Family violence: A microeconomic approach

SOCIAL

SCIENCE

RESEARCH

12, 363-392 (1983)

Family Violence: A Microeconomic Approach SHARON K. LONG,

ANN D. WITTE,

University

of North

AND PATRICE KARR

Carolina

A model of violence between adult family members is developed by integrating material from the sociological theories of family violence and social exchange, and the economic theories of crime and the family. Based on this model a decrease in the dictator’s internal sanctions against violence would be expected to increase the amount of time allocated to violence by the dictator. Further, if the level of fines and other monetary costs imposed by external agencies (e.g., the courts) as a result of the family violence do not vary with the level of violence, then the model indicates that an increase in such monetary sanctions will cause a reduction in the amount of time the dictator allocates to violence. If both the dictator and victim are risk neutral, an increase in the probability of external intervention will decrease the time allocated to violence. In addition, it is found that increases in the opportunities available to the victim outside the marriage will tend to improve the well-being of the victim in the marriage even if it has no effect on the time allocated to violence by the dictator. The model also provides insights for empirical work in family violence such as (1) suggestions of relevant independent variables, (2) the specification of a functional form for estimation, and (3) the specification of an error structure for the empirical model.

In recent years, there has been increasing awareness that the family, like so many other American institutions, can be and often is a setting for violence. Some researchers believe that violent acts “are so much a part of the way family members relate to each other that they are not even thought of as violence” (Straus, 1980). Available evidence seems to indicate that some kind of violence occurs annually in 10 to 20% of The research presented in this paper was supported by a grant from the National Institute of Mental Health (NIMH). We gratefully acknowledge this support; however, NIMH’s support for this work should in no way indicate the organization’s agreement with our approach or conclusions. We thank Helen Tauchen for mathematical and other assistance, Richard Berk, Sarah Berk, and Donileen Loeske, and two referees for their valuable comments and suggestions. An earlier draft of this paper was presented at the National Conference for Family Violence Research at The University of New Hampshire, Durham, in July 1981. Requests for reprints should be sent to Sharon K. Long, Department of Economics, University of North Carolina, Chapel Hill, NC 27514. 363 0049-089X/83 $3.00 CopyrIght 0 1983 by Academic Press, Inc. All rights of reproduction in any form reserved.

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all families, and that perhaps as many as 50% of all couples participate in at least one violent encounter during their relationship.’ The growing awareness of the level and frequency of domestic violence has led to efforts in various fields of the social sciences to better understand the causes of violence within the family. Although this research has alerted us to the extent of family violence and has provided us with broad philosophical perspectives, it has not thus far provided rigorously developed theoretical models designed to explore the causes of this type of violence. Gelles (1982) describes the current theoretical work as “proposing theoretical models based on post hoc examinations of data.” In this paper we develop a formal model of violence between adult family members. Our model combines insights from the family studies and family violence literatures of sociology with concepts drawn from the economic literatures on crime and the family. We use traditional economic techniques and assumptions (e.g., rational utility maximization) in developing our model, but are guided by sociological research when portraying interpersonal relations. The next section of the paper briefly reviews the relevant literatures from sociology and economics which form the basis of our model. In section 2, we present the model and in section 3 we discuss the results which emanate from it. Section 4 contains a discussion of the implications of our model for empirical work and suggests empirical measures for the model’s theoretical constructs. The final section contains our summary and conclusions. In the appendix we indicate the manner in which we derive results for the model. 1. THE RELEVANT

LITERATURES FROM SOCIOLOGY AND ECONOMICS

To date most research on family violence has been broadly descriptive rather than analytic, and has taken one of two general approaches to the problem.’ The first perspective, the “family disturbances” approach, is quite broad and firmly based in traditional sociology.3 This approach views the behavior of individual family members as reflecting the external environment of society. Violence within the family is seen as best understood by examining the societal factors which impact on the family. The second perspective, gender-linked violence, sees as distinct the male ’ See Straus, Gelles, and Steinmetz (1980) for a review of family violence literature. Relevant work includes Gelles (1974), O’Brien (1971), Levinger (1966), Whitehurst (1971), Parnus (1967), Pittman and Handy (1964), and Bondouris (1971). Most of this work, while intriguing, has serious methodological flaws. * See Berk et al. (1980) and Gelles and Straus (1979) for a discussion of the family violence literature. 3 Work by Gelles (1974), Steinmetz (1978), and Straus (1978) is guided by this approach.

