Taxes, Welfare and the Resources Parents Allocate to Children Marianne Bruins∗ 21 September 2017

Abstract This paper examines the impact of the US tax and welfare system on the resources that parents allocate to children. I estimate a collective model that allows parents to allocate resources to children in the form of consumption, parental time and the provision of a public good. I find that for each dollar spent on cash welfare, children receive an additional 18 cents’ worth of tangible household resources, on average. I also find that those welfare programmes that have the least depressing effect on parents’ labour supply are relatively more efficient at promoting the welfare of the children of sole mothers. Redesigning existing programmes with this in mind would bring substantial benefits to these children, while having little or no adverse effect on children in twoparent households.

∗

Email: [email protected]. Nuffield College and Department of Economics, University of Oxford. This research has greatly benefited from discussions with J. Altonji, R. Blundell, S. Bond, M. Browning, J. Duffy, J. Lise, C. Meghir, R. McKibbin and comments from seminar participants at the University of Leuven, University of Rochester, University of Bristol, Barcelona Summer Forum (Family Economics), ISER (Essex), Econometric Society Meeting (Summer 2016, Geneva), Family Economics Conference University of Notre Dame (London Campus), Chicago (HCEO) and IFS/UCL. This work was supported by the HPC facilities operated by, and the staff of, the Yale Center for Research Computing.

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Contents 1 Introduction

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2 Preliminary estimates

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3 Model 3.1 Household preferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Calculation of after-tax earnings: taxes and welfare . . . . . . . . . . . . . .

7 7 11 12

4 Estimation procedure 4.1 Data sources . . . . . . . . . . . . . . . . . . . . 4.2 Simulated method of moments . . . . . . . . . . 4.3 Imputation (via simulation) of missing variables 4.4 Sample moments: selection and construction . . 4.5 Identification of structural parameters . . . . .

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5 Estimates and model fit 5.1 Parameter estimates . . . . . . . . . . 5.2 Model fit . . . . . . . . . . . . . . . . . 5.3 Wage and income elasticities . . . . . . 5.4 Model fit: external validation . . . . . 5.5 Identification of preference parameters

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6 Cash welfare and children’s welfare

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7 Conclusion

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8 References

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Appendices (for online publication) A Data sources

A1

B Calculation of taxes and welfare payments

A4

C Estimation procedure: further details

A7

D Parameter estimates

A12

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1

Introduction

Every year over a hundred billion dollars are spent by the US government on cash welfare (Isaacs, Toran, Hahn, Fortuny, and Steuerle, 2012). But our understanding of how this affects households’ allocation of resources to children remains limited. This is on the one hand surprising, given that disadvantaged children are the focus of many other policy initiatives, and the extensive evidence that what goes on within the household is important for child outcomes. On the other hand, a satisfactory answer to this question requires sophisticated modelling, which places heavy demands on the available data. This paper estimates a structural model with the aim of determining how much spending on cash welfare gets through to children, and what scope there might be for redesigning existing cash welfare programmes so as to better target household resources to children. My work connects with a broad literature aimed at understanding how household income and family type (whether a sole- or two-parent household) relate to child outcomes (such as test scores and lifetime earnings); both in terms of mere correlation, and – at least in more recent studies – in a deeper, causal sense (see, for example Blau, 1999; Milligan and Stabile, 2009; Løken, 2010; Dahl and Lochner, 2011; Løken, Mogstad, and Wiswall, 2012).1 This relationship can be broken into two parts: first, the effect of household income on the resources that children receive; and second, the effect of those resources on child outcomes. Considerable work has been devoted to modelling the second of these mechanisms, often in the context of the child production function (Todd and Wolpin, 2007; Cunha and Heckman, 2008; Todd and Wolpin, 2010; Cunha, Heckman, and Schennach, 2010). But rather less is known about the first mechanism – and particularly the influence that the tax and welfare system has on the allocation of household resources. The present work focuses on this mechanism, and unlike two recent papers on the subject (Cherchye, De Rock, and Vermeulen, 2012; Del Boca, Flinn, and Wiswall, 2014), does so in the context of a detailed modelling of the US tax and welfare system. This paper is the first to quantify the proportion of cash welfare spending that actually gets though to children, in the sense of increasing the household resources that they receive. I estimate that 18 per cent of total (net) government spending on three major cash welfare programmes – the Earned Income Tax Credit (EITC), the Child Tax Credit (CTC) and the Supplementary Nutritional Assistance Program (SNAP) – translates into additional resources allocated to children. To calculate this figure, I estimate a collective household model with children, embedding a realistic detailed model of the US tax and welfare system, 1

Adult outcomes for children who grew up in sole-parent households are significantly worse than those in two-parent households (Abbott, 2015).

1

and drawing on both time use and consumption data. In the model, children benefit from consumption, parental time and a home-produced public good. The model provides a framework for evaluating the resources that children receive, and for computing a money-metric measure of the change in their welfare that results from a policy intervention. By comparing this with (net) government spending on welfare programmes, I am able to calculate the proportion of (net) expenditure on these programmes that gets through to children – what I term the pass-through rate. (My calculation of ‘net spending’ fully accounts for welfare programmes’ secondary effects on government tax revenues, due to households’ behavioural responses.) I use these estimated pass-through rates to examine how alternative transfer programmes compare in terms of their relative efficiency in targeting household resources to children. For sole-parent households, there is considerable variation in these rates across the three programmes – ranging from 17 per cent for SNAP to 46 per cent for CTC – but there is no such variation for two-parent households. This suggests that the transmission of welfare payments to children is much more sensitive to the shape of the benefits schedule – how these payments vary with household earnings and parents’ labour supply – in sole-parent households than it is in two-parent households. Further evidence for this is provided by a counterfactual experiment in which each welfare programme is removed and replaced by a lump-sum transfer of equal value, eliminating its distortionary effects on the (after-tax) returns to labour supply. In this scenario, the differences in pass-through rates across policies, for sole-parent households, almost entirely disappear. These discrepancies between sole- and two-parent households may be accounted for by the differing responses of parents’ time use. The three programmes considered vary markedly in this respect: for example, SNAP induces a significant reduction in parents’ hours worked (8–11 hr/wk, per parent, in both sole- and two-parent households), while CTC has a practically negligible effect. In sole-parent households, the principal result of such distortions is to reduce household (and therefore child) consumption, with relatively little of parents’ time being reallocated to children. In two-parent households, in contrast, a substantial portion of parents’ time is thus reallocated, so that children are effectively compensated for any reduction in their consumption. Consequently, all programmes have similar pass-through rates for two-parent households – whereas for sole-parent households, SNAP has a pass-through rate that is only one-third that of CTC. The main implication of these results for policymakers is that cash welfare programmes that promote maternal labour supply are relatively more beneficial for the children of sole mothers. Taking sole mothers out of the workforce does not help children, because little of time thereby freed is reallocated to children – but such programmes are costly to the state 2

because of their negative effects on (net) tax revenues. Programmes that raise household income, and thence consumption, by promoting sole mothers’ labour supply are therefore much more efficient at improving their children’s welfare. Redesigning existing programmes with this in mind would bring substantial benefits to the children of sole mothers, while having little or no adverse effect on children in two-parent households. At a methodological level, because I am interested in child welfare – and in particular how children may benefit from parental time spent in activities aside from market work – I develop and estimate a structural model that provides a detailed treatment of parents’ time-use, as well as consumption. Estimation of such a model requires both disaggregated time use and consumption data, and in this paper I demonstrate how the model may be successfully estimated even when these are drawn from multiple datasets (principally, the American Time Use Survey, the Consumer Expenditure Survey, and the Current Population Survey). This is important, because to the best of my knowledge the only datasets that provide both types of data have very small sample sizes, which makes their use problematic due to the well-known measurement errors issues associated with expenditure and especially time-use surveys. I subsequently verify, empirically, that the use of time-use data greatly improves the precision which which the model’s parameters are estimated. Outline The remainder of this paper is organised as follows. Section 2 provides some preliminary estimates, using two-stage least squares (2SLS) with simulated instruments, of the causal effect of household income on the allocation of household resources. These estimates indicate that tax and welfare policy has indeed influenced the allocation of household resources, thereby demonstrating that there is a relationship present in the data for the structural model to explain. They also indicate significant differences between sole- and two-parent households, with respect to the response of consumption and parents’ time use to (exogenous) variation in household income, which are subsequently shown to drive some of the key findings of the paper. The empirical specification of the collective labour supply model used in this paper, and my modelling of the tax and welfare system, are then described in Section 3. The data and simulated method of moments (SMM) procedure are discussed in Section 4, where I also provide a heuristic discussion of how the moments matched by SMM help ensure the identification of the model’s parameters. Paralleling the 2SLS estimates, I find some substantial differences between the estimates obtained for sole- and two-parent households; the elasticities implied by the structural model also match the signs of the statistically significant 2SLS estimates. I show that the model performs well, both by in-sample and out-of-sample criteria. I further investigate empirically how different subsets of the selected moments con3

tribute to the identification of the model’s parameters, which I quantify by computing how the precision of the parameter estimates would be affected if certain moments were ignored by the estimation procedure. The results indicate that both the time-use moments and the correlations among household choices contribute significantly to the identification of the structural parameters. Section 6 reports the results of a range of counterfactual policy experiments that I conduct using the model, which provide the basis for the major policy-relevant findings of this paper, as noted above. Section 7 concludes.

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Preliminary estimates

I first discuss some estimates of the relationship between after-tax earnings and the intrahousehold allocation of resources, obtained without the use of the structural model that will be developed subsequently. Though the structural model is the principal focus of this paper, the estimates presented in this section are of interest for two reasons: firstly, because these show that tax and welfare policy has indeed influenced the allocation of household resources, thereby demonstrating that there is a relationship present in the data for the model to explain; and secondly, because they may be used to provide an external check for the model’s validity, by comparing these estimates with the elasticities implied by the model (see Section 5.3 below).2 I here estimate a collection of linear models, in which the l.h.s. variable is either the time a parent spends in a selected activity, or the value of a selected class of consumption expenditures. The r.h.s. variables include after-tax earnings, along with: the parents’ ages, dummies for the parents’ educational attainment (whether only high school, some tertiary, college, or postgraduate), the age of the youngest child, dummies for the number of children under 16 and under 18 (whether one child, two children, or more than two children), the state unemployment rate, and state and year dummies. I use the American Time Use Survey (ATUS; 2003–08) and the Consumer Expenditure Survey (CE; 1993–2008) for the time-use and consumption regressions, respectively (see Section 4.1 below and Appendix A for further details). To the extent that after-tax earnings may be correlated with parents’ unobservable preferences – in particular, with their propensity to devote resources to children – they are potentially endogenous. I therefore estimate these models by 2SLS, instrumenting for aftertax earnings using so-called ‘simulated instruments’ (Milligan and Stabile, 2009; Dahl and 2

As discussed in that section, those elasticities are able to match the signs of the statistically significant responses found here, though not their relative magnitudes.

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Table 1: 2SLS with Simulated Instruments Response to $1000/wk increase in after-tax earnings (year 2000 dollars)

Couples

Sole

Father

Mother

Mother

Time use (2003–08; hr/wk) 1

Time with children

-0.39 ) (0.29)

1.06∗∗ ) (0.45)

-0.11∗∗ ) (0.05)

2

Educational time with children

-0.02 ) (0.09)

0.33∗∗ ) (0.13)

-0.02 ) (0.02)

3

Housework

-0.18 ) (0.21)

0.43∗ ) (0.27)

0.08 ) (0.09)

4

Paid work

0.02 ) (0.01)

0.01 ) (0.02)

-0.01 ) (0.01)

Household consumption (1993–2008; $/wk) 5

Entertainment

62∗∗ ) (4)

5 ) (11)

6

Pets and toys

20∗ ) (11)

2 ) (3)

7

Food

34∗∗ ) (6)

20 ) (24)

8

Housing

-5 ) (16)

26∗∗ ) (11)

9

Public consumption

58∗∗ ) (20)

137∗ ) (86)

Total consumption

489∗∗ ) (32)

188 ) (154)

10

•

Standard errors in parentheses. ∗ denotes significance at the 10% level, ∗∗ at the 5% level. F -statistics for joint significance of excluded instruments in all first-stage regressions exceed 50. Sources: ATUS (2003–08) for time-use data; CE (1993–2008) for consumption data; simulated instruments constructed using the CPS (1993–2008).

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Lochner, 2011). These instruments are intended to provide a measure of the generosity of the tax and welfare system prevailing in each state–year pair present in the sample, which should be exogenous to the household. They are constructed by taking all households in the Current Population Survey (CPS) for 1993–2008, and (counterfactually) computing their after-tax earnings under the tax and welfare system for every possible state–year pair (the CPS is used here due to its large sample size). Variation in the distribution of after-tax earnings thus calculated is therefore purged of any regional or temporal heterogeneity in household characteristics, and reflects only variation in the generosity of the tax and welfare system. From these ‘simulated’ distributions I take selected quantiles (0.1, 0.2, . . . , 0.9). For each household in the ATUS and CE, the values of the simulated instruments are given by these quantiles, for the state–year pair appropriate to that household. The results appear in Table 1, scaled so as to give the response of time use (in hr/wk) and consumption (in $/wk) to a $1000/wk increase in household after-tax earnings (in year 2000 dollars). For sole-parent households, expenditure on housing and public consumption is estimated to be relatively more responsive to after-tax earnings, whereas in two-parent households it is parents’ time use, especially mother’s time use, that is most responsive. For example, in response to a $1000/wk increase in after-tax earnings, mother’s time with children increases by 1 hr/wk in two-parent households, and time in housework by 25 min/wk – whereas in sole parent households the corresponding figures are only about 5 min/wk (see rows 1 and 3). On the other hand, in sole-parent households a $1000/wk-increase in aftertax earnings raises expenditure on public consumption by $130/wk, but by less than half this amount in two-parent households (for which there is a proportionately much greater increase in expenditure on private consumption goods such as food, entertainment, and pets and toys; see rows 9 and 10). While these results are indicative of the relationship between after-tax earnings and the allocation of household resources, they should be interpreted with some caution, because there are good reasons to believe that the simulated instruments only partially resolve the endogeneity problems noted above. Changes to the tax and welfare system will affect not only after-tax earnings but also the returns to market work. The simulated instruments will thus be correlated with hours worked and with variation in the returns to the different possible uses of a parent’s time. In view of this, it seems difficult to sustain the exclusion restriction necessary for the validity of these instruments. Thus one reason for estimating a structural model – aside from it enabling me to conduct counterfactual experiments and evaluate children’s welfare – is that it allows these problems to be circumvented, by explicitly modelling household members’ preferences over alternative uses of the households’ resources (time and income), and the tax and welfare system that the household faces. 6

3

Model

In this paper I am particularly concerned with the manner in which cash welfare impacts affects child well-being. To address this question I estimate a structural model of household decision-making that provides a detailed treatment of the allocation of parents’ time and income within the household. In the model, children benefit from the time they spend with parents, a private consumption good, and the public good (which is produced from public consumption expenditures and parents’ housework). The model differs from others estimated in the literature in two important respects (see e.g. Apps and Rees, 1996; Blundell and Shephard, 2011; Cherchye, De Rock, and Vermeulen, 2012). Firstly, children’s utility is modelled separately from parents’ utility and the home production of a public good. Since the model is estimated using only data on household choices, the estimates of the parameters of the children’s utility function are driven by parents’ decisions about the allocation of resources to children. Thus a key assumption underlying the counterfactual welfare analyses of Section 6, which relies on the children’s utility function as a measure of child welfare, is that parents are altruistic towards their children. The other respect in which the model departs from previous work is in its detailed treatment of parents’ time use. The inclusion of parents’ time with children in the model is motivated by the strong empirical evidence for its contribution to children’s cognitive and non-cognitive development (see e.g. Brooks-Gunn and Markman, 2005; Phillips, 2011; Fiorini and Keane, 2014). The importance of incorporating household production into models of the intra-household allocation has been stressed in the work of Apps and Rees (1997, 2002). Indeed, more than half of all household expenditures in the CE are classified as public, in the sense of being inputs to the production of the domestic public good, and the time parents typically devote to housework in the ATUS is comparable to the time they spend with their children (see Table 2 below). I estimate the model separately for two-parent and sole-parent families. (As is further discussed in Section 5.1 below, I impose no restrictions connecting the parameters of the sole-parent and two-parent household model.) Due to the relatively small number of male sole parents (only 17% of sole parents in the CE are male), I restrict the sample to female sole parents.

3.1

Household preferences

Two-parent households (couples) I employ the collective labour supply model with children, as developed by Chiappori (1988, 1992), Blundell, Chiappori, and Meghir (2005), 7

and Chiappori and Ekeland (2006), to describe the parents’ decision problem. This model offers greater flexibility than the unitary model and has been shown to perform better empirically (see e.g. Lundberg, Pollak, and Wales, 1997; Duflo, 2000). Assuming that parents have egotistic preferences (they care about their own welfare but not their partner’s), and that the household allocation is Pareto optimal, that allocation may be characterised as the constrained maximiser of U := λUm + (1 − λ)Uf , (3.1) subject to the budget and time constraints given below (see (3.11) and (3.10)). Here λ ∈ [0, 1] is a Pareto weight, Um denotes the mother’s utility and Uf the father’s utility. The Pareto weight summarises the relative influence that each parent has over the household’s decision making; that is, their ‘bargaining power’. It is allowed to vary with a vector xλ of distribution factors according to λ := Λ(βλ0 xλ + σλ λ ),

(3.2)

where Λ is the standard logistic cdf, and λ is an i.i.d. Gaussian disturbance. In this paper, I employ four distribution factors: the wage ratio (mother’s wage divided by father’s wage); the average wage of the two parents; the age difference (mother’s age less father’s age); and the relative sex ratio.3 These variables have been used as distribution factors in several previous studies, on the grounds that they correlate with how well-off each parent would be, in relative terms, if the household were to dissolve (Chiappori, Fortin, and Lacroix, 2002; Blundell, Chiappori, Magnac, and Meghir, 2007; Lise and Seitz, 2011; Cherchye, De Rock, and Vermeulen, 2012; Aizer, 2012; Bruins, 2015). Each parent’s utility decomposes as a sum of private utility ui and the child’s utility K, Ui := ui + δik K

(3.3)

for i ∈ {m, f }. Private utility takes the form 1−ηi ηi 1−ηi ηi 1−ηi ηi 1/ηi ui (ci , li , q) := log(γi,c ci + γi,l li + γi,q q )

(3.4)

where ci denotes parent i’s consumption, li his or her leisure, and q the public good (γi,c +

3

The wages of non-working mothers are imputed (via simulation) using an estimated wage equation, see Section 4.3 below; families with non-working fathers are excluded from the sample. The sex ratio is defined as the ratio of men to women in an individual’s state of the same age and race; the relative sex ratio is calculated as the sex ratio faced by the mother multiplied by the sex ratio faced by the father.

