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Child support and non-resident fathers’ contact with their children

Abstract

The paper presents a model of a non-resident father’s child support and contact with his child, which combines the public good treatment of “child quality” with “trade” in father–child contact time in a setting of non-cooperative interaction. It predicts that father’s income and mother’s non-labour income should have exactly the same effect on the frequency of father–child contact if he chooses to make lump sum payments to the mother. If he does not or there is a binding child support payment order, they have effects opposite in direction. A higher binding support order reduces father–child contact but may well raise “child quality”.

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Notes

  1. Children themselves point to loss of contact with their non-resident parent as the most upsetting aspect of their parents’ divorce/separation (Kelly 1993). Non-resident parent–child contact is associated with higher psychological scores, greater self-esteem, fewer behavioural problems, higher academic achievement and better peer relationships (e.g. Amato and Rezac 1994; Peterson and Zill 1986).

  2. There is empirical research on frequency of contact of non-resident fathers with their children and the relation between contact and monetary transfers. Cooksey and Craig (1998) provide a descriptive analysis of frequency of fathers’ contact but do not analyse child support. Bradshaw et al. (1999) and Manning et al. (2003) analyse both, and Manning and Smock (1999) study the dynamics of father’s contact when they form new families. See Beller and Graham (1993) for an early treatment of the analysis of child support.

  3. The papers cited in note 2 other than Bradshaw et al. (1999) use the National Survey of Families and Households from the U.S. Bradshaw et al. (1999) analyze their own survey of non-custodial fathers.

  4. Weiss and Willis (1985) incorporate the determination of custody into their analysis.

  5. In Section 4 below, we discuss a model in which some parents cooperate.

  6. The situation did not improve after the 2003 reform. As of July 2006, there was a backlog of 330,000 cases waiting assessments and unrecovered child support payments of £3.5bn (The Economist, 29 July 2006, p.30 and The Guardian, 25 July 2006, p.7).

  7. In the model of Del Boca and Flinn (1995), fathers are assumed to have varying costs of non-compliance with the order. In this paper, we are saying that they are low for most fathers.

  8. Under the 2006 reform proposals, parents will be encouraged to make their own arrangements, with the government supposedly providing tougher enforcement of these voluntary agreements.

  9. The model ignores fixed costs of working, the market for childcare and the existence of non-convex budget constraints for mothers with unemployed partners.

  10. During 2003, a “child support disregard” of £10 per week was introduced into the IS scheme.

  11. Note that if t does not affect either parent’s utility directly, then, at equilibrium, −U Q /U x =V Q /V x =p; that is, the mother’s marginal utility of child quality is negative at the optimum.

  12. Note that in Del Boca and Ribero (2001), child support payments are pt, as s = 0 because of the absence of a public good.

  13. The stability condition is now \(h^{0}_{p} - {\left( {g_{F} + g_{p} } \right)} < 0.\)

  14. Flinn (2000) argues that, in the context of a repeated game, implementing the particular cooperative outcome associated with the ordered transfer is a best response if the only alternative is the non-cooperative outcome.

  15. I am grateful to Chiara Pronzato for constructing these data. See Ermisch and Pronzato (2006) for an analysis of child support payments using the panel nature of these data.

  16. Literally, “visited, saw or had contact with”. These data over-sample Scotland and Wales because of booster samples for them since 1999, but the weighting in Table 1 takes this into account. The estimates of the models in Tables 5, 6 and 9 are based on unweighted data.

  17. Very similar correlation estimates are obtained from a sample of about 300 mothers of dependent children who separate from their partner during the BHPS panel. For these women, their partner’s monthly income in the annual interview preceding the separation is observed.

  18. Other income in the father’s household never had a statistically significant effect on frequency of contact; furthermore, it may be endogenous. For this reason it is not included in the empirical analysis.

  19. As a consequence, 23 fathers who report no personal income are excluded. Two fifths of these fathers report no contact with their children.

  20. The top category (“shared care”) has been grouped with “almost every day” because only six fathers report this arrangement.