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offender-female victim and female offender-male victim relationships.4 Physiological, social, and economic factors are seen as operating to place the woman in a more vulnerable position as victim than as offender. Under this view violent behavior within the family must be understood in terms of the specific male-female context in which it occurs. While research using both of these approaches has highlighted the importance of understanding the causes of violence in the family, neither approach has provided an analytic framework for determining the relative effects of various factors on that violence. In order to provide a more rigorous model of family violence we draw from the analytic structure of economic theory for the base of our study. The underpinnings of economic theory parallel many of the propositions of the theory of social exchange’ from sociology. In both cases the individual is viewed as evaluating the costs and gains from various behavior choices and determining the actions which will maximize his or her own level of welfare. Thus, under both theories, individuals will use violence within the family if the costs of the violent behavior are less than the benefits which are gained from the violence. A principle difference between the two theoretical approaches is in the unit of analysis. Social exchange theory focuses on individual behavior in the context of relationships in which power is unequally distributed, while traditional economic theory has focused on the behavior of the individual independent of his or her relationships. However, there is a growing literature in economics, the “economics of the family,” which considers the individual behavior within the context of family relationships and family power structures. We will utilize the mathematical modeling techniques of economic theory to derive testable hypotheses concerning the way in which various factors affect the level of family violence. There are two literatures in economics which we believe provide a useful framework for modeling family violence. These are the literatures of The the “economics of crime” and the “economics of the family.” economics of crime literature seeks to analyze the factors affecting the individual’s decision to participate in illegal activities. According to this literature, the individual “commits an offense if the expected utility to him exceeds the utility he could get by using his time and other resources at other activities” (Becker, 1968, p. 176). At present there are two variants of this model of crime. One approach, based on Becker’s pioneering article, views all returns from legal and illegal activities, and all costs 4 Work by Dobash and Dobash (1979), Walker (1979), and Martin (1976) develop this view. ’ See Homans (1974) and Goode (1973) for a general discussion of social exchange theory, Emerson (1976) for a discussion of the relationship between social exchange theory and economic theory, and Gelles (1981) for an application of exchange theory to family violence.

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from illegal activities (e.g., internal and external sanctions for criminal behavior) as either monetary or capable of being monetized.h In addition, it is assumed that the individual gains satisfaction directly only from his or her level of income or wealth. In contrast, the second approach assumes either that not all the costs of illegal activities can be monetized, or that there are factors other than income or wealth (e.g., the way in which one uses one’s time) which directly affect the individual’s level of satisfaction.7 We believe that the first approach is most appropriate for traditional property crimes (e.g., burglary) and some white collar offenses (e.g., tax evasion), while the second approach is more relevant for modeling family violence as it will allow for the explicit consideration of the moral or ethical position of the offender. Although the economic model of crime provides for an explicit analysis of the effects of changed incentives (e.g., increased criminal justice sanctions, stronger “tastes for violence,” higher wages) on family violence, it does not provide a means of incorporating the intimate, nonviolent interactions which occur within the family and the way in which these family interactions affect both the nature and extent of family violence. For aid in developing this portion of our model we turn to the economics of the family literature. There are two distinct models which have been developed in this area. The first approach (Becker, 1974a, 1981) assumes that the head of the household derives positive utility from the satisfaction of other household members. In this way, the welfare of the entire family is maximized when the head of the household maximizes his or her own utility. This approach assumes that the household head is a benevolent and knowledgeable dictator who is able to compensate any member of the family made unhappy by the decisions which result from the dictator’s utility maximization. However, the model is not directly applicable to family violence since the dictator in the violent family often gains satisfaction from inflicting harm while the victim suffers from the violence. In violent families, the violent individual, while often dictatorial, is unlikely to be considered benevolent by the victim. The second approach to modeling family decision making, referred to as the economics of marriage,8 examines the conditions which must hold in order for a marriage to remain intact. The marriage is assumed to continue to exist so long as each partner is as well off in the marriage as they would be in their next best alternative situation. This approach is similar to a model in social exchange theory (Thibaut and Kelley, 6 Ehrlich (1970, 1973, 1975) is primarily responsible for the development of this approach. ’ This approach has been developed by Block and Lind (1975a, 1975b), Block and Heineke (1975) and Heineke (1978). ’ Becker (1973, 1974b), Keely (1974), Manser and Brown (1979), and Homey and McElroy (1981) develop this approach.