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γi,t + γi,q = 1; and γi,c , γi,t , γi,q ∈ [0, 1], ηi ≤ 1).4 Children’s utility has the nested CES form, 1−ηk ηk 1/ηk 1−ηk ηk 1−ηk ηk q ) K(ck , t, q) := αlog(γk,c ck + γk,t t + γk,q

t(tf , tm ) :=

1−ηt ηt (γt,m tm

+

1−ηt ηt 1/ηt tf ) γt,f

(3.5) (3.6)

where tm (resp. tf ) denotes mother’s (father’s) time with children, and ck the children’s private consumption (γk,c + γk,t + γk,q = 1; γt,m + γt,f = 1; and γk,c , γk,t , γk,q , γt,m , γt,f ∈ [0, 1], ηk ≤ 1, ηt ≤ 1, α ≥ 0). The CES functional form adopted here nests (at ηk = 0) the Cobb-Douglas specification that has been used to model children in some previous work (see e.g. Del Boca, Flinn, and Wiswall, 2014). The quantity of the public good produced is given by the (constant returns to scale) nested CES production function 1−η

η

1−η

q(cq , a) := (γq,c q cq q + γq,a q aηq )1/ηq

(3.7)

1−ηa 1−ηa a(am , af ) := (γa,m am + γa,f af )1/ηa

(3.8)

where am (resp. af ) denotes mother’s (father’s) time devoted to housework, and cq the expenditure on consumption goods that are used as inputs in the production of the public good, which I refer to as ‘public consumption’ (γq,c + γq,t = 1; γa,m + γa,f = 1; and γq,c , γq,t , γa,m , γa,f ∈ [0, 1], ηq ≤ 1, ηa ≤ 1). Such a CES specification has been used in prior work to model home production (see, e.g. Aguiar and Hurst, 2007). Sole mothers The model specified for these households is identical to that for two-parent households, except that the father’s variables are eliminated (and λ = 0). (The parameters of the two models are freely estimated, without any requirement that corresponding parameters should agree across the two models.) The children’s utility and public good production functions simplify in obvious ways: since only maternal time inputs are available, t = tm and a = am in the ‘outer’ CES functions (3.5) and (3.7) respectively; the ‘inner’ CES functions in (3.6) and (3.8) are no longer needed. Preference heterogeneity and parametrisation To permit heterogeneity in preferences, a number of parameters are allowed to vary with both observables and unobservables. In the following paragraphs,  signifies an i.i.d. standard normal disturbance (all of which are mutually uncorrelated), and x a vector of observed household characteristics. 4

Regarding the units of these variables: so as to render all inputs roughly the same order of magnitude, consumption goods always enter utilities measured in tens of dollars per week, and time in hours per week.

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Consider first the parameters of parent i’s (i ∈ {m, f }) private utility, (3.4). For the elasticity parameter, I specify ηi = 1 − exp(x0i,η βi,η ) where xi,η includes a constant, an education dummy (an indicator for whether parent i has some tertiary eduction), and a child age dummy (an indicator for whether there is a child aged 0–5). To ensure that the weight parameters γi,· always satisfy the stipulated adding-up and range constraints, I first specify γ˜i,c = exp(x0i,c βi,c + σi,q i,c )

γ˜i,l = exp(x0i,l βi,l + σi,l i,l )

γ˜i,q = 1

where xi,c and xi,l both include a constant and an education dummy; and xi,l additionally includes the residual from parent i’s wage equation (see Section 4.3). i,c and i,l are mutually independent, i.i.d., standard Gaussian disturbances; βi,c , βi,l , σi,c and σi,l are the underlying free parameters that are to be estimated. The utility weights are then constructed from the P γ˜i,· as per γi,j = γ˜i,j / k∈{c,l,q} γ˜i,k . Because δim , δif and α cannot be separately identified, I normalise δim = 1 and specify α = exp(x0α βα + σα α )

(3.9)

where xα includes a constant, an education dummy and a child age dummy. The elasticity parameters for the nested CES used for children’s utility are permitted to vary as ηk = 1 − exp(x0k,η βk,η )

ηt = 1 − exp(x0t,η βt,η )

where both xk,η and xt,η include a constant and a child age dummy, and xk,η additionally includes a dummy for household size (an indicator for whether there are two or more children). Analogously to the weight parameters for the parents’ utilities, I first set γ˜k,i = exp(x0k,i βk,i + σk,i k,i ), γ˜t,m =

i ∈ {c, t}

exp(x0t,m βt,m )

γ˜k,q = 1 γ˜t,f = 1

– where x·,· in all cases includes a constant and a child age dummy – and thence compute P P γk,i = γ˜k,i / j∈{c,t,q} γ˜k,j for i ∈ {c, t, q}, and γt,i = γ˜t,i / j∈{m,f } γ˜t,j for i ∈ {m, f }. Finally, regarding the public good production function: its elasticity parameters are specified as ηq = 1 − exp(x0q,η βq,η )

ηa = 1 − exp(x0a,η βa,η ) 10

where xq,η and xa,η include a constant and a child age dummy. For the weight parameters, I set γ˜q,c = exp(x0q,c βq,c + σq,c q,c )

γ˜q,a = 1

γ˜a,m = exp(x0a,m βa,m + σa,m a,m )

γ˜q,f = 1,

where xq,c and xa,m include a constant and a child age dummy. I then compute γq,i = P P γ˜q,i / j∈{c,a} γ˜q,j for i ∈ {c, a}, and γa,i = γ˜a,i / j∈{m,f } γ˜a,j for i ∈ {m, f }.

3.2

Constraints

The household’s objective U in (3.1) is maximised subject to the household’s time and budget constraints, given below. (For sole mothers, Um is maximised subject to her time and budget constraints.) I discretise the allowable hours choices, so that mothers and fathers choose their hours of work from the sets Hm = {0, 20, 30, 40, 55} and Hf = {20, 30, 40, 50, 60} respectively. These restrictions may be justified by the presence of frictions in the labour market (see Hoynes, 1996; Blundell and Shephard, 2011) and simplify the computation of household’s optimal choices, which would otherwise be greatly complicated by the non-convexity of the budget set. Each parent’s weekly time constraint is given by li + hi + ti + qi ≤ 105,

(3.10)

for i ∈ {m, f } (nine hours per day are excluded to allow for sleep). The household budget constraint is cm + cf + ck + cq + no5 po5 [min(hf , hm ) − fo5 − 30]+ + nu5 pu5 [min(hf , hm ) − fu5 ]+ {z } | {z } | {z } | non-durables consumption

cost of childcare for children over 5

cost of childcare for children 5 and under

≤ e(hf , hm ; wf , wm ) + y − s (3.11) where [x]+ := max{x, 0}; wm and hm (resp. wf and hf ) denote the mother’s (father’s) wage and hours worked; and po5 , fo5 and no5 denote the price of childcare, the hours of free childcare available, and the number of children, for children aged 6–13 (and pu5 , fu5 and nu5 similarly for children aged 0–5). e(hf , hm ; wf , wm ) is the household’s after-tax earnings, the calculation of which accounts for income taxes and a range of welfare programmes, at the federal and state levels (as discussed in the next section). y − s denotes non-labour income less saving. 11

Similarly to Blundell and Shephard (2011), I include hours of free childcare in the model so as to account for unpaid childcare that may be provided by friends and relatives. Thus, for example, [min(hf , hm ) − fo5 − 30]+ gives the hours of childcare that the household needs to purchase for each child aged 6–13; it is non-zero if the time spent in work by both parents exceeds the sum of: the hours of free childcare available fo5 , and the hours these children spend in school (30 hr/wk). (It is assumed that the time both parents spend at work overlaps as completely as possible.)

3.3

Calculation of after-tax earnings: taxes and welfare

A key component of my model – and one whose design is of particular interest in this paper – is the tax and welfare system faced by the household. To limit the extent to which estimates of behavioural parameters might be confounded by unmodelled programmes, I attempt to provide as comprehensive a model of this system as possible (an approach recommended by Blundell and Preston, 1998, and Fang and Keane, 2004). As well as modelling the entire state and federal income tax system, I also account for the most significant cash welfare programmes. Isaacs, Toran, Hahn, Fortuny, and Steuerle (2012) show that, exclusive of healthcare programmes, the four largest government programmes (by total expenditure) that are targeted at children are: the Child Tax Credit (CTC), the Supplemental Nutrition Assistance Program (SNAP), the Dependent Exemption, and the Earned Income Tax Credit (EITC, at both the state and federal level). My model incorporates each of these, and: the Child and Dependent Care Tax Credit (CDCTC) at the state and federal levels; Temporary Assistance for Needy Families (TANF, 1996–2008); and Aid to Families with Dependent Children (AFDC, 1993–1995). Altogether, over $170 billion was spent on these programs in 2011 (Isaacs, Toran, Hahn, Fortuny, and Steuerle, 2012). As these programmes are means tested, they each affect the shape and position of the household budget set. All of these programmes are incorporated into the tax calculations used by my model: see Appendix B for details. Couples are always assumed to file jointly, because of the (usually) significant financial advantages from doing so. As illustrated in Figure 1, the generosity of the modelled welfare programmes varied substantially over the sample period (1993–2008). This was largely due to: the Personal Responsibility and Work Opportunity Act of 1996, which replaced AFDC with TANF; the Tax Payer Relief Act of 1997, which introduced the CTC; and the Bush tax cuts (the Economic Growth and Tax Relief Reconciliation Act of 2001 and the Job Relief and Reconciliation Act of 2003) which, in addition to reducing taxes for high-income earners, expanded both the federal EITC and the CTC. Notably, the maximum benefit afforded by the CTC approximately

12

Figure 1: Federal tax and welfare programmes Weekly income and transfers (2000 dollars) (a) EITC

(b) CTC

80

40 1993

70

1994

35

1996

1996

2000 60

2000 30

2004

2004

2008

2008 25

2012

Net transfer

Net transfer

50 40

15

20

10

10

5

0

100

200

300

400

500 600 Income

700

800

900

0

1000

(c) SNAP

0

500

1000

1500 Income

2000

2500

(d) Change in federal net transfer, relative to 1993

3000

∗

250

200

1996 2000 2004 2008 2012 Earn2000

1994

180

1996 160

200 Net transfer change from 1993

2000 2004

140 Net transfer

20

30

0

2012

2008 2012

120 100 80 60 40

150

100

50

0

20 0

• ∗

0

50

100

150

200

250 300 Income

350

400

450

500

−50 0

500

1000

1500

2000 Income

2500

3000

3500

4000

All calculations are for a two-parent household with two children aged under 16, with no childcare expenses. Change in the net transfer received by the household from the federal tax and welfare system, relative to the year 1993. Earn2000 is the density of earnings for one-parent families in 2000 (source: CPS).

13

Figure 2: Change in net transfer, relative to 1993, all programmes Weekly after-tax income (2000 dollars) (a) Delaware

(b) Nebraska

250

250 1996 2000 2004 2008 2012 Earn2000

150

200 Net transfer change from 1993

Net transfer change from 1993

200

100

50

0

−50 0

1996 2000 2004 2008 2012 Earn2000

150

100

50

0

500

1000

1500

2000 Income

2500

3000

3500

−50 0

4000

500

1000

(c) California 1996 2000 2004 2008 2012 Earn2000

3000

3500

4000

150

1996 2000 2004 2008 2012 Earn2000

200 Net transfer change from 1993

Net transfer change from 1993

2500

250

200

100

50

0

•

2000 Income

(d) New York

250

−50 0

1500

150

100

50

0

500

1000

1500

2000 Income

2500

3000

3500

−50 0

4000

500

1000

1500

2000 Income

2500

3000

3500

4000

Change in the net transfer received by the household from the tax and welfare system in the state indicated (includes federal taxes and welfare), relative to the year 1993. All calculations are for a two-parent household with two children aged under 16, with no childcare expenses. Earn2000 is the density of earnings for one-parent families in 2000 (source: CPS).

14

doubled to $20 per week ($20/wk) for each child in 2001; as can be seen in see panel (b) of Figure 1 which plots weekly CTC benefits for a two-parent, two-child family as a function of weekly household income. In 2000, such a family earning $1000/wk would have been eligible for a credit of just under $20/wk from the CTC (i.e. $10/wk for each child), but by 2004 this had risen to $37/wk (in 2000 dollars). At the other end of the income scale, the significant reduction in income tax rates enjoyed by high-income households due to the Bush tax cuts is clearly apparent in panel (d) of Figure 1, which displays the change in the (weekly) net transfer received by a two-parent, two-child family, relative to that received in 1993. The considerable variation tax regimes across states is evident in Figure 2: for example, taxes on high-income families fell in Delaware and Nebraska during this epoch (compare the plots for 2012 in panels (a) and (b) of Figure 2, to the corresponding plot in panel (d) in Figure 1). As discussed in Section 4.5 below, this substantial variation in tax and welfare policy over the sample period provides a source of exogenous variation that aids the identification of the parameters of the structural model.

4

Estimation procedure

In this section, I describe how a collection of suitable datasets may be combined to estimate the structural model by simulated method of moments (SMM). Because I will subsequently consider counterfactual policy interventions that differentially affect households according to their income level, the model needs to accurately describe household behaviour across a wide range of incomes. Accordingly, as discussed in Section 3.1 above, the model’s parametrisation gives scope to a great deal of preference heterogeneity. Estimation of such a richly parametrised model by SMM requires a collection of sample moments that is highly informative of the distribution of household resources. This, in turn, requires detailed household-level data on both consumption and time use. I draw this from several datasets (see Section 4.1 below), because the use of only one would require that dataset to record both household time use and consumption. To the best of my knowledge, there are only two relatively small datasets that meet this requirement, the Panel Study of Income Dynamics (Child Development Supplement) and the Longitudinal Internet Studies for the Social Sciences (a Dutch panel). The mechanics the of the estimation procedure are described in Sections 4.2–4.3. The choice of sample moments, and the identification of the model parameters, are discussed in Sections 4.4–4.5.

15

4.1

Data sources

Estimating the model requires data on each of the household’s choices (private and public consumption, hours worked, time with children, and housework) as well as data on nonlabour income, the wages of both parents, childcare prices and the availability of free childcare. As no single US dataset contains all the required variables, I draw principally upon four separate datasets: the American Time Use Survey (ATUS; 2003–08) for parents’ time-use decisions; the Consumer Expenditure Survey (CE; 1993–2008) for household consumption expenditures, wages and non-labour income; the Current Population Survey (CPS; 1993–2008) for hours worked; and the Survey of Income and Program Participation (SIPP; 1996, 2001, 2004, and 2008) for data on childcare. In addition, data on housework (in 1995) is drawn from the American Heritage Time Use Survey (AHTUS). These surveys also record a common set of household demographic variables (denoted by x in Section 3 above). The sex ratio (see Section 3.1 above) is computed using census data. In addition to the discussion that follows, further details of these datasets can be found in Appendix A. The A(H)TUS and the CE record time use and consumption expenditures at a much finer level of detail than is needed for the broad categories (leisure, time with children, etc.) to which the model refers. This has the considerable advantage of permitting me to exercise discretion as to how different uses of time and expenditure, should be classified. For example, I construct a measure of mother’s time with children that is focussed on those activities that have been found to be be beneficial for child development, as discussed in Section 4.4 below. Likewise, my measure of ‘public consumption’ includes expenditure on goods that are not ‘public’ in the strict sense, such as food consumed at home, because it is intended to capture all forms of expenditure that, together the mother’s housework, are inputs to home production. The ATUS provides a single 24-hour time diary for one randomly selected parent from each household, on a randomly selected day of the week. These time diaries are used to construct measures of parents’ time with children, and time spent in housework. The ATUS also records data on hours worked per week by both parents. My measure of time with children is an aggregate of time spent in activities that have been found beneficial for child development, including: reading to children (Scarborough and Dobrich, 1994), talking with parents (Tamis-LeMonda, Bornstein, and Baumwell, 2001), mealtime conversation (Snow and Beals, 2006), and novel experiences and places (Phillips, 2011); see Appendix A.1. The CE provides a finely detailed record of each household’s consumption expenditures. My classification of (non-durables) expenditures as either public or private is described in Appendix A.3. Public expenditures include such categories as food consumed at home, utilities, mortgage interest and rent (imputed for homeowners). It thus includes goods 16

that are not ‘public’ in the strict sense, such as food consumed at home, because it is intended to capture all forms of expenditure that, together the parents’ housework, are inputs to home production. I follow Blundell and Walker (1986) by constructing non-labour income less savings (y − s in (3.11) above) as non-durables consumption (including childcare expenditures) less after-tax earnings, which – as discussed in that paper – allows the static model estimated here to be related to a simple intertemporal model of household decisionmaking. For all datasets, I restrict the sample by retaining only households in which: both parents are aged of 18–58 (or 25–58, for the CE); neither parent is self-employed nor in the armed forces; the parents are the only adults (individuals aged 18 or over) living in the household’s residence (this restriction is not applied to the SIPP); and both parents’ wages (if observed) lie in the range $2.50–$250 per hour (in 2000 dollars). The sample of two-parent households is further limited to those in which the father works. I additionally restrict the SIPP sample (used to estimate equations for the price of childcare and availability of free childcare, see Section 4.3 below) so as to include only households in which both parents work, since otherwise the household would not need to purchase childcare (see Appendix A.5). Descriptive statistics for the ATUS and CE samples appear in Table 2. These indicate that single mothers spend less time with their children, have a lower educational attainment, and spend more time working than do married mothers (see rows 9, 1, and 11 respectively).