  21. This reflects the strong effect of father’s income in the middle third of the distribution on the probability of child support payment (see Table 6) and the negative correlation between the error term in the selection (into the non-paying group) equation and that in the contact equation.

  22. The pattern of coefficients in these equations is similar to that in the probability of weekly contact equations discussed in the previous paragraph.

  23. The theoretical model indicates that among fathers who have no contact, (t = 0), ∂s/∂y f > 0. In the sample, 31% of fathers who never see their children paid some child support.

  24. Other household income is defined as her household income minus her personal income of all kinds, including benefits. Frequency of contact also rises significantly with the mother’s educational qualification.

  25. A small lump sum payment from the father may also escape tax, but this is equivalent to a larger b.

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Acknowledgement

The research was supported by the Economic and Social Research Council. I am grateful to two referees for valuable comments on an earlier version and to Chiara Pronzato for providing the “demographic history” data that allow matching of separated mothers and fathers and provide a check on self-reports by fathers of children living elsewhere.

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Appendix

Appendix

Appendix 1

Substituting the father’s child support function into the father–child contact-time supply function, we can derive an expression for the “supply price” of father–child contact time:

$$dp^{s} = \frac{{dt - g_{F} k_{{\text{f}}} d{\left( {y_{f} + v_{{\text{m}}} + w} \right)} - {\left( {g_{w} + g_{F} k_{w} } \right)}dw}}{{g_{p} + g_{F} {\left( {k_{{\text{f}}} + k_{p} } \right)}}}$$
(A1)

Inverting the father–child contact time demand function, we obtain an expression for the “demand price”:

$$dp^{d} = \frac{{dt - h_{F} d{\left( {y_{{\text{f}}} + v_{{\text{m}}} + w} \right)} - h_{w} dw}}{{h_{p} + h_{F} }}$$
(A2)

Equating dp s=dp d, we obtain an equation for changes in the equilibrium amount of father–child contact time:

$$dt = \frac{{{\left\{ \begin{aligned} & {\left[ {{\left( {h_{p} + h_{F} } \right)}g_{F} k_{f} - h_{F} {\left( {g_{F} {\left( {k_{p} + k_{F} } \right)} + g_{p} } \right)}} \right]}d{\left( {y_{f} + v_{{\text{m}}} + w} \right)} + \\ & \left[ {{\left( {h_{p} + h_{F} } \right)}{\left( {g_{w} + g_{F} k_{w} } \right)}} \right. - h_{w} {\left[ {g_{F} {\left( {k_{f} + k_{p} } \right)} + g_{p} } \right]}dw \\ \end{aligned} \right\}}}}{{h_{F} + h_{p} - g_{F} {\left( {k_{f} + k_{p} } \right)} - g_{p} }}$$
(A3)

On the plausible assumption that price dynamics are such that the temporal change in price is a positive function of excess demand for father–child contact time, t dt s, convergence to equilibrium (family market stability) requires that \(h_{F} - g_{F} {\left( {k_{{\text{f}}} + k_{p} } \right)} + h_{p} - g_{p} < 0.\) Furthermore,

$$\frac{{\partial p}}{{\partial y_{{\text{f}}} }} = \frac{{\partial p}}{{\partial v_{{\text{m}}} }} = \frac{{g_{F} k_{{\text{f}}} - h_{F} }}{{h_{F} - g_{F} {\left( {k_{{\text{f}}} + k_{p} } \right)} + h_{p} - g_{p} }}$$
(A4)
$$\frac{{\partial p}}{{\partial w}} = \frac{{g_{F} k_{{\text{f}}} - h_{F} + g_{w} + g_{F} k_{w} - h_{w} }}{{h_{F} - g_{F} {\left( {k_{{\text{f}}} + k_{p} } \right)} + h_{p} - g_{p} }}$$
(A5)

We have seen that k f > 0, h F  > 0 and g F  < 0, and so Eq. (A4) indicates that ∂p/∂y f=∂p/∂v m > 0. The sign of ∂p/∂w is ambiguous.