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1959; Nye, 1979) in which the comparison of satisfaction levels within the marriage to alternative arrangements may lead to the dissolution of the marriage. Under the economics of the marriage model, the family decision maker’s utility maximizing choices are constrained by the “threat” that other family members may withdraw from the family as a result of his or her choices.’ In this way, the model incorporates the satisfaction level of each individual member of the family, not just that of the family decision maker. For our purposes, this type of model provides a means of examining how inflicting harm afffects the utility of both the assailant and the victim. However, this model does not, we believe, realistically portray the nature of interpersonal relations in violent families. The empirical literature indicates that many, if not most, violent families are not egalitarian. The power structure of a marriage is often related to the social and economic status, as well as relative physical sizes, of the two members. Since men generally hold higher paying jobs with more occupational prestige, have traditional roles of dominance within society, and are typically stronger than women, they have a greater tendency toward a position of power within the marriage. (See Blood and Wolfe (1960), Bart (1971), Millman (1971), Rossi (1964, 1965), and Gillespie (1971) for discussions of status and power in marriage.) Furthermore, Straus (1973) and Straus, Gelles, and Steinmetz (1980) have found evidence of greater levels of domestic violence the greater the position of power of the male in the family system. Therefore, in modeling those families in which violence does occur we must consider an extension of the model of family decision making to a less than egalitarian power structure. Manser and Brown (1979) extend the previous economic models to incorporate a dictatorial family structure. They model a family with a single, dominant decision maker. While we do not believe that the power structure in most families is either completely egalitarian or completely nonegalitarian, we do feel that this nonegalitarian model will more closely approximate the typical relationship between the offender and victim in families in which violence does occur. 2. THE MODEL

In modeling families in which violence occurs we will assume that the family consists of a dominant member, who is the decision maker (hereafter referred to as the “dictator”), and one other adult family member (hereafter called the “victim”). The family may also contain any number of children who are assumed to have no role in the family decision making process. This view of family decision making is intentionally simplistic (i.e., one ’ This type of decision making is generally modeled using a Nash-Bargaining technique in which each individual’s decisions are constrained by the potential actions of the others. Thus, the outcome will depend on the interactions of the members involved.