4.2

Simulated method of moments

Marshallian demands Let Y = (y − s, wm , wf , pu5 , po5 , fu5 , fo5 ) collect all the variables relevant to the household’s budget constraint, and φ the household’s preference parameters. (Though it would be tedious to enumerate φ in its entirety, it records e.g. the values of the parameters λ in (3.1), δik in (3.3), and γi,c , γi,l , γi,l and ηi in (3.4), etc.) Recall that to permit preference heterogeneity, some elements of φ are allowed to vary parametrically across households. I accordingly write φ = φ(x, ξ; θ), where x denotes the household demographic variables (age, education, race etc.), which I regard as exogenous, ξ the random disturbances capturing the unobservable preference heterogeneity in the model, and θ the parameters indexing φ(·). Thus, with α = exp(x0α βα + σα α ) specified as per (3.9) above, α is an element of ξ, and βα a subvector of θ. The feasible set for the household’s choices C = (cm , cf , ck , cq , lm , hm , tm , am , lf , hf , tf , af ) is determined by the time and budget constraints ((3.10) and (3.11) above), and the requirement that (hm , hf ) ∈ Hm × Hf . With preferences indexed by φ, maximising C over this set

17

Table 2: Descriptive statistics Couples Father

Sole

Mother

Mother

Demographics and earnings: ATUS sample (2003–08) 1

College degree (%)

32

28

18

Parents’ age

40

38

39

Hourly wages (workers only)

20

14

13

14525

4979

Sample size (# households)

Demographics and earnings: CE sample (1993–2008) 5

College degree (%)

35

32

16

Parents’ age

40

38

38

Hourly wages (workers only)

22

13

11

4903

1472

14

15

6

Housework (hr/wk)

9

14

12

Paid work (hr/wk)

45

28

34

377

322

608 201 238 168

372 220 153

-97

176

Sample size (# households)

Time use (hr/wk; ATUS, 2003–08) 9

Time with children (hr/wk)

Consumption expenditures ($/wk; CE, 1993–2008) 12

Public consumption Private Private Private Private

consumption: consumption: consumption: consumption:

total children mother father

Non-labour income less saving∗ • ∗

Sample means of indicated variables. See Appendix A for details of the samples drawn from the ATUS and CE. Units: all value magnitudes in year 2000 dollars Total non-durables consumption expenditure less after-tax earnings (see Appendix A.3).

18

yields the Marshallian demands C ∗ (Y; φ) = C ∗ [Y; φ(x, ξ; θ)] = g(Y, x, ξ; θ)

(4.1)

In principle, (4.1) should allow one to derive the density of C = C ∗ conditional on (Y, x), and thence to estimate θ by maximum likelihood. In practice, there are two obstacles to this route: (i) the evaluation of C ∗ requires solving the household’s problem numerically (no analytical solution for the household’s problem is available), which makes calculation of the likelihood challenging; and (ii) more significantly, as discussed in the introduction to this section, the joint distribution of C is available in only a few small datasets. Construction of the moments For these reasons I estimate the model by SMM. Since this does not require the recovery of the entire joint (conditional) distribution of C, I am able to combine several datasets to build up an accurate picture of the observed allocation of household resources, as described in more detail in Section 4.4 below. Regarding the selection of suitable moments, these could ordinarily be drawn from the household’s firstorder conditions: but with the present model, most of these would involve non-separable functions of variables appearing in different datasets. I therefore instead select the means, standard deviations and correlations of household choices, computed both unconditionally and conditional on the household characteristics noted in Section 4.4 below. More formally, the simulated moments are constructed by averaging across households i ∈ {1, . . . , n} and simulation draws s ∈ {1, . . . , S}, as per n S 1 X πi X m[g(Y(i) , x(i) , ξ(is) ; θ); x(i) ], m ˆ n (θ) := n i=1 S s=1

(4.2)

P where the πi ’s are the inverse population weights from the CE, normalised so that ni=1 πi = n; m is a vector-valued transformation, chosen so as to deliver a vector of sample means of the levels, squares and cross products of elements of Y; and S is the number of simulation draws. The final estimates are computed with S = 10. A further transformation puts the moments into their required form (as means, standard deviations, and correlations); I denote this by µ ˆn (θ) = ϕ[m ˆ n (θ)]. An estimate of θ is then obtained by minimising Qn (θ) =

K X (ˆ µnk (θ) − µnk )2 = kˆ µn (θ) − µn k2Wˆ 2 ω ˆk k=1

(4.3)

where µn = ϕ(mn ) denotes the corresponding vector of sample moments, constructed from 19

ˆ is a diagonal matrix with kth diagonal element ω data on household’s actual choices; W ˆ k−2 ; and kxkA := x0 Ax. The inverse weights ω ˆ k2 are computed from the estimated asymptotic d variance of the sample moment µnk ; that is, as an estimate of ωk2 in n1/2 (µnk −µk ) → N [0, ωk2 ].5 Although this does not correspond to the theoretically optimal weighting, it at least ensures that sample moments from the same dataset are weighted proportionally to the precision with which they are estimated, and renders Qn invariant to the units in which the households’ choices are measured. As discussed in Section 4.3 below, the estimation procedure described by (4.2) and (4.3) has to be modified slightly, to take account of the imputation of those elements of Y that are missing from the CE. Computation of θˆ The minimisation of Qn is undertaken as follows (separately for both the sole- and two-parent household models). I first draw a large number (20,000) of candidate values of θ, uniformly from a large, bounded subset of the parameter space (chosen conservatively, so as to span all the economically plausible values of the parameters). I evaluate Qn at each of these points, and retain θ(k) ’s corresponding to the 1000 smallest values of Qn (θ(k) ) thus obtained. From each of these points, I run 300 iterations of a Gauss-Newton optimisation routine, and then retain the 500 best-performing optimisations, following which I run a further 600 iterations of Gauss-Newton. (I use the implementation of the GaussNewton routine provided by version 10 of the Artelys Knitro software package.) I then take the best 100 points remaining, and iterate Gauss-Newton from these until convergence. The estimates reported in this paper correspond to the parameters delivering the minimum value of Qn achieved by this algorithm. As a check on my results, I also ran ten independent simulated annealing chains, each of 1 million draws in length, starting from the ten best randomly-sampled values (as drawn at the first stage of the preceding algorithm), followed by a Gauss-Newton optimisation, iterated to convergence: but in no case did this beat the estimates obtained by the preceding algorithm. Inference Under the assumption that the model is correctly specified (with parameters θ0 ), and the number of simulation draws S is fixed as n → ∞, standard results on minimum-distance-type estimators (see e.g. Gourieroux, Monfort, and Renault, 1993) imply

5

Note that ωk2 does not depend on the size of the sample used to estimate µnk : were I to instead weight the moments by their approximate finite-sample variances ω ˆ k2 /n, I would grossly over-weight moments constructed from the CPS, which has a much larger sample size than either the CE or the A(H)TUS (see Appendix A).

20

d that n1/2 (θˆ − θ0 ) → N [0, V ], where

V = (D0 W D)−1 D0 W ΣW D(D0 W D)−1

(4.4)

for D = ∂θ µ(θ0 ) the Jacobian of µ(·) = plim µ ˆn (·) at θ0 , Σ the limiting variance of n1/2 [ˆ µn (θ0 )− ˆ . (Note that V depends on S through Σ.) The estimation of V requires µn ], and W = plim W estimators for D and Σ. D can be estimated by numerically differentiating µ ˆn with respect ˆ The estimation of Σ is somewhat complicated by the use of multiple datasets, and to θ at θ. so a discussion of this is deferred to Appendix C.1. (Note that inferences based on V ignore the additional variability introduced by the imputation procedure described in the following section.)

4.3

Imputation (via simulation) of missing variables

Recall that I use the CE (1993–2008) as to construct the ‘base sample’ of households in the model: this provides me with data on x, wm (for most households), wf (only households in which the father works are retained) and y − s; this last being computed as the excess of total consumption expenditures (including childcare) over after-tax earnings (see (3.11) above). However, wm is not observed for working mothers, while childcare prices and the availability of informal care, (pu5 , po5 , au5 , ao5 ), are not observed for any household in the CE. I therefore estimate – prior to estimating the structural model – a collection of models that allow values of these missing variables to be simulated. To describe these, let each of xw,m , xw,f , xy , xp , xf,u5 and xf,o5 denote subvectors of the household demographic variables x; and each of y , w,m , w,f , p,u5 , p,o5 , f,u5 , and f,o5 i.i.d. standard Gaussian disturbances. Wages and non-labour income (less savings) These are modelled according to   0    log wf xw,f βw,f w,f       0  y − s  =  xy βy  + L  y  log wm x0w,m βw,m w,m 

(4.5)

where L is a lower triangular matrix and wm the mother’s market wage (observed only if the mother participates in the labour force). The right hand side (r.h.s.) variables in the two wage equations are: education dummies (high school, some college, college, more than college), two race dummies (black and white), parent’s age and age-squared, and time and region (nine) fixed effects. The equation for y−s involves the same r.h.s. variables as the male wage equation, along with the Case-Shiller house price index interacted with an indicator

21

for whether the household owns their residence (similarly to Lise and Seitz, 2011). (4.5) is estimated using the CE (1993–2008). The equation for wf is first estimated by OLS. The residual ˆw,f from this equation is added as a regressor to the equation for y − s (to permit estimation of L), which is itself then also estimated by OLS. Finally, the equation for wm is estimated by augmenting the regressors xw,m with the residuals (ˆw,f , ˆy ), and using a Heckman selection correction to account for the mother’s labour force participation decision. To aid identification, the first-stage probit for mother’s participation includes an urban dummy, in addition to all the r.h.s. variables from the mother’s wage equation (including (ˆw,f , ˆy )). Childcare Equations for the missing childcare variables are estimated using the SIPP (1996, 2001, 2004, and 2008), which provides a detailed record of the cost of childcare, hours spent in childcare and the availability of free informal childcare. The price of childcare for children aged 6–13 is modelled as po5 = pmin + max{x0p βp,o5 + σp,o5 p,o5 , 0},

(4.6)

where pmin = 2, and xp includes: age of the mother; two race dummies (white and black); a dummy for college education for each of the mother and father; and (nine) region dummies. (4.6) is estimated via a censored (Tobit) regression. To account for the fact that po5 is only observed for households that purchase childcare, when estimating (4.6) I augment the r.h.s. by the residual from a censored regression for the hours of childcare purchased (see Appendix C.2 for further details). To aid identification, the r.h.s. of that equation includes, in addition to xp , a Hispanic dummy and the age of the youngest child. The Hispanic dummy is significant at the 1 per cent level, and is included because Hispanic extended families tend to be larger and live closer together (Kimmel and Connelly (2007) use the presence of additional adults in a household for a similar purpose). The hours of free childcare available to the household (i.e. care provided by relatives) for children aged 6–13 are given by fo5 = max{x0f,o5 βf,o5 + σf,o5 f,o5 , 0},

(4.7)

where xf,o5 includes the same variables as xp , and additionally an indicator for whether the family owns their residence. fo5 is subjected to further censoring and sample selection, since only min{fo5 , hf , hm } is observed, and that only if hm > 0 (recall that hf is always nonzero in my sample). Accordingly, I estimate (4.7) by a doubly-censored regression regression, with 0 as the left and min{hm , hf } as the right censor point (see Appendix C.2). To control 22

for the right censoring and sample selection, I augment the r.h.s. of (4.7) with the residual from an estimated censored regression for min{hm , hf } (the r.h.s. of that equation includes xa , a Hispanic dummy and the age of the youngest child). Analogous equations to (4.6) and (4.7) hold for children aged 0–5, and are estimated in the same way (see Appendix C.2). For sole-parent households, the same approach is used. The main differences between the models estimated in this case, and those for twoparent households, are the absence of a male wage equation, and that hm now plays the role of min{hm , hf } as the right censor point for the censored regression used to estimate the parameters of (4.7). Imputation via simulation Estimates of the parameters of (4.5)–(4.7) are reported in Appendix D.1. Given these estimates, and data on (y − s)(i) and x(i) for the ith household, I can use these equations to simulate values for those elements of Y that are missing, by drawing values for the disturbances (w,m , p,u5 , p,o5 , f,u5 , f,o5 ) from a multivariate standard normal. (Since wf and y−s is always observed, the residuals (ˆw,f , ˆy ) from equation (4.5) are used to construct a simulated value for wm .) Let Y(is) denote the simulated values obtained for the sth simulation applied of the ith household, where Y(is) records the observed values of (y − s)(i) and – when these are available – wm,(i) . Then the simulated moments at θ can be written as n S 1 X πi X := m[g(Y(is) , x(i) , ξ(is) ; θ); x(i) ], (4.8) m ˆ n (θ) n i=1 S s=1 so that the only difference between the preceding and equation (4.2) is that the (partially unobserved) Y(i) has been replaced by the (partially simulated) Y(is) .

4.4

Sample moments: selection and construction

As summarised in Table 3, the moments used to fit the two-parent household model consist of selected means, standard deviations, and correlations of the household’s choices. Since my specification allows household preferences to vary according to certain (observable) demographic characteristics (see Section 3.1 above), all moments are computed both unconditionally, and conditional on: parents’ educational attainment (having some tertiary qualification, having a college degree); the age of the youngest child (whether 0–2, 3–5 or 6–17); the number of children (whether this is more than one); and the sum of parents’ wages (imputed for non-working mothers) being in the bottom quintile of the sample. Moments for parents’ time use include the (conditional) means and standard deviations of housework and time with children (leisure is excluded, since all time uses must sum to

23

Table 3: Moments used to estimate the model, two-parent households tm

tf

am

af

hm

hf

cm

cf

ck

cq

cpr

Source∗

A

A

A

A

C

C

E

E

E

E

E

Means

?

?

?

?

?

?

?

?

Std. dev.

?

?

?

?

?

?

Corr. with

hm

hf

hm

hf

hm , cpr

hm , cq

Threshold (≥)‡ Year groups§ •

∗ † ‡ §

?

?

?

?

1, 30, 50

30, 60

?

?

Est.† Est.† Est.†

For the means and standard deviations marked ?, and for all correlations and thresholds, I match both the unconditional moment, and moments conditioned on: parents’ educational attainment (having some tertiary qualification, having a college degree); the age of the youngest child (whether 0–2, 3–5 or 6–17); the number of children (whether this is more than one); and the sum of parents’ wages (imputed for non-working mothers) being in the bottom quintile of the sample. Dataset used to compute moments listed below: A = A(H)TUS, C = CPS, E = CE. Estimates of means for these variables constructed using Dunbar, Lewbel, and Pendakur (2013): see Appendix C.3. Only the unconditional mean is matched. Proportion of the sample for which that variable is (weakly) greater than the nominated threshold. I match means conditional on the year groupings: 1993–95, 1996–99, 2000–02, 2003–05, and 2006–08. Due to data unavailability, tm and tf are only observed from 2003 onwards, while am and af are not observed in 1996–2002.

105 hr/wk). The (conditional) means of hours worked are also matched; however, because of the difficulties posed by the large group of women working zero hours, and of workers of both gender working 40 hours per week, I attempt to fit the probability that hours worked exceeds certain nominated thresholds (1, 30 or 50 hr/wk for mothers; 30 or 60 hr/wk for fathers), rather than the standard deviation of hours worked. The (conditional) means and standard deviations of public and total private consumption expenditures are also used to estimate the model. Three additional moments help to calibrate the breakdown of private consumption between household members. Since it is only possible to unambiguously assign expenditures on a few commodities recorded in the CE (e.g. clothing) to either the mother or her children, I use the procedure developed by Dunbar, Lewbel, and Pendakur (2013) to infer the average breakdown of total private consumption expenditures, between parents and children (see Appendix C.3 for further details). This provides me with estimates of the unconditional means of each parent’s and of children’s private consumption, which are matched by the estimation procedure; note that no conditional means are matched, so this only contributes three additional moments (out of a total of 285, for two-parent households). The correlations between household choices that may be matched is limited by the availability of these choices in a common dataset; since hours worked appears in all datasets, I

24

am able to match the (conditional) correlations of mothers’ hours worked with: time with children, time spent in housework, public consumption and private consumption. I also match the (conditional) correlations between private and public consumption expenditures. For sole parents, the selected moments are identical to those for two-parent households, except that all moments involving father’s choices are necessarily excluded. There are a total of 285 moments (compared with 54 free parameters) for two-parent households, and 140 moments (compared with 28 free parameters) for sole-parents households.

4.5

Identification of structural parameters

The evolution of tax and welfare policies over the sample epoch provides the main source of exogenous variation in households’ budgets, which should help to identify the parameters of the model. Substantial policy variation – both over time and across states – is clearly illustrated by Figures 1 and 2. That this has influenced household behaviour is evident from the highly significant coefficient estimates obtained in a (first stage) regression of after-tax earnings on the simulated instruments discussed in Section 2 above: these instruments are jointly significant (controlling for state and year effects, and parents’ characteristics) at the one per cent level, in all cases. Since none of the moments matched by SMM condition on location, the estimation procedure does not exploit any region-specific trends in household resource allocation that might have been induced by this policy variation. Nonetheless, I show in Section 5.4 below that the estimated model is able to match region-specific trends in household time use and consumption over 1993–2008 with a high degree of accuracy. I interpret the model’s success in capturing how households respond to changes to regionspecific variation in welfare policies as implying that variation in these policies must be highly influential in pinning down the values of the estimated parameters. With regards the ability of the selected moments to identify the model parameters, it may be noted that in contrast to much of the prior empirical literature that estimates intrahousehold models using only consumption data (Apps and Rees, 1996; Lise and Seitz, 2011; Dunbar, Lewbel, and Pendakur, 2013), I additionally have time use at my disposal for this purpose. The correlations among household choices, and between those choices and wages, ought to be particularly informative as to the trade-offs households face when deciding between alternative possible uses of their resources, and thus aid the identification of the model parameters. Although it is not possible for me to include all such correlations – because not all variables appear in a common dataset – I am able to include correlations between all household choices and hours worked (see Table 3 above). A detailed numerical analysis of the extent to which these and other moments contribute to parameter identification is given

25

in Section 5.5 below.

