The first model describes domestic violence against children by parents when they have bargaining power. This model is inspired by Farmer & Tiefenthaler (1997) . In my scenario, parents’ utility of violence is formalized as $U^P=U^P\left(R\left(V\right),U^C,C^P,\delta \right)$ where $R\left(V\right)$ denotes all variables that are strictly increasing in violence, such as the relief of stress, and enhanced control over the children. $U^C$ is the child’s utility. I place children’s utility within parents’ utility because parents may not be fully self-interested in their children. Parents extract utility from children through violence, but if children’s utility is extremely low, it could threaten the intact family because a bystander could intervene and involve a third party that will start interventions. Therefore, parents consider their children’s utility when making decisions regarding violence and money transfer. $C^P$ denotes the consumption of the parents, which is a normal good that creates positive utility. δ denotes family capital, such as emotional bonds, social standing, and family cohesion. From the children’s side, the utility function for them can be written as $U^C=U^C\left(V,C^C,\gamma \right)$. V denotes the violence children endure from their parents. $C^C$ is children’s consumption¹, which is determined by money transfers from parents. γ represents all factors in the family that affect the child’s psychological and emotional well-being.

Several factors bind the utility functions of parents and their children. Parents’ consumption $C^P$ is limited by their total income $I^P$ and money transferred to the children $C^C$. Besides, parents should also ensure that the children’s utility remains at least their threat point $\overline{U^C}\left(\theta \right)$ which is determined by external resources θ such as children welfare institutions. Note that γ represents capital that offers children utility only when they stay with their parents. If police engage and arrest their parents, children will lose family capital γ but receive shelter service capital θ. θ represents all service resources that will be made available and accessible to the children, including but not limited to housing services, financial aid, trauma support, and legal representation. This can ensure that violence from parents will not cause children’s payoffs to drop below a critical level. Therefore, the maximum utility equations can be written as

$\underset{V,C^P}{\max }{U}^P\left(R\left(V\right),U^C,C^P,\delta \right)$ subject to $\overline{U^C}\le U^C$ and $C^P\le I^P-C^C$ (1)

$\underset{V,C^C}{\max }{U}^C\left(V,C^C,\gamma \right)$ subject to $\overline{U^C}\le U^C$ (2)

$\overline{U^C}=\overline{U^C}\left(\theta \right)$ (3)

Figure 1 is a graphical illustration of my domestic violence against children model when children have bargaining power and rely on outside service protection. The horizontal axis represents the consumption of parents $C^P$. The vertical axis measures violence level V imposed by parents against children. The curve at the top is the utility indifference curve $U^P$ for the parents. Below is the payoff curve $U^C$ for children. Children’s utility function is concave downward because, as parental consumption increases, represented by the movement to the right on the x-axis, the amount of money available for transfer to children decreases, making children less willing to suffer more pain. The two utility functions will intercept at point E at equilibrium.

To extract the highest level of utility for parents themselves, at equilibrium, children have the exact amount of utility compared to what they can get outside the home $\overline{U^C}\left(\theta \right)$. Household consumption is bound by two vertical dash lines. The vertical dash line $l_1$ on the left is the parents’ subsistence consumption level $C_{SC}^P$, and the vertical dash line $l_2$ on the right is the maximum parental expenditure, equal to their income $I^P$. Equilibrium point E cannot fall outside the range. The marginal rate of substitution (MRS) of violence and consumption at point E for children and parents are the same, where:

$\frac{U_{V^{\prime }}^P}{U_{C^{P^{\prime }}}^P}=-\frac{U_{V^{\prime }}^C}{U_{C^{C^{\prime }}}^C}$² (4)

The model above describes a scenario in which both parents and children have bargaining power. However, such equilibrium relies on these social services θ being available and accessible to children. If there are no local services that provide children with such support, or there are barriers preventing children from accessing shelter resources, then $\theta \approx 0$ and $\overline{U^C}\left(\theta \right)\approx 0$. Children lose their bargaining power in the game, and parents are relaxed from the constraint of the children’s outside utility $\overline{U^C}\left(\theta \right)$.

In my second model, when children lose bargaining power, parents are restricted by two factors: the maximum level of violence against children, $V^{\text{*}}$ and minimum spending on children, $C_{SC}^{C\text{*}}$. These two restrictions ensure that children are not seriously injured or do not lack the supplies necessary to survive. If any restrictions have been violated, parents can no longer hide evidence of their abusive behavior and thus lose the chance to exploit their children. For example, if a parent strikes a child with extreme violence $V>V^{\text{*}}$, excessive bleeding or fainting may occur. Parents are unable to handle such injuries themselves. To prevent a fatality, they choose to call an ambulance. The police arrive, investigate, and find that the injuries are caused by domestic violence, leading to arrest. Therefore, parents tend to inflict violence within a controllable range to avoid external intervention. They maximize their utility functions under boundaries of $V^{\text{*}}$ and $C_{SC}^{C\text{*}}$. The optimum violence level and parental consumption will be enforced on children because they have no bargaining power and, therefore, can only act as passive receivers. The utility function can be written as