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all-powerful decision maker) in order to allow us to examine the structure of violence in a nonegalitarian marriage. We are interested in examining the factors which affect the dictator’s allocation of his or her time among three possible uses-work, leisure, and violence.‘” We are assuming that the dictator gains satisfaction directly from the time spent in violence and leisure, and from the income produced from time spent at work. Like most economic models, we do not address the question of why these various activities give satisfaction. Rather, we simply assume that if an individual participates in any activity he or she must be gaining satisfaction from doing so. In this model the dictator is seen as maximizing his or her own level of satisfaction by choosing to allocate the amount of time which he or she has available among the three alternative uses. Given that the total amount of time available to the dictator is fixed, the choice of the amount of time to allocate to any two uses will determine the remaining time available for the third use. We have modeled the dictator as choosing directly the amount of time to allocate to leisure activities and to violence. Consequently, the amount of time spent in work is treated as a residual claimant on the dictator’s time. The dictator’s time allocation choices will be constrained by two factors- the level of family income and the threat of the dissolution of the marriage. In addition, the dictator’s decisions will be affected by the possibility of external intervention and sanctions for violent behavior. The family’s income will consist of the labor income of the dictator, as determined by the amount of time spent working and the wage rate; and any income from the labor of other family members, from family assets (e.g., stocks, bonds, real property), or from transfer payments (e.g., AFDC payments). In maximizing his or her own utility, the level of family income will act as a constraint on the dictator’s behavior since the dictator’s level of utility is a function of the income available to the family. While we do not explicitly model the dictator’s consumption choices, we do assume that the dictator allocates the family income among alternative goods in order to maximize his or her utility. Thus, the level of family income which is available acts as a constraint on the dictator’s consumption choices. An important assumption of the family income constraint is that the wage rate which the dictator can earn is beyond his or her own control (i.e., it is exogenous) so the dictator’s only means of affecting family income is through the allocation of his or her time to work. The second constraint on the dictator’s time allocation choices is the ” The concept of time in violence does not truly reflect the frequency and severity of the harm which is inflicted. This is a simplified approach to the issue of family violence which allows us to explore one aspect of the problem. This framework assumes that the level of violence is monotonically related to the time spent in violence.

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threat of the dissolution of the marriage. We refer to this constraint as the “intact marriage constraint.” The decision by the victim to leave the marriage, whether temporarily or permanently, is a complex issue. Research by Martin (1976), Walker (1979), and Dobash and Dobash (1979) indicates that women do not stay in violent relationships because they like the brutality; rather, there are complicated forces acting to keep them there. According to Walker: Many stay because of economic, legal and social dependence. Others are afraid to leave because they have no safe place to go. Police, courts, hospitals, and social services do not offer adequate protection. Psychologists tend to counsel them to keep the family together at any cost, which often turns out to be their mental health and sometimes their lives. Both the batterer and the battered woman fear they cannot survive alone. (1979, p. 43)

In addition there may be positive characteristics of the spouse and the marriage which make the choice to end the marriage a difficult one. Pagelow (1981a, 1981b) uses survey data in an attempt to empirically examine the victim’s decision to stay in the battering relationship. While inconclusive, her results do suggest that there are a number of complex economic, sociological, and psychological factors involved in the choice of whether or not to stay. In our model we take a first step in incorporating the victim’s option to leave the violent relationship. However, our focus is to examine the extent to which the responsibilities and resources of the victim if he or she should leave the relationship affect the level of violence within the marriage. These factors enter the model in the form of a constraint placed on the dictator’s behavior. The intact marriage constraint requires that, if the marriage is to continue to exist, the dictator must make choices which provide the victim with a level of satisfaction within the marriage which is at least as great as the level of utility which the victim could receive outside the marriage. The victim’s level of welfare within the family is assumed to depend on the level of family income and the amount of violence which the dictator inflicts. The victim’s comparison level of satisfaction outside the marriage will depend on the level of disposable income which the victim can achieve outside the relationship. It is not possible to directly observe the level of disposable income which would be available to the victim if he or she were not in the marriage. However, we can proxy the level of disposable income using measures of the victim’s labor market skills, the social support system available to the victim (e.g., the proximity of relatives and friends, the availability of shelters), and the number of other individuals for whom the victim would be responsible if he or she left the marriage (e.g., the number of children). If, when the intact marriage constraint is binding, the dictator should decide to increase the level of violence within the family, he or she can only keep the marriage