5

Estimates and model fit

This section begins with a discussion of the estimates of the structural parameters (Section 5.1). Paralleling the 2SLS estimates reported in Section 2 above, I find some substantial differences between the estimates obtained for sole- and two-parent households, which are discussed in some detail below. I then show that the model performs well according to three criteria. Firstly, the model does a good job matching the conditional moments fitted by the SMM procedure: in particular, it is able to reproduce observed variation in household behaviour across demographic subgroups (Section 5.2). Secondly, elasticity estimates from the model match the signs of the 2SLS estimates reported in Section 2 above. Thirdly, the model is able to match ‘out-ofsample’ variation in household choices: that is, variation observed during the sample period, but which is not exploited by the estimation procedure. More specifically, I show that the observed region-specific trends in household choices correlate closely with those predicted by the model, even though none of the moments used to estimate the model condition on location (Section 5.4). The good performance of the model in this respect lends greater credibility to the counterfactual exercises subsequently conducted in Section 6, which involve changes to welfare policy different from those observed historically. I also investigate empirically how different subsets of the selected moments contribute to the identification of the model’s parameters (Section 5.5), to complement the heuristic discussion of parameter identification given above. I quantify this by computing how the precision of the parameter estimates would be affected if certain moments were ignored by the estimation procedure. The results indicate that time-use moments and correlations among household choices contribute significantly to the identification of the structural parameters.

5.1

Parameter estimates

Means of behavioural parameters The average values of the behavioural parameters implied by the estimates, conditional on certain household characteristics, are displayed in Tables 4 and 5. (The underlying parameter estimates are given, along with standard errors, in Appendix D.2.) The inputs to home production (of the public good) are much more complementary in sole-parent than in two-parent households: the mean elasticity of substitution is estimated at 0.22 for the former and 0.89 for the latter (see rows 21–22 of Table 4 and rows 11–12

26

Table 4: Mean parameter estimates, two-parent households Param.

Subgroup

Est.

(s.e.)

Pareto weight on mother’s utility ∗ 1

λ

Full sample High: sex ratio† High: wage ratio High: average wage High: age difference

Param.

Subgroup

Est.

(s.e.)

Scaling of children’s utility ∗ 1.00∗ 1.00 1.12 0.94 1.00

α (0.07) (0.08) (0.07) (0.05)

Mother

Full sample Child aged 0–5 More than 1 child

1.00∗ 1.61 (0.17) 1.16 (0.08)

Father

6

γm,c

Full sample College degree

0.19 0.18

(0.02) (0.02)

γf,c

Full sample College degree

0.25 0.24

(0.06) (0.07)

8

γm,l

Full sample College degree

0.39 0.42

(0.04) (0.05)

γf,l

Full sample College degree

0.68 0.73

(0.14) (0.15)

10

ςm ‡

Full sample College degree

1.11 1.12

(0.26) (0.32)

ςf ‡

Full sample College degree

1.19 1.18

(0.35) (0.56)

γt,m

Full sample Child aged 0–5

0.48 0.50

(0.02) (0.01)

Children 12

γk,c

Full sample Child aged 0–5

0.35 0.20

(0.15) (0.05)

14

γk,t

Full sample Child aged 0–5

0.43 0.67

(0.21) (0.19)

16

ςk ‡

Full sample Child aged 0–5 More than 1 child

1.05 1.16 0.88

(0.39) (0.30) (0.36)

ςt ‡

Full sample Child aged 0–5 More than 1 child

0.26 0.28 0.26

(0.23) (0.10) (0.21)

Public good production 19

γq,c

Full sample Child aged 0–5

0.61 0.64

(0.01) (0.01)

γa,m

Full sample Child aged 0–5

0.56 0.61

(0.01) (0.02)

21

ςq ‡

Full sample Child aged 0–5

0.22 0.20

(0.20) (0.26)

ςa ‡

Full sample Child aged 0–5

0.21 0.21

(0.28) (0.22)

•

∗

† ‡

Subgroup means of for estimated values of selected behavioural parameters; standard errors computed via the delta method. Subgroups: child aged 0–5 collects households with at least one child aged aged 0–5; more than 1 child households with two or more children; college degree households in which the mother or father (as appropriate) has at least a college degree. Because the Pareto weights and scale parameter for children’s utility do not have intrinsically meaningful units, conditional means are expressed relative to the average values of these parameters over the full sample (λ = 0.67; α = 0.17). Subgroup of households for which the indicated variable exceeds the 75th percentile. ςi denotes the elasticity of substitution corresponding to the underlying curvature parameter ηi ; these are related through ςi = 1/(1 − ηi ).

27

Table 5: Mean parameter estimates, sole-parent households Param.

Subgroup

Est.

(s.e.)

Param.

Mother

Subgroup

Est.

(s.e.)

Children

1

γm,c

Full sample College degree

0.18 0.21

(0.02) (0.02)

γk,c

Full sample Child aged 0–5

0.58 0.69

(0.07) (0.08)

3

γm,l

Full sample College degree

0.24 0.28

(0.03) (0.02)

γk,t

Full sample Child aged 0–5

0.11 0.13

(0.01) (0.03)

5

ςm †

Full sample College degree

2.40 1.50

(0.45) (0.27)

ςk †

Full sample Child aged 0–5 More than 1 child

1.66 2.19 1.75

(0.40) (0.54) (0.40)

Scaling of children’s utility ∗

Public good production 8

γq,c

Full sample Child aged 0–5

0.75 0.83

(0.01) (0.04)

11

ςq †

Full sample Child aged 0–5

0.89 2.09

(0.25) (0.61)

•

∗

†

α

Full sample Child aged 0–5 More than 1 child

1.00∗ 1.35 (0.12) 1.19 (0.05)

Subgroup means of for estimated values of selected behavioural parameters; standard errors computed via the delta method. Subgroups: child aged 0–5 collects households with at least one child aged aged 0–5; more than 1 child households with two or more children; college degree households in which the mother has at least a college degree. Because the scale parameter for children’s utility does not have intrinsically meaningful units, the conditional means are expressed relative to the average value of this parameter over the full sample (α = 0.17). ςi denotes the elasticity of substitution corresponding to the underlying curvature parameter ηi ; these are related through ςi = 1/(1 − ηi ).

28

of Table 5). The 2SLS estimates suggest a possible explanation: as sole-parent households vary their public consumption expenditures, this was estimated to have relatively more of an effect on expenditure on housing than on other categories of public consumption, compared with two-parent households (see rows 8–9 of Table 1). Thus for sole-parent households, movement along an isoquant of public good production function, by varying public consumption, mostly entails varying expenditure on housing. Insofar as this is more complementary with housework – a larger house requires more housework – than are other forms of public consumption, the isoquant tends to be more concave for sole-parent than for two-parent households. On the other hand, the inputs to the children’s utility function are estimated to be less complementary in sole-parent households than in two-parent households, the mean elasticity estimates being 1.66 and 0.26 respectively. Previous work has found that parents in two-parent households spend more of their ‘time with children’, relative to sole parents, in activities that are more sharply focused on their children’s cognitive development (see e.g. Lundberg and Pollak, 2007). Such time is accordingly less substitutable with children’s consumption and the public good, which may account for the different elasticities of substitution estimated for sole- and two-parent households, and provides a further justification of the separate estimation of models for these two household types. The children’s utility function places a relatively greater weight on public consumption in sole-parent households, and on parents’ time with children in two-parent households. The average weight on public consumption is estimated at 0.30 and 0.22 in sole- and two-parent households respectively (this coefficient is calculated as 1 − γk,c − γk,t , see rows 12 and 14 in Table 4, and rows 1 and 3 in Table 5). The difference in the average weights on parents’ time with children is more pronounced, being respectively 0.11 and 0.43 (see row 14 in Table 4 and row 1 in Table 5). Modelling the trade-offs between household resources This paper is particularly concerned with the manner in which households may trade off resources, and how this is altered by the tax and welfare system. For example, parents can choose either to spend more time with their children, or to work more so as to provide their children with more consumption. The estimates highlight the importance of three aspects of the model which differ from those estimated in the previous literature. Firstly, the more flexible CES specification – which nests the the Cobb-Douglas specification employed in previous work (see e.g. Del Boca, Flinn, and Wiswall, 2014) – allows for varying degrees of substitutability among inputs. For example, relative to the unit elasticity of substitution assumed by Cobb-Douglas preferences, the inputs to children’s utility are 29

far more substitutable in sole-parent households, and more complementary in two-parent households (see rows 16–18, col 2 in Table 4, and rows 5–7, col 2 in Table 5). Secondly, I have imposed no restrictions relating the parameters of the sole- and twoparent household models. This is intended to allow the model to better reflect the different nature of the trade-offs between resources in families with one or two parents. These differences are very evident, in particular, from the estimates of the parameters of the children’s utility function that I obtain for both family types. Finally, whereas previous work has typically subsumed children within the modelling of home production, I treat these separately, since I am interested in how children benefit from the home-produced public good. My approach may be justified by the markedly different estimates of the elasticities of substitution that I obtain for the children’s utility and home production functions.

5.2

Model fit

Tables 6–8 summarise the model’s fit to time-use and consumption moments. Overall, these are closely fit: the model is thus evidently capable of tracking the demographic patterns in household behaviour that are embodied in the conditional moments. (A complete listing of all moments is not given here because of its very considerable length, but is available from the author on request.) Time-use moments The unconditional means of time-use moments appear in row 1 and 8 of Tables 6–7: these are fit very well, with most simulated moments falling within the 95 per cent confidence interval of the corresponding sample moments. The largest discrepancies arise for fathers’ time use in two-parent households, and for the moments pertaining to maternal labour supply. Fathers’ time with children is underestimated by 1 hr/wk (row 8, cols 1–2 in Table 6), while mothers’ labour supply in both two-parent and sole-parent households is underestimated by around 2.5 hr/wk (row 1, cols 5–6 in Tables 6 and 7). The conditional means also appear to be reasonably well fit. The worst-fitting moments are again those relating to mothers’ labour supply, but even for these the model is able to track, at least qualitatively, some of the demographic variation in hours worked. For maternal time with children, this variation is matched much more closely: for example, the model is able to exactly match the almost 10-hour difference between the unconditional mean, and the mean for households with children aged 0–5 (compare rows 1 and 4, cols 1–2 of Table 6). For sole mothers, the 3.5-hour difference between these means is also matched (compare rows 1 and 4, cols 1–2 of Table 7). Demographic variation in housework is fit less well. 30

Table 6: Selected time-use moments: two-parent household Time w/children (hr/wk)

Housework (hr/wk)

Hours worked (hr/wk)

Participation rate

Siml.

Smpl.

Siml.

Smpl.

Siml.

Smpl.

Siml.

Smpl.

1

2

3

4

5

6

7

8

Mother 1

Full sample

14.70

14.96 (0.22)

13.79

14.02 (0.20)

25.27

27.88 (0.04)

0.78

0.77 (0.00)

2

Some tertiary education

14.39

13.37 (0.39)

13.45

12.88 (0.34)

25.12

28.81 (0.08)

0.78

0.81 (0.00)

3

College degree

14.50

18.12 (0.40)

14.18

13.34 (0.30)

27.72

30.46 (0.08)

0.83

0.83 (0.00)

4

Child aged 0–5

24.17

23.91 (0.41)

13.66

16.15 (0.32)

20.09

25.28 (0.08)

0.68

0.71 (0.00)

5

More than one child

18.18

16.22 (0.28)

13.07

15.07 (0.26)

23.56

26.20 (0.06)

0.75

0.75 (0.00)

6

Lowest wage quintile

17.86

15.95 (0.49)

13.29

16.83 (0.50)

21.90

23.92 (0.68)

0.70

0.65 (0.01)

7

Standard deviation

11.15

11.32 (0.09)

5.67

5.21 (0.04)

Father 8

Full sample

12.77

13.74 (0.21)

13.79

14.02 (0.20)

44.24

44.18 (0.02)

9

Some tertiary education

12.48

13.58 (0.40)

13.45

12.88 (0.34)

43.63

44.11 (0.04)

10

College degree

12.44

16.24 (0.34)

14.18

13.34 (0.30)

44.24

45.48 (0.04)

11

Child aged 0–5

18.97

16.39 (0.32)

13.66

16.15 (0.32)

42.65

44.11 (0.03)

12

More than one child

15.68

14.79 (0.26)

13.07

15.07 (0.26)

43.52

44.36 (0.02)

13

Lowest wage quintile

15.61

12.15 (0.45)

13.29

16.83 (0.50)

43.06

44.24 (0.33)

14

Standard deviation

7.59

7.59 (0.07)

4.49

4.27 (0.05)

•

Simulated moments (Siml.) corresponding to the model parameter estimates, and sample moments (Smpl.) computed from the data. Standard errors of sample moments in parentheses. All entries except standard deviation report the mean for the time-use category given in the column, conditional on the subgroup indicated in the row; standard deviation is computed for the full sample.

31

Table 7: Selected time-use moments: sole mother Time w/children (hr/wk)

Housework (hr/wk)

Hours worked (hr/wk)

Participation rate

Siml.

Smpl.

Siml.

Smpl.

Siml.

Smpl.

Siml.

Smpl.

1

2

3

4

5

6

7

8

1

Full sample

5.83

5.83 (0.18)

11.69

11.63 (0.27)

28.23

30.63 (0.08)

0.87

0.80 (0.00)

2

Some tertiary education

6.12

6.05 (0.31)

10.75

10.80 (0.39)

30.00

33.09 (0.13)

0.92

0.86 (0.00)

3

College degree

4.87

4.88 (0.35)

12.82

11.44 (0.60)

35.36

37.66 (0.17)

0.95

0.93 (0.00)

4

Child aged 0–5

9.16

9.46 (0.38)

11.92

11.96 (0.45)

24.36

27.28 (0.17)

0.81

0.74 (0.00)

5

More than one child

7.14

6.76 (0.30)

11.74

11.78 (0.43)

27.11

29.36 (0.14)

0.87

0.78 (0.00)

6

Lowest wage quintile

7.74

7.83 (0.46)

12.46

12.27 (0.61)

21.52

24.71 (1.37)

0.75

0.67 (0.03)

7

Standard deviation

4.29

4.34 (0.08)

7.48

7.63 (0.11)

•

Simulated moments (Siml.) corresponding to the model parameter estimates, and sample moments (Smpl.) computed from the data. Standard errors of sample moments in parentheses. All entries except standard deviation report the mean for the time-use category given in the column, conditional on the subgroup indicated in the row; standard deviation is computed for the full sample.

32

Table 8: Consumption moments ($/wk) Two parent household

Sole-parent household

Public

Public

Private

Siml. Smpl. Siml. Smpl.

Siml. Smpl.

Private Siml.

Smpl.

1

2

3

4

5

6

7

8

1

Full sample

405.2

381.8 (3.2)

635.3

603.8 (6.4)

278.6

267.0 (3.5)

349.3

339.5 (6.1)

2

Some tertiary education

403.1

380.3 (5.8)

640.8

623.2 (11.7)

266.5

272.0 (5.9)

366.7

340.8 (8.7)

3

College degree

446.2

444.5 (6.6)

780.1

738.8 (12.9)

361.4

341.9 (9.0)

498.0

511.9 (15.4)

4

Child aged 0–5

407.6

370.2 (4.8)

574.1

567.3 (9.3)

268.1

253.3 (8.1)

342.2

304.4 (13.9)

5

More than one child

388.0

384.6 (4.0)

640.9

595.6 (7.5)

286.0

274.0 (5.5)

362.5

341.7 (9.8)

6

Lowest wage quintile

360.2

308.4 (5.1)

453.0

420.2 (9.8)

224.0

211.2 (5.8)

254.2

218.2 (8.8)

7

Standard deviation

156.3

191.7 (7.3)

351.4

408.0 (11.2)

122.3

113.4 (4.7)

197.2

196.5 (5.7)

•

Simulated moments (Siml.) corresponding to the model parameter estimates, and sample moments (Smpl.) computed from the data. Standard errors of sample moments in parentheses. All entries except standard deviation report the mean for the consumption category given in the column, conditional on the subgroup indicated in the row; standard deviation is computed for the full sample.

Consumption moments While the model is able to produce the same sort of demographic variation observed in the data, it does not fit these moments quite as closely. It does, for example, replicate the large differences between the average consumption expenditures of low-wage individuals, and the average across the full sample. For example, the difference in total private consumption of $200/wk for two-parent households, and approximately $100/wk for sole-parent households, is closely reflected by the model (compare rows 1 and 6, cols 3–4 and 7–8, of Table 8). Matching these moments is particularly important in view of this paper’s concern with the effect of policies that are mostly affect – and indeed, are principally targeted at – low-income households. Aside from this, there is relatively little demographic variation in public and private consumption – with the exception of private consumption for two-parent households in which the mother has a college degree, something which is also tracked by the model.