Appendix 2

Example: Stone–Geary preferences

Ignoring home production so that Q=C, let mother’s preferences be represented by the utility function U=a 1 ln(C)+a 2ln(x m)+a 3ln(1−t)+a 4ln(L), and mother’s total time available for work or leisure is normalized to unity. The father’s preferences are represented by the utility function V=b 1 ln(Cγ C )+b 2ln(x fγ x )+a 3ln(tγ t ), γ j  ≥ 0, j=C,x,t. In this case,

$$C = f{\left( {F_{{\text{m}}} + s,{\text{ }}p,{\text{ }}w} \right)} = a_{1} {\left( {v_{{\text{m}}} + s + p + w} \right)}$$
$$t^{s} = g{\left( {F_{{\text{m}}} + s,{\text{ }}p,{\text{ }}w} \right)} = {{\left[ { - a_{3} {\left( {v_{{\text{m}}} + s + p + w} \right)} + p} \right]}} \mathord{\left/ {\vphantom {{{\left[ { - a_{3} {\left( {v_{{\text{m}}} + s + p + w} \right)} + p} \right]}} p}} \right. \kern-\nulldelimiterspace} p$$
$$s = k{\left( {y_{{\text{f}}} ,{\text{ }}F_{{\text{m}}} } \right)} = b_{1} {\left( {y_{{\text{f}}} - \gamma _{x} - p\gamma _{t} } \right)} - {\left( {1 - b_{1} } \right)}{\left( {{v_{{\text{m}}} + p + w - \gamma _{C} } \mathord{\left/ {\vphantom {{v_{{\text{m}}} + p + w - \gamma _{C} } {a_{1} }}} \right. \kern-\nulldelimiterspace} {a_{1} }} \right)}$$
$$t^{d} {\text{ = }}h{\left( {F,{\text{ }}p,{\text{ }}w} \right)} = {b_{3} {\left( {{y_{{\text{f}}} + v_{{\text{m}}} + p + w - \gamma _{x} - \gamma _{C} } \mathord{\left/ {\vphantom {{y_{{\text{f}}} + v_{{\text{m}}} + p + w - \gamma _{x} - \gamma _{C} } {a_{1} }}} \right. \kern-\nulldelimiterspace} {a_{1} }} \right)}} \mathord{\left/ {\vphantom {{b_{3} {\left( {{y_{{\text{f}}} + v_{{\text{m}}} + p + w - \gamma _{x} - \gamma _{C} } \mathord{\left/ {\vphantom {{y_{{\text{f}}} + v_{{\text{m}}} + p + w - \gamma _{x} - \gamma _{C} } {a_{1} }}} \right. \kern-\nulldelimiterspace} {a_{1} }} \right)}} p}} \right. \kern-\nulldelimiterspace} p + \gamma _{t} {{\left( {b_{3} b_{1} + b_{2} } \right)}} \mathord{\left/ {\vphantom {{{\left( {b_{3} b_{1} + b_{2} } \right)}} {{\left( {1 - b_{1} } \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {1 - b_{1} } \right)}}$$

Solving for the equilibrium price by equating t d and t s,

$$p = {{\left[ {{{\left( {b_{3} + a_{3} b_{1} } \right)}{\left( {y_{{\text{f}}} + y_{{\text{m}}} + w - \gamma _{x} } \right)} + {\left( {a_{3} - b_{3} - a_{3} b_{1} } \right)}\gamma _{C} } \mathord{\left/ {\vphantom {{{\left( {b_{3} + a_{3} b_{1} } \right)}{\left( {y_{{\text{f}}} + y_{{\text{m}}} + w - \gamma _{x} } \right)} + {\left( {a_{3} - b_{3} - a_{3} b_{1} } \right)}\gamma _{C} } {a_{1} }}} \right. \kern-\nulldelimiterspace} {a_{1} }} \right]}} \mathord{\left/ {\vphantom {{{\left[ {{{\left( {b_{3} + a_{3} b_{1} } \right)}{\left( {y_{{\text{f}}} + y_{{\text{m}}} + w - \gamma _{x} } \right)} + {\left( {a_{3} - b_{3} - a_{3} b_{1} } \right)}\gamma _{C} } \mathord{\left/ {\vphantom {{{\left( {b_{3} + a_{3} b_{1} } \right)}{\left( {y_{{\text{f}}} + y_{{\text{m}}} + w - \gamma _{x} } \right)} + {\left( {a_{3} - b_{3} - a_{3} b_{1} } \right)}\gamma _{C} } {a_{1} }}} \right. \kern-\nulldelimiterspace} {a_{1} }} \right]}} q}} \right. \kern-\nulldelimiterspace} q$$