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intact by providing to the victim enough additional benefits to maintain him or her at a constant level of well-being. This can only be achieved if the dictator increases the family’s income by allocating more of his or her time to work. Given that other factors remain unchanged, the dictator must offer the victim a more satisfying family situation if the victim (1) would have few responsibilities outside the marriage, (2) has many labor market skills, or (3) if there is a strong system of support for the victim outside the marriage. The victim’s level of abilities, the availability of social support systems for the victims, and the level of responsibilities of the victim outside the marriage are beyond the control of the dictator in this model. The dictator can only affect the intact marriage constraint through his or her choices of time spent in violence or in work. Finally, the dictator’s time allocation choices are affected by the possibility of external sanctions or outside intervention as a result of family violence. This intervention may take a number of forms, ranging from neighbors breaking up a fight to formal processing through the criminal justice system, and may impose either a time cost, a money cost, or both on the dictator. For example, a frequent police action in family violence situations is to take the individual who is inflicting the violence to the police station for a “cooling-off” period. This action has a time cost to the dictator as it reduces the amount of time which he or she is able to freely allocate. In a more serious external intervention situation, the police may formally charge the dictator with a criminal violation. This may lead to a fine or the payment of court costs in addition to the time costs involved. The probability of external intervention as well as the nature of the intervention are both beyond the dictator’s control in the present model. The possibility of outside intervention greatly complicates the dictator’s utility maximization problem as he or she, can no longer know with certainty the level of satisfaction to be gained as a result of a particular time allocation choice. The actual level of utility obtained by the dictator will be determined by his or her time allocation choices and by the occurrence or nonoccurrence of external sanctions as a result of violent behavior. The existence of two possible states of the world (i.e., the occurrence or nonoccurrence of external sanctions) means the dictator will no longer simply maximize his or her level of satisfaction. Instead, the uncertainty as to whether outside intervention will occur leads the dictator to choose the time allocation which will result in his or her highest expected level of utility. In the body of the paper we briefly outline the formal maximization problem facing the dictator. For those readers who desire a more explicit account we have provided a mathematical appendix. The dictator’s expected level of utility (denoted EU) will be a probabilistic function of

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the individual’s level of utility if no outside intervention occurs (hereafter referred to as the successful state) and the level of utility if outside intervention does occur (hereafter referred to as the unsuccessful state). The dictator’s level of utility in these two states of the world will differ because of variation in the level of disposable family income. The level of disposable family income when outside intervention does occur (denoted I”) is lower than the level of disposable family income when there is no outside intervention (denoted I”) because of the time and money costs of the external intervention. If we denote the probability of outside intervention occurring as p we can write the dictator’s expected utility as EU = (1 - p) U(Z”, t,, t, ; (u) + pU(Z”, t, , t, ; cz),

(1)

where I” and I” are as previously defined; I, and t, are time spent in violence and leisure, respectively; and (Y is a shift parameter which reflects the dictator’s level of internal sanctions against domestic violence. For the reader’s convenience, we define all symbols used to develop the model in Table 1. The dictator’s level of internal sanctions against domestic violence is a function of his or her family background and social and cultural environment. The level of internal inhibitions can adjust in response to changes in the cultural and societal mores regarding domestic violence. If an aggressive act is committed, with no punishment resulting, the inhibitions against aggression of both the perpetrator and those viewing the act will be lessened. (See Megaree (1982) for a discussion of this issue.) The level of internal sanctions of the dictator and changes in the level of these internal sanctions will affect the dictator’s time allocation to violence. There are two factors which constrain the dictator’s time allocation choices. These are the family income constraint and the intact marriage constraint. The family income constraint simply states that the dictator’s gain in utility from income, and thus from consumption, is limited by the level of family income which is available. As was discussed earlier, family income consists of the labor income of the dictator, which is determined by the wage rate (denoted W) and the time spent working (denoted to), and other income of the family (denoted A). In the event of outside intervention occurring as a result of the violence, the level of family income will be reduced by the monetary value of the time costs and money costs of outside intervention. The time costs of external intervention (denoted t,,) are assumed to be valued at the wage rate. The monetary costs of external intervention (e.g., fines, court costs) are represented by a sanction function, S(t,; /3), where p is an exogenous shift parameter reflecting changes in the level of direct monetary costs of external intervention (e.g., an increase in the level of fines). As a