33

5.3

Wage and income elasticities

Estimated elasticities for selected household choices appear in Table 9, along with the estimated responses to a $1/wk increase in non-labour income (because non-labour income may be negative, I do not convert this to an elasticity). Consistent with other studies (see the review by Blundell and MaCurdy, 1999), own-wage labour supply elasticities, and the responsiveness to a change in nonlabour income, are much larger for married women than for their husbands (see rows 1, 3, and 5, cols 3 and 4 of Table 9). The estimated own-wage elasticities for married women are towards the lower end of the range of estimates reported by Blundell and MaCurdy (1999). Relative to sole-parent households, in two-parent households parents’ time with children is more responsive, and consumption expenditure (devoted to the public good and children) less responsive, to changes in wages and non-labour income. In two-parent households, parents’ total time with children increases by over an hour in response to a $100/wk increase in non-labour income, while in sole-parent households, mothers’ time with children actually decreases slightly (see rows 5 and 6, cols 1 and 2). In contrast, in sole–parent households, public consumption increases by almost $50/wk in response to a $100/wk increase in nonlabour income, as compared with only $10/wk in two-parent households (see rows 5 and 6, col. 7). Similarly, the response of children’s consumption, which increases by almost $30/wk in sole-parent households, is an order of magnitude larger than its response in two-parent households (see rows 5 and 6, col. 8). Insofar as they are consistent with the signs of the (statistically significant) 2SLS estimates presented in Section 2, the elasticity estimates reported here provide some further, ‘external’ evidence for the validity of the model. (The failure of the magnitudes to agree can be explained partly by some of the latter being elasticities, whereas the former are marginal effects, and partly by the endogeneity problems that still affect the 2SLS estimates, as were noted in Section 2 above.) Both sets of estimates imply that: in two-parent households, mothers’ time with children and housework would respond positively in response to an increase in non-labour income; whereas in sole-parent households, mothers’ time with children would respond negatively to an increase in mothers’ wages or non-labour income. Both suggest that public expenditure is more responsive to non-labour income and wages in sole-parent households than it is in two-parent households. The negative response of mothers’ time with children to income in sole-parent households might at first seem surprising, but can be accounted for as follows. As was noted in Section 5.1 above, the 2SLS estimates indicate that sole-parent households respond to an increase in their income by disproportionately raising expenditure on housing (see row 8 in Table 1). This is complementary with housework, and so tends also to raise the latter: although the 2SLS 34

Table 9: Elasticities and derivatives Time w/children

Hours worked

Housework

Consumption

tm

tf

hm

hf

am

af

cq

ck

1

2

3

4

5

6

7

8

Mother’s wage elasticity Couples Sole parents

1

-0.37 -0.02

-0.10

2.44 0.03

-0.44

0.05 0.01

0.27

0.33 0.28

0.40 0.02

-0.01

-0.03

0.01

0.01

0.00

0.01

0.13

0.56 1.39

0.17

11.01 49.43

2.08 29.24

Father’s wage elasticity Couples

3

0.00

Response to a $100/wk increase in non-labour income Couples Sole parents

5

•

0.58 -0.17

0.48

-3.88 0.00

0.00

Numerical elasticities and derivatives (aggregate) for model specified in Section 3, at estimated parameter values. Figures reflect the mean response to a perturbation to either parent’s wage or the household’s non-labour income, as appropriate.

estimate for the response of housework is is not precisely estimated, the model estimates imply a large positive response of housework (of more than 80 min/wk in response to a $1000/wk increase in income). The result is that the mother has less time available to devote to her children, and as a result this falls somewhat (by about 10min/wk).

5.4

Model fit: external validation

I next examine the model’s ability to match variation in household behaviour not targeted by the estimation procedure (‘out-of-sample’ variation). Specifically, for a number of household choices – hours worked, and public and private consumption – I compute the mean change between 1993–95 and 2004–08, for each of nine regions in both sole- and two-parent households (see Appendix A.6 for the definitions of these regions). Since the SMM estimator does not match moments that condition on location, these region-specific trends in household behaviour do not directly inform the parameter estimates. Nonetheless, the observed trends in household choices across different regions are closely tracked the model. This is particularly true for public consumption: the correlation between the observed and predicted changes being 0.86 and 0.73 for sole- and two-parent households respectively. The model also has some success in matching regional variation in hours worked, the corresponding 35

correlation for maternal hours being 0.66 and 0.48 for sole- and two-parent households respectively, though for paternal hours worked this is only 0.30. Regional trends in private consumption are matched closely in two-parent households, with a correlation of 0.53, but not in sole-parent households (the correlation is only 0.13). Insofar as these trends in household behaviour have been shaped by the regional differences in the evolution of tax and welfare policies over the sample period, the model’s close fitting of these trends suggests that the model accurately describes households’ responses to variation in the tax and welfare system. That variation in welfare policy across states and time has substantially impacted household behaviour was established in Section 2 where such variation was enough to provide strong instrumentation for household income (this specification controls for state and year fixed effects). Panels (c) and (d) in Figure 1 also suggest that there exists substantial variation across states and time in the generosity of the tax and welfare system. This enhances the credibility of the counterfactual exercises performed using the model (see Section 6 below), which rely on it accurately predicting households’ responses to policy variation that is ‘out of sample’ by construction. The model’s success in tracking households’ responses to ‘out-of-sample’ variation in the tax and welfare system also indicates that ‘in-sample’ variation in the tax and welfare system – that is, the variation captured by the moments used to estimate the model – must be highly influential in pinning down the estimated values of the model’s parameters. (Note that the policy variation in the tax and welfare system over this period is substantial, see Section 2. This variation occurs not only across different periods, but also across families with different structures and earnings levels, and thus would have a substantial impact on the values of different moments.)

5.5

Identification of preference parameters

It is often difficult to theoretically establish the identification of complex structural models. The usual approach to this problem is to discuss heuristically how the moments selected identify individual parameters, as per Section 4.4 above. Here I complement this approach, by evaluating numerically which moments convey the information that most substantially reduces the uncertainty with which parameters are estimated. The results of this exercise confirm that correlations between household choices are particularly important for pinning down the parameter estimates, as are the moments computed from time-use data (drawn from A(H)TUS). The approach taken here involves computing the asymptotic variance according to the formula given in (4.4) above, but with the weight matrix W replaced by another in which the entries corresponding to certain moments have been zeroed out. Comparing this with

36

Table 10: Percentage increase in asymptotic standard error when indicated moments are excluded

#

λ

Public good

1

2

3

Mother’s Father’s Child. utility utility utility 4

5

6

Two-parent households Means and standard deviations 1

tm , am tf , af

30 30

8 6

52 6

32 13

9 27

80 2

3

hm hf

44 32

16 10

56 -11

11 13

19 28

10 16

5

cm , cf , ck cpr , cq

3 28

-1 0

4 0

35 1

-6 0

6 0

14 28 28

6 3 -1

40 119 1

24 -21 -25

7 28 9

15 4 29

Distribution factors

50

46

42

36

7

58

A(H)TUS moments

88

30

442

78

76

147

Correlations 7

11

hf · hm , cpr · cq hm · (tm , am , cpr , cq ) hf · (tf , qf , cpr , cq )

Sole mothers Means and standard deviations 12

tm hm am

16 33 16

52 21 150

17 22 6

57 32 -4

15

cm , ck cp cpr

2 14 14

2 2 0

-1 2 -3

13 5 -5

14 14 7

1 53 61

-11 68 65

-1 73 35

46

187

36

94

Correlations 18

21 •

hm · (tm , am ) hm · (cq , cpr ) cq · cpr A(H)TUS moments

For the group of parameters indicated in each column, the table reports the percentage increase in the ‘standard error’ (square root of the the largest eigenvalue of the asymptotic variance submatrix pertaining to those parameters) that would result if the moments indicated in each row were ignored (rather than fitted) by the estimation procedure. ‘#’ denotes the number of moments.

37

the asymptotic variance as originally computed (using W ), indicates how the precision with which the parameters are estimated is sensitive to the matching of those (now-excluded) moments by SMM.6 For individual parameters, it is appropriate to compute how the associated ‘standard error’ – the square root of the appropriate diagonal entry of the asymptotic variance matrix – would increase if a subset of moments were excluded. For groups of related parameters – such as those pertaining to public good production or children’s utility – I define the ‘standard error’ to be the square root of the largest eigenvalue of the submatrix of V pertaining to those parameters. (This can be interpreted as the standard error associated with that linear combination of those parameters that would be least precisely estimated.) Table 10 reports, for the group of parameters indicated in each column, the percentage increase in this ‘standard error’ that would result if the group of moments given in the row label were ignored by the estimation procedure. (Because the optimal weighting is not used by the SMM estimator, these figures need not be positive, though they overwhelmingly are.) Consistent with the arguments given in Section 4.5, the results suggest that correlations between household choices (rows 7–10 and 19–21) contribute substantially to the identification of the structural parameters. This is likely because these moments most clearly convey the trade-offs households face when deciding between alternative uses of their resources. For the parameters of the public good production function (col 3), among the most influential moments are the correlations involving public and private consumption. These include both: (i) the correlation between public and private consumption, and (ii) that between each of these variables and mothers’ hours worked. In sole-parent households, the standard error associated to the public good production function’s parameters would rise by 61 per cent if (i) were ignored, and 53 per cent if (ii) were ignored by the estimation procedure (rows 18–19). In two-parent households, the correlations (ii) are mostly responsible for the 119 per cent increase in the standard error that would result if a broader group of correlations involving maternal hours were ignored (row 8). These correlations may be important because they capture how households trade off public consumption against private consumption.7 6

An alternative measure of local identification is provided by the Jacobian matrix D: as is well known, a sufficient condition for θ0 to be (locally) identified is that D should have full column rank. The further that D is from rank deficiency the better that θ0 will be identified; and so the singular values of D – and of appropriately selected submatrices thereof – could also be used to measure identification. I have preferred to use V (re-computed for alternative groups of moments) rather than D because it is directly interpretable in terms of the precision with which the parameters are estimated. In any case, these two objects are closely related: e.g. it is clearly evident from equation (4.4) that the further D is from rank deficiency, the more precisely that θ0 will be estimated. 7 Consider an increase in the mother’s wage in a two-parent household. This increases hours worked and income, which also leaves less time for other activities, such as housework. The family then needs to decide how much of the additional income should be allocated to either public or private consumption. This will depend on how the household compensates the mother for any decline in her leisure, and the extent of complementarity between housework time and public consumption. In this way, the correlation between

38

Also relevant for these parameters are the mean and standard deviation of mother’s time in housework (see rows 1 and 14). The variation in these moments across subgroups reflects how households with different levels of resources are devoting mother’s time to housework. The results for the children’s utility function (col 6) suggest that information on consumption is important for estimates in sole-parent households, while mother’s time use is important in two-parent households. Recall that the 2SLS estimates in Section 2 showed that consumption is relatively more responsive to changes to household after-tax income in sole-parent households, whereas mothers’ time use is more responsive in two-parent households. This striking analogy suggests that the model estimates are driven by key variation in the data, variation identified in 2SLS, a simpler estimation procedure. Moments involving the distribution factors (row 10) appear to be important for pinning down the estimates of most parameters, not merely those pertaining to the Pareto weights, in two-parent households. Ignoring these moments would appreciably increase the standard errors of the parameters of the public good production function, and the mother’s and children’s utility functions (row 10, cols 3, 4 and 6). Recall from Section 3 that a major innovation in this paper, relative to much of the prior literature (Apps and Rees, 1988, 1996, 1997; Dunbar, Lewbel, and Pendakur, 2013), is the use of both disaggregated consumption and time-use data to estimate a model of intra-household resource allocation. Rows 11 and 21 of the table record how the precision of parameter estimates would be affected if the moments derived from time-use data were removed (88 from a total of 288 for couples, and 46 from 141 for singles). (Note that the figures in the table do not decompose additively, so e.g. those in row 11 cannot be deduced from rows 1–10.) The results clearly indicate that the significant contribution to parameter identification made by the time-use moments. Excluding these moments would raise the standard errors of estimates of the parameters of the home production function by 187 per cent and 442 per cent, and those of the children’s utility function by 94 per cent and 147 per cent, respectively in sole- and two-parent households (rows 11 and 21, cols 3 and 6).

6

Cash welfare and children’s welfare

To assess the efficacy of cash welfare in targeting household resources to children, I use the model to conduct a series of counterfactual experiments, the results of which are displayed in Table 11. In the first three experiments, each of the EITC, CTC and SNAP are respectively eliminated (the baseline) and then reintroduced (the counterfactual), while keeping all other programmes in place (cols 1–3 and 6–8). In the fourth, all three of these programmes are public and private consumption will be informative as to how households respond to these trade-offs.

39

simultaneously eliminated, and then reintroduced (cols 4–5 and 9–10). This section begins by outlining the measures of programme cost and children’s welfare that will be used to measure the proportion of (net) expenditure on these programmes that gets through to children – what I term the pass-through rate. I find that the three programmes, taken together, have an overall pass-through rate of 18 per cent. For children in sole-parent households, the pass-through rate varies widely across the three programmes, ranging from 17 per cent for SNAP to 46 per cent for EITC; but for two-parent households this variation is almost negligible. Some further analysis of this variation in pass-through rates then yields the main policy-relevant finding of this paper: welfare programmes that promote – or at least, do not substantially depress – maternal labour supply are relatively more beneficial for the children of sole mothers. Redesigning existing programmes along these lines would bring substantial benefits to the children of sole mothers, while having little effect on children in two-parent households. I conclude the section by considering the possible impact of these programmes on child cognitive development, which provides the basis for a rough cost-benefit analysis. Measuring programme cost The first two rows in Table 11 provide alternative measures of the cost of the programmes (the EITC, CTC and SNAP), expressed in per household terms for the indicated subgroup (all recipients in cols 1–4 and 6–9; EITC recipients in cols 5 and 10). Nominal benefits refers to the average transfer made to households under the specified programme (row 1; note that cols 1–3 do not add to col 4, because of the different subgroups being referred to). This reflects expenditure on the programme as it would be recorded in the government’s budget, but ignores the effect that the programme’s introduction has had on government tax revenues. These may be substantial, owing to manner in which these programmes distort the returns to market work. To account for this, I compute what I term the net cost of implementing the programme, computed by subtracting government revenues under the counterfactual (with the programme) from those in the baseline (without the programme). This is reported in row 2 of the table, and is my preferred measure of programme cost. Change in children’s welfare To quantify the extent to which each of the EITC, CTC and SNAP improve children’s welfare, in money-metric terms, I use the model to construct a compensating variation type measure of the change in children’s welfare consequent on the introduction of each programme. This is denoted CVk , and is reported in row 3 of the table; a brief discussion of its construction follows. The construction of CVk requires an appropriate vector of prices at which to value the 40

41

§

‡

†

∗

•

47

4 8

0.5

-0.1

SNAP

46 30

6

13 13

73

7 7

-9.6

0.9

SNAP

17 35

18

96 103

3

elig.

91

13 17

-9.6

0.7

SNAP

20

28

126 137

4

elig.

ALL

70

15 19

-11.4

0.9

SNAP

28

33

29 118

5

eitc

ALL

7

18 20

5

22 27

elig.

CTC

14 21

19

85 136

8

elig.

SNAP

9

17

13

62 77

elig.

ALL

17

31

148 183

10

eitc

ALL

20

-10 -35

-5.7 -0.8

1.7 2.0

54

1 5

-0.5 -0.4

0.1 0.1

19

-12 -46

-7.8 -10.8

2.4 2.8

63

-4 -17

-3.9 -4.8

1.1 1.3

20

-16 -59

-11.0 -14.1

3.4 3.8

SNAP SNAP SNAP SNAP SNAP

13 19

13

54 101

6

elig.

EITC

Two parents

Reports the result of counterfactual experiments in which each of the indicated programmes is withdrawn (the baseline), and then reintroduced. ‘ALL’ denotes a counterfactual in which the EITC, CTC and SNAP are simultaneously introduced. Units: all dollar values are expressed in year 2000 dollars; time with children and hours worked are in hr/wk; consumption in $/wk. Subgroup of households for which the displayed statistics are computed: elig. denotes subgroup of all households eligible for the indicated policy (so this varies across cols 1–4); eitc the subgroup of all households eligible for the EITC. Mean for subgroup of households indicated. Policy is the pass-through rate for the policy (CVk expressed as a percentage of net cost); lump sum is the pass-through rate for a lump-sum transfer paying the same to each household as the net cost of the policy. Mean change across indicated subgroup, as a result of the introduction of the policy (baseline outcome less counterfactual).

70

14 14

Consumption: children Consumption: public

10

Recipients (%)

1.1

Hours worked: mother Hours worked: father

8

12

-0.3

Time w/children: mother Time w/children: father

SNAP

36 27

Pass-through: policy Pass-through: lump sum

Household allocation §

22

55 60

2

1

CVk ($/wk)

Children’s welfare ‡

Nominal benefits ($/wk) Net cost ($/wk)

elig.

SNAP

Sole mother CTC

elig.

EITC

6

3

1

Programme cost †

Subgroup

∗

Counterfactual

Household type

Table 11: Counterfactual experiments: introduction of welfare programmes

household resources allocated to children. However, it is not obvious what ‘prices’ should be used here, even for goods for which market prices are available. Owing to budgetary distortions created by the tax and welfare system, and the discreteness imposed on hours worked (recall that I restrict (hm , hf ) ∈ Hm × Hf ), the usual decentralisation results no longer hold. Thus, for example, the marginal value to children of their parents’ time, at the household’s optimal allocation, will not exactly equal the market wage – although it will bear some relationship to it. (But note that e.g. the marginal value of leisure to the mother will still equal the marginal value of her time to her children.) My approach to this problem is to use the ‘prices’ that decentralise the household allocation, which are given by the household members’ marginal rates of substitution (this is similar but not identical to Chiappori and Meghir, 2014). For the public good, this entails using the implied Lindahl prices. The decentralising prices for parents’ time are denoted by (w ˜f , w˜m ) – as distinct from the (pre-tax) market wages (wf , wm ) – while the Lindahl prices for the public good are denoted (˜ pf , p˜m , p˜k ); these are collected into the vector p˜ := (w˜f , w˜m , p˜f , p˜m , p˜k ). The value of the resources allocated to children, at these prices, may then be computed as ρk = ck + p˜k q + w ˜m tm + w ˜ f tf .

