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”.

Appendix

1.1 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 ^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,

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

(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.

1.2 Appendix 2

1.2.1 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(C−γ _C )+b ₂ln(x _f−γ _x )+a ₃ln(t−γ _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,

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

1.3 Appendix 3

1.3.1 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.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.