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TABLE 1 Definition of Symbols A (Y P

EU EV

El IQ hj If&I I” I” [Jl IJI L, L h P St.1 T 11 1” to t, U(.) V(,) VO(.) w

Family’s nonlabor income An exogenous shift parameter which reflects the dictator’s level of internal sanctions against domestic violence An exogenous shift parameter in the penalty function for domestic violence The dictator’s expected utility The victim’s expected utility The bordered Hessian matrix The determinant of [z] The element of the Hessian in the ifh row andj’h column The cofactor associated with element h,, The level of family income when there is no outside intervention The level of family income when there is outside intervention The Jacobian matrix The determinant of [J] The first partial derivative of the Lagrangian with respect to choice variable i The second partial derivative of the Lagrangian with respect to choice variables iandj The Lagrangian multiplier associated with the intact marriage constraint The expected probability that outside intervention will occur as a result of family violence A function which gives the monetary cost of criminal justice sanctions Total time available to the dictator The number of hours of time spent in leisure by the dictator The number of hours of time spent in violent activity by the dictator The number of hours of time spent in work by the dictator The expected amount of the dictator’s time needed to deal with outside intervention if such intervention occurs The dictator’s utility function The victim’s utility function when married The victim’s utility function in his or her next best alternative outside of the marriage The dictator’s market wage

consequence of the effect of external sanctions on disposable income we must consider two potential levels of disposable family incomedisposable income when there is no external intervention (denoted I”) and disposable income when there are external sanctions (denoted I”). We can express these formally as I” = A + w(to)

(2)

I” = A + w(to - tY) - S(t,; p).

(3)

and We know that the total time available to the dictator (denoted T) must be allocated among time in work (denoted to), time in violence (denoted t,), time in leisure (denoted t,) and, if outside intervention occurs, the

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time sanctions for violence (denoted tY). Using the relationship, t,, + t, + t, + t, we can rewrite Eqs. (2) and (3) as I” = A + w(T - t, - t,) I” = A + w(T

- fv - t, - tY) - Sk;

T = (2’)

/3).

(3’)

The final constraint on the dictator’s choices which we are considering is the intact marriage constraint. In order for the marriage to remain intact the victim’s expected utility in the marriage (denoted EV) must at least equal the victim’s utility in the next best alternative outside the marriage (denoted V’). The victim’s utility outside the marriage will be a function of the victim’s level of disposable income when he or she is no longer associated with the dictator (denoted Y). The intact marriage constraint can be written formally as EV - V’(Y) = (1 - p)V(I”, t,) + pV(Z”, t,) - V’(Y) 2 0

(4)

where p, I”, I”, and tv are as defined previously; V is the victim’s utility function while in the marriage and V” is the victim’s utility function outside the marriage. Note that this allows the victim’s “tastes” to differ depending on whether or not the individual is married. The dictator will allocate his or her available time between work, leisure, and violence so as to maximize his or her expected utility (Eq. (1)) subject to the constraints of Eqs. (2’), (3’), and (4). 3. THE ANALYTIC RESULTS On the basis of this model we want to examine the effects that changes in exogenous factors (i.e., those factors which are beyond the dictator’s control) will have on the dictator’s allocation of time to violence. Our results will provide in some cases testable hypotheses as to the effects of various factors on the level of violence within the family. In other instances, they will provide possible interpretations of empirical results. Deriving these analytic results first requires that we determine the mathematical conditions which must hold in order that the dictator maximize his or her expected utility. We are interested in the situations where the dictator allocates some amount of time to all three activities (work, leisure, and violence).” Using these conditions, the implicit function theorem, ‘* and Cramer’s rule, we can derive relationships which indicate ” Specifically we set up the appropriate Lagrangian equation and derive first-order conditions for a maximum. The appendix gives the details of this process. ‘* The implicit function theorem allows one to obtain the derivatives of functions even when they cannot be explicitly solved for the variable of interest. In the current instance it allows us to determine how time in violence will change with a change in the exogenous variables of interest. See Chiang (1974, pp. 219-227) for a discussion of this theorem and for an application (pp. 229-232).