(6.1)

Finally, let p˜0 denote the price vector that decentralises the household’s allocation in the baseline, prior to a policy change (typically, the introduction of a welfare programme); ρ0k the associated value of the children’s allocation at these prices, computed as per (6.1); and 1 K the utility level of the children under the counterfactual policy. The change in children’s welfare resulting from the policy change will be evaluated using  CVk = min ∆ρ ∈ R |

max 0 0

K(ck , tm , tf , q) ≥ K

1

 ,

(6.2)

BC(˜ p ,ρk +∆ρ)

where BC(˜ p, ρ) = {(ck , tm , tf , q) ∈ R4+ | ck + p˜k q + w ˜ m tm + w ˜f tf ≤ ρ}. CVk can thus be interpreted as the amount the children would need to be paid in the baseline, to be as welloff as under the counterfactual policy, if they were able to purchase consumption, time with their parents, and the public good at the prices p˜0 with an ‘income’ of ρ0k + ∆ρ.8

8

Because the household’s objective is separable in children’s utility, a second interpretation of CVk is also available (see Browning, Chiappori, and Weiss, 2011, Ch. 4). Suppose I were to provide the household with an additional lump-sum transfer ∆y, which it could allocate however it liked, but which was chosen 1 so as to be exactly sufficient to raise the children’s utility to K . Then using p˜0 to evaluate the resultant 0 0 change in the child’s resources would yield ∆ck + w ˜m ∆tm + p˜k ∆q = CVk .

42

Relative efficiency: pass-through rates To facilitate a comparison of EITC, CTC and SNAP in terms of their relative efficacy at improving child welfare, I define the pass-through rate of a policy as CVk pass-through rate = , (6.3) net cost (expressed as a percentage). This can be interpreted as the proportion of the welfare spending that gets through to children; it is reported in row 4 of Table 11. As measured by the passthrough rate, I find that 17 per cent of total (net) spending on the EITC, CTC and SNAP translates into tangible household resources allocated to children, for two-parent families (see row 4, col 9); this figure is slightly higher for sole-parent households, at 20 per cent (row 4, col 4). This implies an average pass-through rate of 18 per cent across all households, and thus that of the $121 billion spent on these programmes in 2010 (Isaacs, Toran, Hahn, Fortuny, and Steuerle, 2012), more than $22 billion was transmitted to children. (This is an underestimate, because the $121 billion quoted is an accounting figure that understates the net cost of these programmes.) Among low-income households – identified here as households eligible for the EITC – the pass-through rate rises to 28 per cent for sole-parent households, but remains at 17 per cent for two-parent households (see row 4, cols 5 and 10). By way of comparison, the proportion of full income allocated to children is 45 per cent and 31 per cent in sole- and twoparent households respectively. (Full income is total household consumption plus parents total awake time multiplied by their wages.) This is somewhat higher than the proportion of welfare expenditure that passes through to children, but exhibits a similar discrepancy between sole- and two-parent households. For two-parent households, the pass-through rates for the EITC, CTC and SNAP are all very similar, ranging from 14 per cent for SNAP to 18 per cent for CTC (row 4, cols 1–3). In contrast, for sole-parent households these rates vary over a very considerable range: in particular, SNAP has a pass-through rate of 17 per cent, whereas CTC has a pass-through rate of 46 per cent (row 4, cols 6–8). It is to the possible explanations for these differences, and their implications for policy, that I turn to next. Decoupling income and wage effects Cash welfare alters the household budget constraint in two ways: by providing the household with additional income, and by changing the returns to market work (the ‘after-tax wage rate’). To separate these two effects for the EITC, CTC and SNAP, I conduct the following ‘lump sum’ counterfactual experiments: (i) I first calculate the transfer received by eligible households, when the EITC (or CTC, SNAP) is in place, measuring this by the net cost of the programme (using nominal 43

benefits gives similar results); (ii) I then solve for the household’s choices for the baseline in which the EITC (or CTC, SNAP) is not in place; (iii) Finally, I counterfactually provide each eligible household with a lump-sum transfer of a value equal to the transfer each household received in (i), and re-solve the households’ optimisation problems. Note that the transfers provided in (iii) will differ across households, on the basis of information that would be unobservable to a policymaker. Thus the purpose of this exercise is not to compare the EITC, CTC and SNAP with alternative, feasible lump-sum transfers, but to identify how the terms on which these transfers are provided might influence the household’s allocation of resources. In particular, the lump-sum payments remove the distortionary effect that these programmes have on the returns to market work. The pass-through rates associated with the lump-sum transfers are displayed in row 5 of Table 11. For two-parent households, these pass-through rates differ little from those computed for the corresponding programmes (as displayed in in row 4). For example, the EITC has a pass-through rate of 13 per cent, while the associated lump-sum transfers has a pass-through rate of 19 per cent (see rows 4–5, col 6). In comparison, pass-through rates for sole-parent households change substantially when their benefits are instead provided as a lump sum. For these households, the pass-through rate for the CTC is 46 per cent, but declines to 30 per cent when replaced by a lump-sum transfer (rows 4–5, col 2). On the other hand, repeating this exercise for SNAP raises its pass-through rate from 17 per cent to 35 per cent (rows 4–5, col 3). Indeed, whereas the pass-through rates for CTC and SNAP for sole-parent households differ by almost 30 percentage points, those of the associated lump-sum transfers differ by less than 8 percentage points. These results clearly illustrate that the transmission of welfare payments to children in sole-parent households is much more sensitive to the shape of the benefits schedule – that is, how these payments vary with household earnings and parents’ labour supply – than it is in two-parent households. There is thus considerable scope for redesigning these schedules so as to better target household resources to children in sole-parent families, as is discussed further below. The intra-household allocation Once transfers have been stripped of their distortionary effect on after-tax wage rates, as per the ‘lump-sum’ experiments above, differences in passthrough rates largely disappear. This suggests that the response of parents’ time use to these programmes should be able to account for these variations in pass through. 44

As was discussed in Section 5.3, mothers’ time with children is much less elastic, with respect to wages and non-labour income, in sole-parent households than in two-parent households. This has a counterpart in the counterfactual experiments, with policy interventions that induce large reductions in parental labour supply tending to be associated with significant increases in parents’ time with children only in two-parent households. In sole-parent households, children derive no such benefit to offset the depressing effect that such policies have on household income, and thence consumption; as a result, these policies also tend to achieve the least improvement in children’s welfare. Clear evidence for this thesis is provided by a consideration of SNAP. This has the by far the most negative impact on labour supply of all three programmes, reducing the labour supply of both parents between 8–11 hours per week, in both household types (see row 8, cols 3 and 8). It has the lowest pass-through rate among all programmes for sole-parent households, but its pass-through rate for two-parent households differs little from that of the EITC or CTC. Consistent with the argument made the previous paragraph, SNAP is associated with a large increase in parents’ time with children in two-parent households (of around 2.5 hr/wk for each parent), but not in sole-parent households (for which this is less than 1 hr/wk; see row 6, cols 3 and 8). In two-parent households, this offsets the estimated negative response of children’s consumption; whereas the small increase in children’s consumption in sole-parent households is outweighed by the reduction in parents’ time. The main implication of these results for policymakers is that cash welfare programmes that promote maternal labour supply are relatively more beneficial for the children of sole mothers. Taking sole mothers out of the workforce does not help children, because little of time thereby freed is reallocated to children – but such programmes are costly to the state because of their negative effects on (net) tax revenues. Programmes that raise household income, and thence consumption, by promoting sole mothers’ labour supply are therefore much more efficient at improving children’s welfare. Implications for children’s cognitive outcomes Taken together, the EITC, CTC and SNAP are estimated to have a significantly positive effect on parents’ time with children, particularly among low-income (i.e. EITC-eligible) households (see rows 6–7, cols 5 and 10 of Table 11). Since the measure of ‘time with children’ used to estimate the model is an aggregate of time spent in activities that have been found beneficial for child development, it is natural to ask how this additional time might improve children’s cognitive outcomes. I do this by combining the estimated change in time with children for low-income (i.e. EITC-eligible) households, consequent on the simultaneous introduction of all three welfare programmes, with Fiorini and Keane’s (2014) estimates of the relationship between children’s 45

Table 12: Effect of welfare programmes on child test scores Children in low-income (EITC-eligible) households

Household type

Two parents

Child’s age

5

10

15

5

10

15

10.0

10.8

16.4

1.3

1.3

1.3

8.3

13.6

14.0

2.1 2.2

2.2

Peabody Picture Vocabulary Test Matrix Reasoning Test •

Sole mother

Estimated response of children’s test scores to the simultaneous introduction of the EITC, CTC and SNAP, averaged over children in EITC-eligible households. See the text and Appendix C.4 for details of the calculation. Units: percentage of a standard deviation.

time use and their cognitive development (see Appendix C.4 for details of their model and my minor modifications to their estimation procedure). Since their model is dynamic (past time use influences current outcomes) I assume that the change in parents’ time with children, as estimated to result from the introduction of the welfare programmes, affects the children in every year from their birth. The results are displayed in Table 12. For low-income households, the EITC, CTC and SNAP are estimated to collectively raise child test scores, for children in two-parent families, by around 8–10 per cent of a standard deviation for a child aged 5, and 14–16 per cent of a standard deviation for a child aged 6–15, on the Peabody Picture Vocabulary Test (PPVT) and the Matrix Reasoning Test. The estimated effects for children in low-income, sole-parent households is comparatively slight. Cost–benefit analysis Finally, the preceding may in turn be used to inform a rough cost–benefit analysis of these welfare programmes, along the lines of Kline and Walters (2015). Specifically, I here ask how much welfare spending can be justified in terms of its ultimate effect on children’s lifetime earnings, by computing r · ∆s net cost (per child)

(6.4)

(expressed as a percentage), where: ∆s is the average change in children’s scores (in units of standard deviation) on the PPVT; r is the return, in terms of discounted lifetime earnings from a one standard deviation increase in test scores, and net cost (per child) refers to the net cost of implementing the three welfare programmes, on a per child basis (recall that this accounts for the effect of these programmes on government tax revenues). Both ∆s

46

and the net cost are computed for EITC-eligible households, separately for children in soleand two-parent households. Follow the approach of Kline and Walters (2015, for p, in their notation), and using the average earnings of EITC recipients provided by Athreya, Reilly, and Simpson (2010), I estimate r to be $33,066 in two-parent households, and $38,444 in sole-parent households. Among EITC-eligible households, the ratio equation (6.4) is estimated to be 20 per cent two-parent households, but only 2 per cent for sole-parent households – in keeping with the much smaller response of time with children to the three welfare programmes, among the latter. For two-parent households, this suggests that a substantial proportion of (net) expenditure on cash welfare can be justified solely in terms of its impact on the lifetime earnings of recipients’ children. Indeed, 20 per cent is likely to be an underestimate of this, since the calculation described above only considers the impact that cash welfare has on test scores via its effects on parents’ time with children; it thus ignores the response of consumption and home production, which should also contribute positively to child outcomes. Moreover, since children in recipient households may benefit from these programmes in other ways – e.g. through improvements in health outcomes, as in Aizer, Eli, Ferrie, and Lleras-Muney, 2016 and Hoynes, Schanzenbach, and Almond, 2016 – this represents only a lower bound on total benefits associated with these expenditures.

7

Conclusion

There is a substantial literature examining the effects of the tax and welfare system – and more generally, household income – on child outcomes (e.g. see Bernal and Fruttero, 2008; Milligan and Stabile, 2009; Løken, 2010; Dahl and Lochner, 2011; Black, Devereux, Løken, and Salvanes, 2012; Løken, Mogstad, and Wiswall, 2012). Yet few papers have analysed the mechanisms within the household that underlie these effects. In this paper, I fill this gap by estimating a structural model of household decision-making, which allows me to precisely quantify the effect that welfare programmes have on the resources allocated to children. I find that 18 per cent of total (net) government spending on three major cash welfare programmes – the Earned Income Tax Credit (EITC), the Child Tax Credit (CTC) and the Supplementary Nutritional Assistance Program (SNAP) – translates into additional resources allocated to children. I also find that the transmission of welfare payments to children is much more sensitive to the shape of the benefits schedule – how these payments vary with household earnings and parents’ labour supply – in sole-parent households than it is in two-parent households. Differences across the programmes, in terms of their efficacy in promoting children’s welfare, can be accounted for in terms of their effects on parents’ time 47

use. My findings have clear implications for the design of cash welfare programmes, if these are to be made more effective at targeting household resources to children. In sole-parent households, policies that encourage maternal labour supply will be most beneficial for children. Such policies raise household income, and thus child consumption, without significantly diminishing mothers’ time with children. Redesigning existing programmes with this in mind would bring substantial benefits to the children of sole mothers, while having few adverse effects on children in two-parent households.

8

References

Abbott, B. (2015): “The effect of parental composition on investments in children,” Discussion paper, Working paper, Yale University. Aguiar, M., and E. Hurst (2007): “Life-cycle prices and production,” American Economic Review, 97(5), 1533–1559. Aizer, A. (2012): “The Gender Wage Gap and Domestic Violence,” American Economic Review. Aizer, A., S. Eli, J. P. Ferrie, and A. Lleras-Muney (2016): “The long term impact of cash transfers to poor families,” American Economic Review. Apps, P., and R. Rees (1988): “Taxation and the Household,” Journal of Public Economics. (1996): “Labour Supply, Household Production and Intra-family Welfare Distribution,” Journal of Public Economics. (1997): “Collective Labor Supply and Household Production,” Journal of Political Economy. (2002): “Household production, Full Cconsumption and the Costs of Children,” Labour Economics. Athreya, K., D. Reilly, and N. B. Simpson (2010): “Earned income tax credit recipients: Income, marginal tax rates, wealth, and credit constraints,” FRB Richmond Economic Quarterly, 96(3), 229–272. Bernal, R., and A. Fruttero (2008): “Parental leave policies, intra-household time allocations and children’s human capital,” Journal of Population Economics. Black, S., P. Devereux, K. Løken, and K. Salvanes (2012): “Care or cash? The effect of child care subsidies on student performance,” Discussion paper, National Bureau of Economic Research. Blau, D. M. (1999): “The effect of income on child development,” Review of Economics and Statistics, 81(2), 261–276.

48

Blundell, R., P. Chiappori, T. Magnac, and C. Meghir (2007): “Collective Labour Supply: Heterogeneity and Non-participation,” Review of Economic Studies, 74, 417–445. Blundell, R., P.-A. Chiappori, and C. Meghir (2005): “Collective Labor Supply with Children,” Journal of Political Economy. Blundell, R., and T. MaCurdy (1999): “Labor Supply: A Review of Alternative Approaches,” Handbook of Labor Economics. Blundell, R., and I. Preston (1998): “Consumption inequality and income uncertainty,” The Quarterly Journal of Economics. Blundell, R., and A. Shephard (2011): “Employment, Hours of Work and the Optimal Taxation of Low-Income Families,” Review of Economic Studies. Blundell, R., and I. Walker (1986): “A Life-cycle Consistent Empirical Model of Family Labour Supply Using Cross-Section Data,” Review of Economic Studies. Brooks-Gunn, J., and L. Markman (2005): “The contribution of parenting to ethnic and racial gaps in school readiness,” The Future of Children, 15(1), 139–168. Browning, M., P.-A. Chiappori, and Y. Weiss (2011): Familiy Economics. Online Book. Bruins, M. (2015): “Women’s economics opportunities and the intra-household production of child human capital,” Oxford working paper. Cherchye, L., B. De Rock, and F. Vermeulen (2012): “Married with children: A collective labor supply model with detailed time use and intrahousehold expenditure information,” American Economic Review, 102(7), 3377–3405. Chiappori, P. (1988): “Nash-Bargained Household Decision: A Comment,” International Economic Review, 29, 791–796. (1992): “Collective Labor Supply and Welfare,” Journal of Political Economy, 100, 437– 467. Chiappori, P., and I. Ekeland (2006): “The micro economics of group behavior: General Characterisation,” Journal of Economic Theory, 130, 1–26. Chiappori, P., B. Fortin, and G. Lacroix (2002): “Marriage Market, Divorce Legislation, and Household Labor Supply,” The Journal of Political Economy, 110, 37–72. Chiappori, P.-A., and C. Meghir (2014): “Intra-household Welfare,” Discussion paper, National Bureau of Economic Research. Cunha, F., and J. Heckman (2008): “Formulating, Identifying and Estimating the Techology of Cognitive and Noncognitive Skill Formation,” Journal of Human Resources. Cunha, F., J. Heckman, and S. Schennach (2010): “Estimating the Technology of Cognitive and Noncognitive Skill Formation,” Econometrica. Dahl, G., and L. Lochner (2011): “The impact of family income on child achievement: evidence from changes in the Earned Income Tax Credit,” American Economic Review.