where \(q = {1 - b_{3} - a_{3} b_{1} + {\left[ {{\left( {1 - b_{1} } \right)}a_{3} b_{1} + b_{3} b_{1} + b_{2} } \right]}\gamma _{t} } \mathord{\left/ {\vphantom {{1 - b_{3} - a_{3} b_{1} + {\left[ {{\left( {1 - b_{1} } \right)}a_{3} b_{1} + b_{3} b_{1} + b_{2} } \right]}\gamma _{t} } {{\left( {1 - b_{1} } \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {1 - b_{1} } \right)}}.\) Equilibrium father–child contact time (t) is obtained by substituting for p in the demand or supply function. It depends on y f+v m+w and the preference parameters, including γ j , j=C,x,t. In particular,

$${\partial t} \mathord{\left/ {\vphantom {{\partial t} {\partial _{{y_{{\text{f}}} }} }}} \right. \kern-\nulldelimiterspace} {\partial _{{y_{{\text{f}}} }} } = {\partial t} \mathord{\left/ {\vphantom {{\partial t} {\partial _{{v_{{\text{m}}} }} }}} \right. \kern-\nulldelimiterspace} {\partial _{{v_{{\text{m}}} }} } = {\partial t} \mathord{\left/ {\vphantom {{\partial t} {\partial w}}} \right. \kern-\nulldelimiterspace} {\partial w} = {a_{3} b_{3} \gamma _{C} } \mathord{\left/ {\vphantom {{a_{3} b_{3} \gamma _{C} } {a_{1} p^{2} q}}} \right. \kern-\nulldelimiterspace} {a_{1} p^{2} q} > 0{\text{ for }}\gamma _{C} > 0.$$

In the special case where γ j  = 0, j=C,x,t (Cobb–Douglas preferences),

$$p = {{\left( {b_{3} + a_{3} b_{1} } \right)}{\left( {y_{{\text{f}}} + v_{{\text{m}}} + w} \right)}} \mathord{\left/ {\vphantom {{{\left( {b_{3} + a_{3} b_{1} } \right)}{\left( {y_{{\text{f}}} + v_{{\text{m}}} + w} \right)}} {{\left[ {1 - {\left( {b_{3} + a_{3} b_{1} } \right)}} \right]}}}} \right. \kern-\nulldelimiterspace} {{\left[ {1 - {\left( {b_{3} + a_{3} b_{1} } \right)}} \right]}}$$
$$t = {b_{3} } \mathord{\left/ {\vphantom {{b_{3} } {{\left( {b_{3} + a_{3} b_{1} } \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {b_{3} + a_{3} b_{1} } \right)}}$$

Thus, with these preferences, equilibrium father–child contact time is independent of parents’ incomes—it only depends on parents’ preferences. For example, higher income of the father initially raises the father’s demand for contact (by b 3/p) and lump sum child support transfers to the mother (by b 1), and the latter produces a reduction in the mother’s supply (by a 3 b 1/p). At the initial price, there is excess demand for contact [of (b 1+a 3 b 1)/p], which increases the price of father–child contact, which in turn reduces father’s lump sum transfers. The higher price and lower transfers choke off the excess demand for contact, producing the same contact at a higher price when the father’s income is higher.

