## 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

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).

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.

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

In Section 4 below, we discuss a model in which

*some*parents cooperate.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).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.

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.

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

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

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.Note that in Del Boca and Ribero (2001), child support payments are

*pt*, as*s*= 0 because of the absence of a public good.The stability condition is now \(h^{0}_{p} - {\left( {g_{F} + g_{p} } \right)} < 0.\)

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.

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.

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.

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.

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

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

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.

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

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.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.

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

*b*.

## References

Amato PR, Rezac SJ (1994) Contact with nonresident parents, interparental conflict, and children’s behavior. J Fam Issues 15:191–207

Beller AH, Graham JW (1993) Small change: the economics of child support. Yale University Press, New Haven

Bradshaw J et al (1999) Absent fathers? Routledge, London

Cooksey EC, Craig PH (1998) Parenting from a distance: the effects of paternal characteristics on contact between nonresidential fathers and their children. Demography 35:187–200

Chiappori P-A (1992) Collective labor supply and welfare. J Polit Econ 100:437–467

Del Boca D, Flinn C (1995) Rationalizing child support decisions. Am Econ Rev 85:1241–1262

Del Boca D, Ribero R (1998) Transfers in non-intact households. Struct Chang Econ Dyn 9:469–479

Del Boca D, Ribero R (2001) Evaluation of child support policies. Am Econ Rev Papers and Proceedings 91:130–135

Del Boca D, Ribero R (2003) Visitations and transfers after divorce. Review of the Economics of the Household 1:187–204

Ermisch J, Francesconi M (2000) Patterns of household and family formation. In: Berthoud R, Gershuny J (eds) Seven years in the lives of British families. The Policy Press, Bristol

Ermisch J, Pronzato C (2006) Using fathers’ child support payments to test income pooling. ISER Working Paper 2006–57, University of Essex, Colchester (in press)

Flinn C (2000) Modes of interaction between divorced parents. International Economic Review 41:544–578

Kelly JB (1993) Current research on children’s postdivorce adjustment: no simple answers. Fam Concil Courts Rev 31:29–49

Manning WD, Smock PJ (1999) New families and nonresident father–child visitation. Soc Forces 78:87–116

Manning WD, Stewart S, Smock PJ (2003) The complexity of fathers’ parenting responsibilities and involvement with nonresident children. J Fam Issues 24:645–667

Peterson JL, Zill N (1986) Marital disruption, parent–child relationships, and behavior problems in children. J Marriage Fam 48:295–307

Weiss Y, Willis RJ (1985) Children as collective goods and divorce settlements. J Labor Econ 3:268–292

Weiss Y, Willis RJ (1993) Transfers among divorced couples; evidence and interpretation. J Labor Econ 11:629–679

Wikeley N et al (2001) National Survey of Child Support Agency Claimants. Department for Work and Pensions Research Report No. 152, London

## 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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### Authors and Affiliations

### Corresponding author

## Additional information

*Responsible editor*: Deborah Cobb-Clark

## 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:

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

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

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*
^{d}−*t*
^{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,

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*
_{2}ln(*x*
_{m})+*a*
_{3}ln(1−*t*)+*a*
_{4}ln(*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(*C*−*γ*
_{
C
})+*b*
_{2}ln(*x*
_{f}−*γ*
_{
x
})+*a*
_{3}ln(*t*−*γ*
_{
t
}), *γ*
_{
j
} ≥ 0, *j*=*C*,*x*,*t*. In this case,

Solving for the equilibrium price by equating *t*
^{d} and *t*
^{s},

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,

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

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,

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

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,

### 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 *pt*≤*K*, 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*
_{f}−*Q*
_{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*
_{f}−*Q*
_{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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### Cite this article

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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DOI: https://doi.org/10.1007/s00148-006-0125-4

### Keywords

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

### JEL Classification

- J12