49

Del Boca, D., C. Flinn, and M. Wiswall (2014): “Household Choices and Child Development,” Review of Economic Studies. Duflo, E. (2000): “Child health and household resources in South Africa: Evidence from the Old Age Pension program,” American Economic Review, 90(2), 393–398. Dunbar, G., A. Lewbel, and K. Pendakur (2013): “Children’s Resources in Collective Households: Identification, Estimation and an Application to Child Poverty in Malawi,” American Economic Review. Fang, H., and M. P. Keane (2004): “Assessing the impact of welfare reform on single mothers,” Brookings Papers on Economic Activity, 2004(1), 1–116. Fiorini, M., and M. P. Keane (2014): “How the Allocation of Children”s Time Affects Cognitive and Noncognitive Development,” Journal of Labor Economics, 32(4), 787–836. Gourieroux, C., A. Monfort, and E. Renault (1993): “Indirect inference,” Journal of Applied Econometrics, 8(S1), S85–S118. Hoynes, H. (1996): “Welfare Transfers in Two Parent Families: Labor Supply and Welfare Participation under AFDC-UP,” Econometrica. Hoynes, H. W., D. W. Schanzenbach, and D. Almond (2016): “Long run impacts of childhood access to the safety net,” American Economic Review. Isaacs, J., K. Toran, H. Hahn, K. Fortuny, and C. Steuerle (2012): “Kids’ share 2012: report on federal expenditures on children through 2011,” Discussion paper, The Urban Institute. Kimmel, J., and R. Connelly (2007): “Mothers’ Time Choices: Caregiving, Leisure, Home Production, and Paid Work,” The Journal of Human Resources. Kline, P., and C. Walters (2015): “Evaluating public programs with close substitutes: The case of Head Start,” Discussion paper, National Bureau of Economic Research. Lise, J., and S. Seitz (2011): “Consumption Inequality and Intra-Household Allocations,” Review of Economic Studies. Løken, K. V. (2010): “Family income and children’s education: Using the Norwegian oil boom as a natural experiment,” Labour Economics, 17(1), 118–129. Løken, K. V., M. Mogstad, and M. Wiswall (2012): “What Linear Estimators Miss: The Effects of Family Income on Child Outcomes,” American Economic Journal: Applied Economics, 4(2), 1–35. Lundberg, S., and R. Pollak (2007): “The American Family and Family Economics,” The Journal of Economic Perspectives, 21(2), 3–26. Lundberg, S., R. Pollak, and T. Wales (1997): “Do husbands and wives pool their resources? Evidence from the United Kingdom child benefit,” Journal of Human Resources, pp. 463–480. Milligan, K., and M. Stabile (2009): “Child Benefits, Maternal Employment, and Children’s Health: Evidence from Canadian Child Benefit Expansions,” The American Economic Review, 99(2), 128–132.

50

Phillips, M. (2011): Whither opportunity? Rising Inequality, Schools, and Children’s Life Chanceschap. Parenting, time use, and disparities in academic outcomes, pp. 207–228. Russell Sage Foundation. Scarborough, H. S., and W. Dobrich (1994): “On the efficacy of reading to preschoolers,” Developmental Review, 14(3), 245–302. Snow, C. E., and D. E. Beals (2006): “Mealtime talk that supports literacy development,” New Directions for Child and Adolescent Development, 2006(111), 51–66. Tamis-LeMonda, C. S., M. H. Bornstein, and L. Baumwell (2001): “Maternal responsiveness and children’s achievement of language milestones,” Child development, 72(3), 748–767. Todd, P., and K. Wolpin (2007): “The Production of Cognitive Achievement in Children: Home, School and Racial Test Score Gaps,” Journal of Human Capital. (2010): “On the Specification and Estimation of the Production Function for Cognitive Achievement,” Economic Journal.

51

52

————–for online publication————– A A.1

Data sources ATUS

The American Time Use Survey (ATUS) randomly selects one individual from each family, and asks participants to complete a time diary for one full day, recording their main activity, secondary activity and who the activity was with. All observations with incomplete time diaries are dropped, as per Guryan, Hurst, and Kearney (2008). Sample The sample (2003–08) of single mothers is restricted to households in which the mother (i) is not self-employed or in the armed forces; (ii) is aged between 18 and 58; (iii) has children below the age of 18 who she lives with; (iv) does not share a household with any other family; (v) has all demographic variables required by my estimation procedures; and (vi) has hourly wages between $2.50–$250 (in 2000 dollars), if reported. The final sample has 14525 two-parent and 2979 sole-parent households. Parents’ time with children For each parent, defined as an aggregate of: caring for and helping children, including general care of children; activities related to children’s health, including providing and obtaining medical care for children; activities related to children’s education; time spent on animals and pets with children; time spent using childcare services; time spent eating and drinking with children; time spent socialising and communicating and attending or hosting social events with children; time spent listening to music, playing games and doing hobbies with children; time spent attending performing arts, museums or any arts and entertainment with children; sports, exercise and recreation with children; religious and spiritual activities with children; volunteer activities with children; and travelling associated with any of the above activities.

A1

Parents’ time in housework For each parent, defined as an aggregate of: food/meal preparation; cleaning; laundry and clothes repair; home repairs and vehicle maintenance; domestic work; purchasing goods and services; gardening; and pet care. Variance estimates for time-use measures For time with children and housework, daily time diaries provide a consistent estimate of the average weekly time spent in these activities, but yield an upwardly biased estimate of the weekly variance (Bruins and Duffy, 2015). Using a unique Dutch dataset (Tijdsbestedingsonderzoek, 2005), which provides a complete set of daily time diaries for each respondent (i.e. one for every day of the week), I estimate these variance measures to be upwardly biased by a factor of 1.5. The sample variance moments for time with children and housework are therefore computed by multiplying the (daily) sample variance computed from the ATUS time diaries by a factor of 0.67 (equivalently, the sample standard deviation is scaled down by a factor of 0.82).

A.2

AHTUS

Data on time spent in housework in 1995 is based on time diaries provided by the American Heritage Time Use Study (AHTUS). Bad or incomplete diaries, as indicated in the AHTUS documentation, were dropped; the sample is otherwise constructed exactly as for the ATUS. The final sample has 465 two-parent and 230 sole-parent households. Time spent in housework is aggregated from the same set of activities as for the ATUS.

A.3

CE

In constructing the Consumer Expenditure Survey (CE) sample I follow Blundell, Pistaferri, and Preston (2008). The sample (1993–2008) is constructed identically to that of the ATUS except that, owing to the problems discussed in Blundell, Pistaferri, and Preston (2008), the age range in (ii) is modified to 25–58. The final sample has 4903 two-parent and 1472 sole-parent households. Public consumption expenditure This is defined to include food expenditures, property taxes, mortgage interest, rent, household operations, utilities, fuel, and public services. Non-labour income less saving Recall from the household budget constraint (3.11) that ‘non-labour income less saving’, y − s, is the difference between household non-durables consumption and after-tax earnings. Non-durables expenditure is defined as total consumption expenditure less expenditures on: house furnishings and equipment; outlays for other lodging;

A2

mortgage principal outlays; maintenance, repairs and insurance; educational expenditures; and personal insurance and pensions.

A.4

CPS

The Current Population Survey (CPS) sample is taken from the ‘CPS Merged Outgoing Rotation Groups’ as provided by the NBER (1993-2008). The sample is restricted analogously to the ATUS. The final sample has 211093 two-parent and 58593 sole-parent households.

A.5

SIPP

The Survey of Income and Programme Participation (SIPP) sample (1996, 2001, 2004, and 2008) is constructed identically to the CE, except that I retain only those households in which both parents work (otherwise there is no need for childcare) and (vi) above is not imposed. The final sample has 7291 two-parent and 2986 sole-parent households. For each household, I calculate the total amount spent on childcare, including expenditures on family daycare, sports and club events. The price of childcare per child, per hour, is computed by dividing this expenditure by the total number of hours spent in childcare. These calculations are performed separately for children 5 and under, and those aged 6–13. The amount of free childcare available to the household, per child, is calculated as [min(hf , hm ) − hc,u5 ]+ for children 5 and under, where hf (resp. hm ) denotes father’s (resp. mother’s) hours worked, and hc,u5 the (average) time spent in childcare by children 5 and under ([a]+ = max{a, 0}). For children aged 6–13, the preceding is modified to [min(hf , hm ) − hc,u5 − 30]+ since these children attend school for 30 hours each week.

A.6

Regional groupings

Throughout the paper, I use the following nine regional groupings, taken from the US Census Bureau: New England (CT, ME, MA, NH, RI, VT); Middle Atlantic (NJ, NY, PA); East North Central (IL, IN, MI, OH, WI); West North Central (IA, KS, MN, MO, NE, ND, SD); South Atlantic (DE, DC, FL, GA, MD, NC, SC, VA, WV); East South Central (AL, KY, MS, TN); West South Central (AR, LA, OK, TX); Mountain (AZ, CO, ID, MT, NV, NM, UT, WY); and Pacific (AK, CA, HI, OR, WA).

A3

B

Calculation of taxes and welfare payments

The calculation of household’s after-tax earnings is an important component of the model. (Recall the discussion of the household’s budget constraint given in Section 3.2.) Details of the relevant tax and benefit calculations, and the sources that are drawn upon to make these calculations, are given below. In general, after-tax earnings will vary according to: before-tax earnings; the number of children (aged 0–2, 0–5, 0–13, 0–16 and 0–18) in the household; the age of youngest child; childcare expenditures; and the hours worked by the mother.

B.1

Taxes

The US tax system permits married couples to file either joint or separate tax returns. My modelling assumes that couples always file jointly, because of the (usually) significant financial advantages from doing so. Federal taxes Federal tax parameters are taken from the Urban Institute’s Tax Policy Centre (TPC) database. I first determine the tax-free threshold as the sum of the standard deduction and the dependent exemption (multiplied by the number of dependants). Then the usual bracketing formula is applied to income above this threshold. State taxes State tax parameters, for those states that do have income taxes, are drawn principally from the State Tax Handbook. These either take the form of a flat rate on beforetax incomes (CT, IL, IN, NH, PA), or involve a similar bracketing calculation to that used to compute federal taxes.

B.2

Tax credits

Parameters for the tax credits are drawn from the TPC. Each of these are means tested, and are calculated on the following basis. As before-tax earnings y increases from zero, the value of the credit increases linearly from zero, at rate rin . This is the phase-in region: once a certain income level is reached, denoted y in , the credit stops increasing and remains at b = rin · y in . Finally, beyond a certain threshold y out , the value of the credit declines linearly with income to zero, at rate rout (the phase-out region). Mathematically, the value of the

A4

Figure 3: Federal tax credits Weekly transfers and income (year 2000 dollars) (a) EITC

(b) CTC

80

40 1993

70

1994

35

1996

1996

2000 60

2000 30

2004

2004

2008

40

15

20

10

10

5

0

100

200

300

400

500 600 Income

700

800

900

1000

2012

20

30

0

•

2008 25

2012

Net transfer

Net transfer

50

0

0

500

1000

1500 Income

2000

2500

Calculated for a two-parent family with two children aged 0–16.

credit for which a household is eligible can be expressed as:    r ·y   in credit(y) = b    max{r

out

if y ≤ y in (phase-in) if y ∈ [y in , y out ]

(B.1)

· [b − (y − y out )], 0} if y ≥ y out (phase-out)

The household’s tax liabilities are then reduced by the magnitude of the credit. If the credit is refundable, then the household receives any excess of the credit over its tax liabilities as a payment. Earned Income Tax Credit This credit is refundable. The formula for the federal EITC is as in (B.1); the parameters (b, rin , rout , y in and y out ) vary over time and with family size (the number of children aged 0–16). The value of the credit for a two-parent, two-child family, is displayed in Figure 3. State EITC is an additional credit that is calculated as a proportion of federal EITC benefit; this proportion varies across time and states (and is zero in most states). Child Tax Credit This credit is non-refundable. The formula for the credit is (B.1); again its parameters vary over time and family size. The value of the credit is depicted in

A5

3000

Figure 3, which illustrates that the credit is not as strictly means tested as is the EITC. Child and Dependent Care Tax Credit This credit is non-refundable. Its formula is given by    min{rin · c, cmax } if y ≤ y in   n  o y−y in cdctc(y) = min rin − 0.01 · 2000/52 · c, cmax if y ∈ [y in , y out ]    min{r · c, c } if y ≥ y out max out where c denotes the household’s childcare costs, and cmax the maximum value of the benefit.

B.3

Welfare payments

Supplemental Nutritional Assistance Program Parameters for SNAP are taken from Eslami, Leftin, and Strayer (2012, Appendix G). Letting ynet = 0.8y−dstd , where dstd denotes the standard deduction, the benefits (in the form of food stamps) for which a household is eligible is given by:

snap =

 b − 0.3 · y

if y ≤ 0.3 · ypov and ynet ≤ ypov

0

otherwise

net

where b denotes the maximum benefit payable, and ypov the official poverty threshold. Temporary Assistance for Needy Families Parameters for TANF are taken from the Urban Institute’s Welfare Rules Database. The benefits for which are household is eligible under TANF is calculated, for most states (excepting CT, HI, MO, NV, NJ, ND and WI), from the positive part of tanf = min{b, r · [b − (y − d)]} (B.2) where b denotes the maximum benefit claimable, r is the rateable percentage, b the benefit standard, y before-tax earnings, and d the disregard. Both the benefit standard and the maximum benefit vary with the number of individuals in the family. In most states, TANF can only be claimed for a maximum of five years, and the value of the disregard depends on the number of years for which TANF has already been claimed. To simplify the calculations – and because I do not observe the number of times that an individual has claimed TANF – I set d to the average of these values. The disregard also depends on the amount that the household spends on childcare, which I take account of in my calculations.

A6

Aid to Families with Dependent Children AFDC is calculated using an identical formula to (B.2); its parameters are drawn from the Urban Institute’s Trim3 database.

C

Estimation procedure: further details

C.1

Inference: estimation of Σ

Let n index the sample size in the CE, which therefore also indexes the size of the model, and nA the sample size of the ATUS. (Although some sample moments are constructed from the CPS, this is ignored here to simplify the the exposition.) I suppose that nA = λn, where λ is held constant as n → ∞; the number of simulations draws, S, is also held fixed. ˆ − µ ]. To construct an estimate of Σ, recall from Σ is the limiting variance of n1/2 [ˆ µn (θ) n equation (4.2) and the surrounding text that the simulated moments (at θ) are given by µ ˆn (θ) = ϕ[m ˆ n (θ)], where n S n S n 1X1X 1X 1X1X π m[g(Y(i) , x(i) , ξ(is) ; θ); x(i) ] =: ψ (θ) =: Ψ (θ). m ˆ n (θ) = n i=1 S s=1 i n i=1 S s=1 is n i=1 iS

The sample counterparts mn of these moments are computed using the CE and the ATUS: to reflect this, I partition these as # # " P n ∗ 1 ψ mn1 i1 = n1 Pi=1 mn = nA ∗ mn2 i=1 ψi2 nA "

(CE) (ATUS)

Under the assumption of a correctly specified model, both mn and m ˆ n (θ0 ) will converge in probability to m0 = Emn . Since the CE and ATUS samples are independent, " n1/2

m ˆ n (θ0 ) − m0 mn − m0

#

 1 Pn     [Ψ (θ ) − m ] U R 0 iS 0 0 12 i=1 n P   d    ∗ = n1/2  n1 ni=1 [ψi1 − m01 ]  → N 0, R21 V11 0  PnA ∗ 1 0 0 λ−1 V22 i=1 [ψi2 − m02 ] nA

0 ∗0 ∗0 0 where R12 = R21 , and m0 = (m001 , m002 )0 is partitioned conformably with (ψi1 , ψi2 ) . Let Ω denote the limiting variance matrix on the r.h.s. of the preceding. To estimate the nonzero

A7

blocks of Ω, I use "

# " #" #0 n ˆ ˆ ˆ ˆ ˆ 12 X Uˆ R Ψ ( θ) − m ˆ ( θ) Ψ ( θ) − m ˆ ( θ) 1 iS n iS n = ∗ ∗ ˆ ˆ n i=1 R21 V11 ψi1 − mn1 ψi1 − mn1 nA X 1 ∗ [ψ ∗ − mn2 ][ψi2 − mn2 ]0 , Vˆ22 = nA i=1 i2

ˆ (note that λ is computed as nA /n). Finally, letting which are assembled to construct Ω Jϕ (m) = ∂m ϕ(m) denote the Jacobian of ϕ with respect to m, I can estimate Σ using h i ˆ ˆ Σ = Jϕ (m ˆ n ) −Jϕ (mn ) Ω

C.2

"

0

#

Jϕ (m ˆ n) . −Jϕ (mn )0

Censored regression models for childcare variables

Price of childcare The price of childcare is affected by a sample selection problem, being observed only for those households that purchase childcare. Let c = max{x0c βc + uc , 0} denote the hours of childcare purchased for children aged 0–5. Assuming that (uc , up ) are jointly Gaussian, I may write p − pmin = max{x0p βp + up , 0} = max{x0p βp + ρc uc + σp p , 0} for the price of childcare for children aged 0–5, where p ⊥ ⊥ uc and xc = (x0p , zc0 )0 . xp consists of a constant, the age of the mother, two race dummies (white and black), a dummy for tertiary education, and (nine) region dummies. zc consists of a Hispanic dummy and the age of the youngest child. I adopt the following control function approach to estimating βp : (i) Tobit regression of c on xc : yields residuals uˆc . (ii) Tobit regression of p − pmin on (xp , uˆc ), using the subsample c > 0. To produce separate estimates of βp for children aged 6–13, I run the above procedure a second time, with (c, p) now recording the hours purchased and price of childcare for children aged 6–13.

A8

Hours of free childcare Estimation of the parameters in (4.7) is complicated by sample selection and two-sided censoring. Let h = min(hf , hm ) denote the minimum of mother’s and father’s hours worked; suppose it is generated according to: h = max{x0h βh + uh , 0}. Assuming that (uh , uf,u5 ) are jointly Gaussian, I may write fu5 = max{x0f,u5 βf,u5 + uf,u5 , 0} = max{x0f,u5 βf,u5 + ρh uh + σf,u5 f,u5 , 0}, for the hours of free childcare available, for children aged 0–5, where f,u5 ⊥ ⊥ uh and xh = 0 0 0 (xf,u5 , zh ) . xf,u5 consists of the same variables as xp above, and additionally records whether the family owns their residence. zh consists of a Hispanic dummy and the age of the youngest child. Now note that: (i) fu5 is observed only if h > 0; and (ii) even if h > 0, the observed values of fu5 are censored to lie in the interval [0, h]; i.e. I observe f˜u5 = min{fu5 , h} for these households. I therefore adopt the following control function approach to estimating βf,o5 : (i) Tobit regression of h on xh : yields residuals uˆh . (ii) ‘Double tobit’ regression of f˜u5 on (x, uˆh ), with h as the right censor point (and zero as the left censor point), using the subsample h > 0. To estimate the parameters of (4.7) for children aged 6–13, the same approach is used except that fo5 is observed only if h − 30 > 0, and is censored to lie in the interval [0, h − 30]. Sole parents Exactly the same approach is used to estimate equations for the price of childcare and hours of free childcare, for sole-parent households. The main differences between the models estimated for two-parent households are the absence of a male wage equation, and that h = hm rather than min(hf , hm ).