Taking the father–child contact price into account, in equilibrium,

$$s = {{\left\{ {{\left[ {{\left( {1 - a_{3} } \right)}b_{1} - b_{3} } \right]}y_{{\text{f}}} - {\left( {1 - b_{1} } \right)}\left( {w + v_{{\text{m}}} } \right.} \right\}}} \mathord{\left/ {\vphantom {{{\left\{ {{\left[ {{\left( {1 - a_{3} } \right)}b_{1} - b_{3} } \right]}y_{{\text{f}}} - {\left( {1 - b_{1} } \right)}\left( {w + v_{{\text{m}}} } \right.} \right\}}} {{\left[ {1 - {\left( {b_{3} + a_{3} b_{1} } \right)}} \right]}}}} \right. \kern-\nulldelimiterspace} {{\left[ {1 - {\left( {b_{3} + a_{3} b_{1} } \right)}} \right]}},$$
$$C = {a_{1} b_{1} {\left( {y_{{\text{f}}} + v_{{\text{m}}} + w} \right)}} \mathord{\left/ {\vphantom {{a_{1} b_{1} {\left( {y_{{\text{f}}} + v_{{\text{m}}} + w} \right)}} {{\left[ {1 - {\left( {b_{3} + a_{3} b_{1} } \right)}} \right]}}}} \right. \kern-\nulldelimiterspace} {{\left[ {1 - {\left( {b_{3} + a_{3} b_{1} } \right)}} \right]}},{\text{ }}L = {a_{4} b_{1} {\left( {y_{{\text{f}}} + v_{{\text{m}}} + w} \right)}} \mathord{\left/ {\vphantom {{a_{4} b_{1} {\left( {y_{{\text{f}}} + v_{{\text{m}}} + w} \right)}} {{\left[ {1 - {\left( {b_{3} + a_{3} b_{1} } \right)}} \right]}}}} \right. \kern-\nulldelimiterspace} {{\left[ {1 - {\left( {b_{3} + a_{3} b_{1} } \right)}} \right]}}w$$
$$\begin{aligned} & x_{{\text{m}}} = {a_{2} b_{1} {\left( {y_{{\text{f}}} + v_{{\text{m}}} + w} \right)}} \mathord{\left/ {\vphantom {{a_{2} b_{1} {\left( {y_{{\text{f}}} + v_{{\text{m}}} + w} \right)}} {{\left[ {1 - {\left( {b_{3} + a_{3} b_{1} } \right)}} \right]}}}} \right. \kern-\nulldelimiterspace} {{\left[ {1 - {\left( {b_{3} + a_{3} b_{1} } \right)}} \right]}}{\text{ and }} \\ & x_{{\text{f}}} = {b_{2} {\left( {y_{{\text{f}}} + v_{{\text{m}}} + w} \right)}} \mathord{\left/ {\vphantom {{b_{2} {\left( {y_{{\text{f}}} + v_{{\text{m}}} + w} \right)}} {{\left[ {1 - {\left( {b_{3} + a_{3} b_{1} } \right)}} \right]}}}} \right. \kern-\nulldelimiterspace} {{\left[ {1 - {\left( {b_{3} + a_{3} b_{1} } \right)}} \right]}}. \\ \end{aligned}$$
$$s + pt = {{\left[ {{\left( {1 - a_{3} } \right)}b_{1} y_{{\text{f}}} - b_{2} {\left( {v_{{\text{m}}} + w} \right)}} \right]}} \mathord{\left/ {\vphantom {{{\left[ {{\left( {1 - a_{3} } \right)}b_{1} y_{{\text{f}}} - b_{2} {\left( {v_{{\text{m}}} + w} \right)}} \right]}} {{\left[ {1 - {\left( {b_{3} + a_{3} b_{1} } \right)}} \right]}}}} \right. \kern-\nulldelimiterspace} {{\left[ {1 - {\left( {b_{3} + a_{3} b_{1} } \right)}} \right]}}.$$

