothers participate more than childless women (0.90 against 0.83) and earn less ($13.22 against $18.78), yet work slightly fewer hours (1,620 against 1,639). Net total expenditures are positive for mothers ($9,470) and negative for childless women (−$3,010), reflecting the fact that mothers’ earnings fall short of household spending. Among women in couples, by contrast, participation falls with children.

The estimates satisfy the testable restriction. At the median, a $1 raise adds 27 hours of work a year for a childless working woman and 40 hours for a working mother. $1,000 of extra resources removes 28 and 25 hours respectively (Table 4).

The child-cost equation gives $\hat{b}_t = 0.497$ (s.e. 0.178), about 500 hours of care per child per year. The 2019 ATUS reports 389 hours for comparable single mothers. The income share $\hat{b}_m = 0.089$ (s.e. 0.084) is imprecise (Table 3).

The median annual cost of one child is $21,000 for working mothers and $31,000 for non-working mothers (Table 5). The gap comes from the price of time. For non-working mothers the median reservation wage is $36.5, about twice their market wage of $18.4 (Table 4). At the median across all mothers, the mother’s time is worth about $12,200 a year and goods and services about $14,200. The second figure is close to the USDA estimate for a single parent (Lino et al., 2017).

These costs answer Browning’s (1992) positive question, how children change behavior. They are costs as perceived by parents, not amounts that ought to be compensated.