$\underset{V,C^P}{\max }{U}^P\left(R\left(V\right),U^C,C^P,\delta \right)$ subject to $V\le V^{\text{*}}$ and $C^P\le I^P-C^C\le I^P-C_{SC}^{C\text{*}}$ (5)

$U^C=U^C\left(V,C^C,\gamma \right)$ (6)

I first analyze the equilibrium outcome when abusive parents are purely self-interested (see Figure 2 ). In such cases, the children’s utility does not affect parents’

payoffs. Then, instead of $\underset{V,C^P}{\max }{U}^P\left(R\left(V\right),U^C,C^P,\delta \right)$, parents’ utility function will be $\underset{V,C^P}{\max }{U}^P\left(R\left(V\right),C^P,\delta \right)$. The rectangular areas defined by $V^{\text{*}}$, E, $C^{P^{\prime }}$, and

$C_{SC}^P$³ are the parental consumption and violence level strategy profiles, respectively. Consumption and violence pairs in the rectangular area allow parents and children to maintain a family relationship, and any strategy point outside the rectangular region will lead to third-party interventions, such as police arrest. The optimum choice among the strategy sets is point E, where parents only transfer the exact amount of sustainable consumption $C_{SC}^{C\text{*}}$ to their children, consume all disposable income $C^{P^{\prime }}=I^P-C_{SC}^{C\text{*}}$ for their own utility, and impose the highest level of violence $V^{\text{*}}$.

Now, suppose I do not omit children’s utility in the parent’s utility function. The initial increase in V provides high marginal utility to parents because

$\frac{\partial R\left(V\right)}{\partial V}>0$ and $\frac{{\partial }^{2}R\left(V\right)}{\partial V^2}<0$. As parents impose more violence, the negative impact on children’s utility $U^C$ becomes pronounced, reducing $U^P$ significantly because $\frac{\partial U^C}{\partial V}<0$, $\frac{{\partial }^2{U}^C}{\partial V^2}<0$ and $\frac{\partial U^P}{\partial U^C}>0$. Any further increase in V beyond

the optimal point would reduce $U^P$ because the diminishing return of V on $R\left(V\right)$ makes the additional utility gained from $R\left(V\right)$ cannot be offset by the loss in utility from the decreased $U^C$. To find the equilibrium, set the violence point where additional utility gained from $R\left(V\right)$ is exactly offset by the loss in utility $U^C$ as $\bar{V}$. The equation can be written as:

$U^P=U^P\left(R\left(V\right),U^C,C^P,\delta \right)$ subject to $V\le V^{*}$ and $C^P\le I^P-C^C\le I^P-C_{SC}^{C*}$ (7)

Maximize parents’ utility $U^P$ with respect to V

Set the derivative to zero for optimization:

At the optimum parental utility point:

If $\bar{V}>V^{\text{*}}$ (see Figure 3 ), parents will maximize utility by setting $V=V^{\text{*}}$, because at $V^{\text{*}}$, $\left|\frac{\partial U^P}{\partial R\left(V^{\text{*}}\right)}\cdot \frac{\partial R\left(V^{\text{*}}\right)}{\partial V^{\text{*}}}\right|>\left|\frac{\partial U^P}{\partial U^C}\cdot \frac{\partial U^C}{\partial V^{\text{*}}}\right|$.

If $\bar{V}\le V^{\text{*}}$ (see Figure 4 ), parents will maximize utility by setting $V=\bar{V}$, because $\left|\frac{\partial U^P}{\partial R\left(\bar{V}\right)}\cdot \frac{\partial R\left(\bar{V}\right)}{\partial \bar{V}}\right|=\left|\frac{\partial U^P}{\partial U^C}\cdot \frac{\partial U^C}{\partial \bar{V}}\right|$.

Parents will not abuse children using violence level $V^{\text{*}}$, due to $\left|\frac{\partial U^P}{\partial R\left(V^{\text{*}}\right)}\cdot \frac{\partial R\left(V^{\text{*}}\right)}{\partial V^{\text{*}}}\right|\le \left|\frac{\partial U^P}{\partial U^C}\cdot \frac{\partial U^C}{\partial V^{\text{*}}}\right|$.

The equilibrium results show that parents will not use violence greater than $V^{\text{*}}$ even when their children lose their bargaining power. Parents’ actions are limited by the threat of external interventions that serve to protect children’s rights and constrain the maximum level of abuse that parents can enforce on their children. These findings underscore the crucial role of support services and legal protection in safeguarding against and mitigating domestic violence against children.

However, providing children with accessible external services remains a challenge. Children usually return to their abusers after staying in shelters. Research has found that victims’ choices to return to their situation frustrate service providers, leading to reduced service quality ( Farmer & Tiefenthaler, 1996 ). In recent years, shelters have implemented more rules for community residents, restricting survivor empowerment ( Glenn & Goodman, 2015 ). Such increasing regulations and codes of conduct within the community create isolation and force survivors to leave ( Wood et al., 2020 ). Furthermore, for domestic violence, police involvement is a palliative treatment, and its effect is transitory ( Tauchen & Witte, 1995 ). These challenges emphasize the need for more effective and sustainable interventions to protect children from abuse.