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the effect of various exogenous factors on the dictator’s time allocation to violence. The external factors which we have examined are changes in the family’s nonlabor income, the dictator’s wage rate, the time sanction for violence, the dictator’s level of internal sanctions against violence, the probability of outside intervention, the level of monetary sanctions for violence, and the victim’s disposable income outside the marriage. Our first result, the effect on time in violence of a change in the family’s nonlabor income, follows from standard economic theory. If the dictator’s desire to inflict violence either remains constant or rises as income rises (i.e., if violence is a normal good) then an increase in the family’s nonlabor income will bring an increase in the level of violence. Correspondingly, a decrease in the family’s nonlabor income will lead to a decline in the level of violence. If violence is an inferior good, that is, the dictator’s desire to inflict violence declines as income rises, then an increase in family income will bring a decrease in the level of violence. This result is the total income effect and indicates the way in which an exogenous and equal change in both successful and unsuccessful family income, with no change in relative time valuation, affects the level of violence. The impact of a change in the time sanction for violence on the level of violence is, by analogy to the total income effect above, a partial income effect. An increase in the time sanction for violence reduces family income only in the unsuccessful state of the world, hence, the term partial income effect. If violence is a normal good, we would expect an increase in the time sanction to lead to a decrease in the amount of time spent in violence. Alternatively, if violence is an inferior good this change should lead to an increase in the level of violence. The empirical evidence concerning this issue is decidedly mixed. The extent to which adequate police protection, access to restraining orders, and the facilitation of prosecution will reduce domestic violence is not known. There is some clinical evidence (Walker, 1979) that police intervention leads to greater violence although the quality of the intervention (i.e., the adequacy of enforcement) is highly questionable. Summing up the police and judicial response to family violence, Dobash and Dobash conclude: Police officers and judicial officials are apt to see the offense [husband’s assault of his wife] either as unimportant and not worthy of attention or as a civil and social problem that requires the services of another agency and to use their discretionary powers to avoid arrest and prosecution. (1979, p. 207)

The question of whether improvements in the quality of police and judicial response will reduce the dictator’s level of violence is currently the subject of the Domestic Abuse Intervention Project in Duluth, Minnesota. Early results (Novack and Galaway, 1982) suggest there is reduction in violence following more systematic intervention by police and courts.

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In examining the effect of a change in the dictator’s wage rate on the level of violence we find three forces at work. The first effect which we can partition out is a total income effect. This effect indicates that an increase in the dictator’s wage rate will tend to cause an increase in violence if violence is a normal good and a decrease in violence if it is an inferior good. The second effect is a partial income effect of the same type as caused by an increase in the time sanction for violence and should be negative if violence is a normal good and positive if it is an inferior good. The final term will be positive regardless of whether or not violence is a normal or inferior good. Thus, the overall effect of a change in the dictator’s wage on the level of violence will depend on the relative magnitude of the opposing total and partial income effects, and on whether violence is a normal or inferior good. It would seem most likely that if violence is a normal good, the total income effect would dominate and that an increase in the dictator’s wage rate would lead to an increase in the level of violence. However, we cannot prove that this is the case. Clinical survey studies of family violence have shown that domestic violence occurs in families of all socioeconomic groups; no group is immune. However, there is evidence (Straus et al., 1980; Prescott and Letko, 1977) of direct relationship between levels of violence within a family and financial stress. This would suggest that an increase in the wage rate should reduce violence, making the partial income effect the dominant force, or alternatively indicating that violence is an inferior good. A change in the monetary cost of outside intervention (e.g., an increase in the level of fines or court costs) corresponds to an increase in the “price” to the dictator of inflicting harm. In examining the impact of this type of increase in the price of violence on the amount of time allocated to violence we find two effects. The first effect gives the impact of a marginal increase in the price of violence when the price increase has no effect on real income (i.e., it is a substitution effect) and is unambiguously negative. The second effect is a partial income effect and the sign of this effect will depend upon whether violence is a normal or inferior good. If time in violence is a normal good this term will be negative, making the impact of a change in the severity of the monetary sanction on time in violence unambiguously negative. Thus, our model suggests that the criminal justice system can decrease the amount of time spent in family violence by increasing the fines or other monetary costs for this type of activity. In examining the effects of a change in the level of the dictator’s internal sanctions against violence on the level of violence we find, as we would expect given the assumptions of our model, that a decrease in internal sanctions results in an unambiguous increase in the level of violence. Since internal sanctions reflect the dictator’s social and cultural