C.3

Household members’ private consumption

Although the CE records the total private consumption of the household, it does not provide a breakdown of that consumption across family members. However, it does provide data on

A9

some private assignable goods, in particular clothing, which has been used in prior literature to estimate household members’ individual private consumption. Here I adapt the procedure developed by Dunbar, Lewbel, and Pendakur (2013) to estimate the breakdown of private consumption. This is in turn used to construct (unconditional) sample means for mother’s and children’s consumption, which are matched in estimation (as discussed in Section 4.4). Suppose that there are N consumption goods: let c˜ij denote expenditure on the jth P consumption good (from some set J) for person i ∈ {f, m, k}, and ci = j∈J c˜ij his (or her) total private consumption expenditure. Under certain conditions on individual preferences, given by Dunbar, Lewbel, and Pendakur (2013, Sec. II), the optimal choice of c˜ij will be related to ci via c˜ij = (β0i + β1i log ci ). (C.1) ci Dunbar, Lewbel, and Pendakur (2013) further show that if preferences are ‘similar across types’, in the sense that β1f = β1m = β1k , then c˜ij c˜ij = ηi C ci

(C.2)

for i ∈ {f, m, k} and j ∈ J, where C = cf + cm + ck denotes the household’s total private consumption expenditure. With the aid of (C.1) and (C.2), it is possible to recover ci from observations on C and (˜ cf 1 , c˜m1 , c˜k1 ) alone, i.e. from data on total household private consumption expenditure and on a single assignable good – in my case, clothing. This involves first estimating the slope coefficients in the regression c˜i1 = αi + (βηi ) log C = αi + γi log C, C which follows from substituting (C.1) into (C.2), and recognising that ci = ηi C. I then P compute ηˆi = γˆi / j∈{f,m,k} γˆj , whence ηˆi C provides an estimate of ci . Separate estimates are calculated for families with different numbers of children, and (analogously) for soleparent households.

C.4

Child test scores

Using the same (Australian) data as Fiorini and Keane (2014), I estimate a version of their ‘CV’ specification (‘contemporaneous and lagged inputs, and lagged test scores’; see their Tables 10 and 11), modifying their measure of parents’ time with children so that it is commensurate with that which is used in my model. Since I am interested in the cognitive

A10

development of children in low-income households, I restrict the sample used to estimate their model to households whose income is below 175 per cent of the (Australian) poverty line (see Melbourne Institute, 2006), which corresponds to the point at which households (in the US) become ineligible for the EITC. (My coefficient estimates, which do not differ greatly from those of Fiorini and Keane (2014), are available on request.) As their model allows each possible use of children’s time to separately affect test scores, my calculations assume that any hypothetical increase in parents’ time with children has the effect of proportionally decreasing the time spent in all other activities, where these proportions are taken from the sample averages of children’s time use. Since their model is dynamic, I assume that the change in parents’ time with children, resulting the counterfactual introduction of a welfare programme, occurs in every year for the child from birth. A further complication arises for families with more than one child: the estimates provided by Fiorini and Keane’s (2014) model are on a per-child basis. Since my structural model does not separately identify the children in the household (there is simply a ‘children’s utility function’ in the model), and parents’ time with children is to some extent non-rivalrous, I cannot precisely determine the time that each child spends with their parents. I therefore construct upper and lower bounds for the amount of time spent with each child: the lower bound is calculated on the basis that the parents’ time is spent separately with, and split equally between, their children; the upper bound on the basis that this time is spent simultaneously with all children. The results reported in the paper are an average of these two estimates.

A11

D

Parameter estimates

D.1

First stage parameters Table 13: Log wages: two-parent households log wm

log wf

1993-98

1999-03

2004-08

1993-98

1999-03

2004-08

High school

0.17 (0.08)

0.36 (0.05)

0.35 (0.06)

0.30 (0.03)

0.28 (0.01)

0.24 (0.02)

Some college

0.38 (0.09)

0.63 (0.06)

0.62 (0.07)

0.41 (0.03)

0.43 (0.01)

0.43 (0.02)

College

0.63 (0.09)

0.90 (0.05)

0.93 (0.07)

0.60 (0.04)

0.67 (0.01)

0.73 (0.02)

Post-college

0.87 (0.10)

1.20 (0.07)

1.04 (0.07)

0.79 (0.04)

0.85 (0.02)

0.92 (0.02)

White

-0.07 (0.07)

0.04 (0.04)

-0.17 (0.05)

0.05 (0.04)

0.25 (0.02)

0.19 (0.04)

Black

0.06 (0.10)

0.14 (0.05)

-0.20 (0.07)

-0.00 (0.05)

0.14 (0.03)

-0.01 (0.04)

Age

0.03 (0.03)

0.08 (0.01)

0.05 (0.01)

0.08 (0.01)

0.09 (0.00)

0.04 (0.01)

Age2 /100

-0.07 (0.02)

-0.09 (0.01)

-0.03 (0.01)

-0.03 (0.04)

-0.08 (0.01)

-0.06 (0.02)

Education

Race

•

Coefficient estimates for the wage equations in (4.5). Standard errors in parentheses. Units: wages in $/hr (2000 dollars). All explanatory variables are interacted with dummy variables for the years 1993–98, 1999–2003, 2004–08. Coefficients on region (nine) and year dummies are also estimated (suppressed from the table).

A12

Table 14: Non-labour income less saving: two-parent households y−s 1993-98

1999-03

2004-08

High school

-63.24 (22.79)

-97.51 (13.51)

-75.45 (12.84)

Some college

-75.53 (23.38)

-63.11 (13.50)

-133.53 (12.01)

College

-127.83 (24.24)

-177.23 (15.54)

-270.18 (12.97)

Post-college

-112.00 (31.11)

-211.66 (19.94)

-403.13 (17.72)

White

96.90 (24.63)

-92.42 (17.65)

-202.29 (28.47)

Black

-35.01 (36.79)

-36.60 (26.06)

-111.32 (34.81)

-40.69 (11.00)

-43.80 (4.63)

-13.41 (4.59)

Age2 /100

0.45 (0.13)

0.49 (0.05)

0.07 (0.06)

Case-Shiller house price index

-1.23 (4.53)

-987.74 (122.06)

527.75 (57.32)

Education

Race

Age

•

Coefficient estimates for the y − s equation in (4.5). Standard errors in parentheses. Units: y − s in $/wk (2000 dollars). All explanatory variables are interacted with dummy variables for the years 1993–98, 1999–2003, 2004–08. Coefficients on region (nine) and year dummies are also estimated (suppressed from the table).

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Table 15: Log wages and non-labour income less saving: sole parents y−s

log wm 1993-98

1999-03

2004-08

1993-98

1999-03

2004-08

Education High school

0.28 (0.12)

0.58 (0.12)

0.95 (0.20)

81.95 (35.19)

18.62 (24.99)

-17.33 (40.70)

Some college

0.46 (0.14)

1.05 (0.14)

1.34 (0.26)

75.21 (37.07)

-7.82 (25.47)

-0.31 (40.53)

College

0.84 (0.19)

1.29 (0.16)

0.05 (0.07)

153.50 (58.26)

-37.92 (38.62)

-24.08 (76.90)

Post-college

1.06 (0.20)

-0.11 (0.05)

0.13 (0.03)

108.17 (101.31)

-77.83 (56.02)

-191.41 (128.20)

White

-0.04 (0.09)

0.04 (0.02)

-0.00 (0.00)

52.53 (105.20)

63.37 (42.45)

-21.21 (78.51)

Black

0.15 (0.06)

-0.00 (0.000)

0.02 (0.17)

22.07 (107.90)

41.17 (45.05)

-82.8 (81.73)

Age

-0.00 (0.00)

0.35 (0.15)

0.07 (0.02)

-4.75 (18.18)

5.91 (8.76)

-11.91 (15.98)

Age2 /100

34.89 (9.02)

57.46 (15.69)

-0.08 (0.03)

2.40 (22.89)

-7.73 (11.37)

13.60 (22.00)

-20.20 (5.97)

-1264.99 (448.55)

286.75 (255.98)

Race

Case-Shiller house price index •

Coefficient estimates for (4.5). Standard errors in parentheses. Units: wm in $/hr (2000 dollars), y − s in $/wk (2000 dollars). All explanatory variables are interacted with dummy variables for the years 1993–98, 1999–2003, 2004–08. Coefficients on region (nine) and year dummies are also estimated (suppressed from the table).

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Table 16: Price of childcare Couples pu5 po5

Sole parent pu5 po5

Age Mother

-0.47 (0.04)

0.53 (0.08)

6.75 (0.45)

5.76 (0.44)

Father

0.10 (0.03)

-0.00 (0.07)

Mother

-5.49 (0.38)

2.82 (0.77)

-13.26 (1.60)

-11.66 (1.37)

Father

0.82 (0.39)

-9.61 (0.84)

White

-9.40 (0.44)

3.09 (1.18)

49.29 (2.51)

35.64 (1.45)

Black

-13.0 (0.70)

7.71 (1.45)

44.13 (2.74)

32.14 (1.75)

25.97 -27.80 (1.11) (2.78)

-178.46 (9.29)

-150.77 (8.51)

College educated

Race (mother)

Constant •

Parameter estimates for (4.6). Standard errors in parentheses.

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Table 17: Hours of free childcare available Couples au5 ao5

Sole parent au5 ao5

Mother

0.03 (0.46)

0.17 (0.19)

0.49 (2.41)

0.13 (1.04)

Father

0.04 (0.42)

-0.01 (0.16)

Mother

1.80 (3.78)

1.53 (1.66)

2.77 (6.84)

2.20 (2.39)

Father

0.40 (3.96)

1.61 (2.09)

White

-0.01 (6.87)

0.06 (2.79)

8.04 (12.64)

1.28 (4.42)

Black

0.24 (10.45)

-0.00 (4.13)

8.21 (12.93)

0.84 (4.41)

0.09 (4.29)

0.01 (2.41)

-0.40 (6.45)

-2.40 (3.23)

-2.66 (13.52)

-0.99 (7.03)

-23.27 (43.63)

-7.44 (20.78)

Age

College educated

Race (mother)

Hispanic Own home Constant •

Parameter estimates for (4.7). Standard errors in parentheses.

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D.2

Structural parameters Table 18: Parameter estimates, two-parent households Pareto weights λ = Λ(βλ0 xλ + σλ λ ) βλ : constant

-0.89 (0.33)

βλ : mean of parents’ wages

0.12 (0.08)

βλ : sex ratio

0.06 (2.32)

βλ : mother’s age less father’s age

-0.01 (0.03)

βλ : ratio of mother’s to father’s wage

-0.47 (0.10)

σλ

0.03 (0.14)

Home production γ ˜q,c = exp(x0q,c βq,c + σq,c q,c ) βq,c : constant

0.42 (0.07)

σq,c

0.63 (0.02)

βq,c : youngest child aged 5 or under

0.22 (0.05)

βq,η : youngest child aged aged 0–5

-0.57 (1.67)

βa,m : youngest child aged aged 0–5

0.30 (0.04)

βa,η : youngest child aged 5 or under

0.08 (2.18)

ηq = 1 − exp(x0q,η βq,η ) βq,η : constant

-3.36 (1.27)

γ ˜a,m = exp(x0a,m βa,m + σa,m a,m ) βa,m : constant

0.13 (0.04)

σa,m

0.03 (0.13)

ηa = 1 − exp(x0a,η βa,η ) βa,η : constant •

-3.88 (1.85)

Standard errors in parentheses.  always denotes an i.i.d. standard Gaussian disturbance.

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Table 19: Parameter estimates, two-parent households Mother’s preferences γ ˜m,c = exp(x0m βm,c + σm,c m,c ) βm,c : constant

0.81 (0.19)

σm,c

0.07 (0.05)

βm,c : some tertiary education

-0.03 (0.15)

γ ˜m,l = exp(x0m βm,l + σm,l m,l ) βm,l : constant

0.54 (0.14)

βm,l : some tertiary education

0.13 (0.09)

βm,l : wage equation residual

0.97 (0.04)

σm,l

0.04 (0.11)

βm,η : constant

-0.14 (0.06)

βm,η : some tertiary education

-0.04 (0.06)

βm,η : youngest child aged 0–5

0.02 (0.07)

ηm = 1 − exp(x0m,η βm,η )

Father’s preferences γ ˜f,c = exp(x0f βf,c + σf,c f,c ) βf,c : constant

-0.83 (0.75)

σf,c

0.52 (0.29)

βf,c : some tertiary education

-1.51 (2.09)

γ ˜f,l = exp(x0f βf,l + σf,l f,l ) βf,l : constant

0.84 (0.13)

βf,l : some tertiary education

0.30 (0.11)

βf,l : wage equation residual

0.70 (0.08)

σf,l

0.03 (0.55)

βf,η : constant

-0.20 (0.13)

βf,η : some tertiary education

0.04 (0.18)

βf,η : youngest child aged 0–5

-0.17 (0.05)

ηf = 1 − exp(x0f,η βf,η )

•

Standard errors in parentheses.  always denotes an i.i.d. standard Gaussian disturbance.

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Table 20: Parameter estimates, two-parent households Children’s preferences α = exp(x0α βα + σα α ) βα : constant

-2.35 (0.19)

βα : youngest child aged 0–5

0.77 (0.06)

βα : two or more children

0.32 (0.07)

σα

0.05 (0.13)

βk,c : youngest child aged 0–5

-0.08 (0.55)

βk,t : youngest child aged 0–5

1.41 (0.58)

ηk : youngest child aged 0–5

-0.30 (0.29)

0.72 (0.17)

βt,m : youngest child aged 0–5

-1.52 (0.51)

-2.54 (0.39)

βa,η : youngest child aged 0–5

-0.54 (1.31)

γ ˜k,c = exp(x0k,c βk,c + σk,c k,c ) βk,c : constant

0.53 (0.39)

σk,c

0.03 (0.16)

γ ˜k,t = exp(x0k,t βk,t + σk,t k,t ) βk,t : constant

0.23 (0.46)

σk,t

0.03 (0.22)

ηk = 1 − exp(x0k,η βk,η ) ηk : constant

-0.35 (0.25)

ηk : two or more children

0.57 (0.06)

γ ˜t,m = exp(x0t,m βt,m ) βt,m : constant ηa = 1 − exp(x0a,η βa,η ) βa,η : constant δf,k •

3.58 (0.62)

Standard errors in parentheses.  always denotes an i.i.d. standard Gaussian disturbance.

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Table 21: Parameter estimates, sole-parent households Mother’s preferences γ ˜m,c = exp(x0m βm,c + σm,c m,c ) βm,c : constant

0.21 (0.31)

σm,c

0.07 (0.12)

βm,c : some tertiary education

0.09 (0.20)

γ ˜m,l = exp(x0m βm,l + σm,l m,l ) βm,l : constant

1.23 (0.23)

βm,l : some tertiary education

-0.35 (0.14)

βm,l : wage equation residual

1.81 (0.19)

σm,l

0.28 (0.12)

βm,η : constant

-1.32 (0.13)

βm,η : some tertiary education

0.91 (0.11)

βm,η : youngest child aged 0–5

-0.50 (0.16)

ηm = 1 − exp(x0m,η βm,η )

Home production γ ˜q,c = exp(x0q,c βq,c + σq,c q,c ) βq,c : constant

1.14 (0.20)

σq,c

0.74 (0.11)

βq,c : youngest child aged 5 or under

0.63 (0.36)

βq,η : youngest child aged aged 0–5

1.15 (0.09)

ηq = 1 − exp(x0q,η βq,η ) βq,η : constant •

-0.63 (0.38)

Standard errors in parentheses.  always denotes an i.i.d. standard Gaussian disturbance.

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Table 22: Parameter estimates, sole-parent households Children’s preferences α = exp(x0α βα + σα α ) βα : constant

-1.07 (0.10)

βα : youngest child aged 5 or under

0.25 (0.03)

βα : two or more children

0.28 (0.11)

σα

0.36 (0.04)

βk,c : youngest child aged 5 or under

1.19 (0.54)

βk,t : youngest child aged 5 or under

1.21 (0.42)

ηk : youngest child aged 5 or under

-0.39 (0.16)

γ ˜k,c = exp(x0k,c βk,c + σk,c k,c ) βk,c : constant

1.02 (0.29)

σk,c

0.18 (0.10)

γ ˜k,t = exp(x0k,t βk,t + σk,t k,t ) βk,t : constant

-0.80 (0.28)

σk,t

0.30 (0.29)

ηk = 1 − exp(x0k,η βk,η )

•

ηk : constant

-0.50 (0.22)

ηk : two or more children

-0.02 (0.13)

Standard errors in parentheses.  always denotes an i.i.d. standard Gaussian disturbance.

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References for appendices Blundell, R., L. Pistaferri, and I. Preston (2008): “Consumption inequality and partial insurance,” American Economic Review, 98(5), 1887–1921. Bruins, M., and J. Duffy (2015): “The econometrics of time-use data,” Working paper. Dunbar, G., A. Lewbel, and K. Pendakur (2013): “Children’s Resources in Collective Households: Identification, Estimation and an Application to Child Poverty in Malawi,” American Economic Review. Eslami, E., J. Leftin, and M. Strayer (2012): “Supplemental Nutrition Assistance Program Participation Rates: Fiscal Year 2010,” Discussion paper, United States Department of Agriculture, Appendix G. Fiorini, M., and M. P. Keane (2014): “How the Allocation of Children”s Time Affects Cognitive and Noncognitive Development,” Journal of Labor Economics, 32(4), 787–836. Guryan, J., E. Hurst, and M. Kearney (2008): “Parental Education and Parental Time with Children,” Journal of Economic Perspectives. Melbourne Institute (2006): “Poverty Lines: Australia December Quarter 2006,” Working Paper.

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