In the case of Cobb–Douglas preferences, the efficient outcomes for C and t are given by:

$$C^{{\text{e}}} = {{\left( {b_{1} + \mu a_{1} } \right)}{\left( {y_{{\text{f}}} + v_{{\text{m}}} + w} \right)}} \mathord{\left/ {\vphantom {{{\left( {b_{1} + \mu a_{1} } \right)}{\left( {y_{{\text{f}}} + v_{{\text{m}}} + w} \right)}} {{\left[ {1 - b_{3} + \mu {\left( {1 - a_{3} } \right)}} \right]}}}} \right. \kern-\nulldelimiterspace} {{\left[ {1 - b_{3} + \mu {\left( {1 - a_{3} } \right)}} \right]}}$$
$$t^{{\text{e}}} = {b_{3} } \mathord{\left/ {\vphantom {{b_{3} } {{\left( {b_{3} + \mu a_{3} } \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {b_{3} + \mu a_{3} } \right)}}$$

where μ is the weight given the mother’s preferences relative to the father’s. If, for example, they are given equal weight (μ = 1), t is above the efficient level in the non-cooperative equilibrium because b 3+a 3 b 1<b 3+a 3.

When s = 0,

$$t = {b_{3} {\left( {1 - a_{3} } \right)}y_{{\text{f}}} } \mathord{\left/ {\vphantom {{b_{3} {\left( {1 - a_{3} } \right)}y_{{\text{f}}} } {{\left[ {b_{3} y_{{\text{f}}} + a_{3} {\left( {1 - b_{1} } \right)}{\left( {v_{{\text{m}}} + w} \right)}} \right]}}}} \right. \kern-\nulldelimiterspace} {{\left[ {b_{3} y_{{\text{f}}} + a_{3} {\left( {1 - b_{1} } \right)}{\left( {v_{{\text{m}}} + w} \right)}} \right]}}$$

Appendix 3

Side payment constraint

Suppose that payments pt up to K escape the attention of the benefits agency, then mothers receiving IS choose C and t to maximize U=U(Q, L, x m, t) subject to \(Q = Q{\left( {C,{\text{ }}H{\text{ }},t,{\text{ }}1 - t} \right)},{\text{ }}T = H + L\) and \(b + pt = x_{{\text{m}}} + C\) and ptK, where b is IS benefits.Footnote 25 This implies U Q Q C =U x , U L =U Q Q H and pU x ≥−U t U Q (Q fQ m). The strict inequality in the latter holds when pt=K; that is, the mother would like to increase the supply of contact time to obtain more income, but she is constrained by the benefit system from doing so. With K > 0, the mother’s effective supply function is the side payment constraint, and ∂t s/∂b = 0 and \({\partial t^{s} } \mathord{\left/ {\vphantom {{\partial t^{s} } {\partial p}}} \right. \kern-\nulldelimiterspace} {\partial p} = { - t} \mathord{\left/ {\vphantom {{ - t} {p < 0}}} \right. \kern-\nulldelimiterspace} {p < 0}.\)

The only decision variable for fathers whose ex-partners receive IS is their contact time with their child, with their choice satisfying V t +V Q (Q fQ m)≤pV x . His contact demand function takes the form h(y f,p) for t > 0, with h y  > 0 and h p  < 0. Equilibrium when the side payment constraint is binding with K > 0 is given by the intersection of the demand curve with the pt=K constraint. Higher income for the father would reduce contact in this case.

Equilibrium when not receiving IS is given by the intersection of the supply and demand functions compared with intersection with the pt=K constraint when she receives IS. Thus, father–child contact is lower when she receives IS than when she does not. The father’s demand curve for contact may also be lower when the mother receives IS than when she does not because we have seen that lower mother’s income reduces his demand for contact when he is making lump sum payments (s > 0). This would dampen the decline in contact.

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Ermisch, J. Child support and non-resident fathers’ contact with their children. J Popul Econ 21, 827–853 (2008). https://doi.org/10.1007/s00148-006-0125-4

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Keywords

  • Child support
  • Non-resident fathers
  • Father–child contact

JEL Classification

  • J12