Led by the pioneering work of Gary Becker (1965, 1973, 1981), economists have been studying the family for thirty years. However, the work by economists has focused on families that exhibit cooperation and altruism. Initial studies of the family treated marriage as a cooperative institution in which both spouses behave as if they are altruistic, allocating goods and time to satisfy a common set of family preferences. In contrast, the bargaining models developed by Manser and Brown (1979, 1980) and McElroy and Horney (1981) treat each spouse as an independent party with distinct preferences. In doing so, they arrive at a cooperative family utility function which incorporates the preferences of both individuals. Although the optimization is over a cooperative utility function, the solution must provide each spouse with a level of utility at least as great as that which could be achieved outside of the marriage. This constitutes the individual's threat point in the bargaining relationship. This cooperative bargaining model provides the initial framework for a model of marriage that is not entirely altruistic in nature. In considering marriage in these terms, it becomes possible to consider the threat points available to both spouses and the impact of differing incomes within and outside of marriage.
Studies of families which don't fit into the cooperative framework have been relatively ignored in the literature even though they are not uncommon. Households characterized by domestic violence are examples of families which don't fit the cooperative mold. Studies on the incidence of domestic violence indicate that between three and four million women in the US are beaten by their partners each year (Heise 1992: 2). These beatings take place at extreme cost to both the victims and society. Noncooperative models of the family are necessary to adequately characterize the behavior of households where violence occurs. Although there has been little study by economists on the incidence of violence, Tauchen, Witte, and Long (1991) model the determinants of violence within the family. In their model, the man punishes the woman with violence for behavior of which he does not approve and the woman adjusts her behavior in an attempt to minimize this violence. Both spouses have the ability to make transfers to the other, and each has a threat point level of utility that must be maintained. The empirical work analyzes the impact of income on the level of violence. In all but the highest income households in which the woman is the primary source of earnings, a rise in her income diminishes the violence while a rise in the husband's income increases the level of violence.
This paper presents a model of the strategic interaction between a man and a woman in a relationship with domestic violence. Using a game theoretic model, we analyze the equilibrium solution to a game in which each spouse, with independent preferences and threat points, maximizes utility given the behavior and the threat point of the other. Although the setup is noncooperative, altruistic behavior is not precluded. If a man or woman derives utility from the spouse's happiness, this is accounted for in each spouse's own optimization problem. The man maximizes his utility by choosing the optimal transfer and level of violence. The woman's threat point determines the level of violence she will tolerate for a given transfer from the man. As her income (and consumption) increases, the marginal utility she receives from an additional unit of consumption declines; therefore, the man's ability to "buy" violence from her decreases and the violence falls. In addition to formalizing the strategic interaction in the relationship, this paper explicitly models the effects of outside alternatives (welfare, services such as shelters, extended family resources, and divorce settlements) on the woman's utility and the level of violence.
From the solution to this strategic model, the comparative statics yield clear predictions on the impact of the woman's income and external services on the violence in the household. …