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environment this result is consistent with the finding that individuals who themselves were abused as children are likely to inflict violence in the marriage (Straus et al., 1980). This result is also consistent with the greater levels of violence which occur within marriages as a result of a perception of the marriage license as a “hitting license.” Efforts to decrease the cultural acceptance of domestic violence should increase the internal sanctions against violence and, therefore, decrease violent behavior. Next consider the impact of a change in the victim’s disposable income outside the marriage. If the victim’s situation outside the marriage improves, perhaps because of improvements in the victim’s labor market skills or because of improved community support, then the victim will choose to leave the marriage if the dictator’s behavior remains unchanged. In order to maintain the marriage intact the dictator must increase the level of well-being available to the victim within the relationship. This can be achieved by decreasing the level of violence, by increasing the time spent at work and thereby increasing family income. or by some combination of the two. Regardless of which of these actions the dictator chooses, we know that so long as the marriage remains intact the improvement in the victim’s opportunities outside the marriage will lead to a higher level of well-being for the victim within the relationship. At present, there is no empirical work in family violence which addresses the effect of changes in the victim’s opportunities on the level of violence in the marriage; however, survey studies by Gelles (1976) and Pagelow (198la, 1981b) suggest that women with lower levels of education and occupational status are more likely to stay in the violent relationship. Given that the victim does remain in the relationship we cannot unambiguously determine the impact of the improvement in the victim’s outside opportunities on the level of violence. If time in violence is a normal good, the impact of any combination of changes in time in work or time in violence will result in partial income effects which tend to offset the direct effect on the level of violence. This occurs because monetary sanctions and, thus, the level of family income in the unsuccessful state depend on the dictator’s time in violence. If the level of monetary sanction does not depend on the time in violence then the countervailing partial income effect is eliminated and we can say unambiguously that an improvement in the victim’s opportunities outside the marriage will decrease the level of violence within the marriage. The final result which we have examined is the effect of a change in the probability of outside intervention on the time spent in violence. An increase in the probability of external intervention increases the probability that the less desirable state of the world will occur. In order to maintain the marriage intact the dictator must compensate the victim by either

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decreasing the level of violence, increasing income, or some combination of the two. However, we cannot determine unambiguously the effect of an increase in p on the level of violence since the compensation could take the form of only an increase in income. The only way in which the effect on time in violence of changing this probability can be unambiguously signed is by making an assumption equivalent to the assumption of risk neutrality when income is the only argument in the utility function. The assumption of risk neutrality is equivalent to assuming that individuals are indifferent when choosing between risky and safe alternatives of equal expected value. Wtih this assumption we can say unambiguously that an increase in the probability of outside intervention leads to a decrease in violence. Unfortunately, risk aversion, not risk neutrality, is the usual assumption about human behavior. That is, we normally believe that individuals dislike risk and will choose a safe rather than a risky alternative of equal value. If we assume the dictator and victim are risk averse we cannot unambiguously determine the effect of a change in the probability of external intervention on the level of violence. 4. IMPLICATIONS OF THE MODEL FOR EMPIRICAL WORK The model we developed in the last section has a number of implications for empirical work. Specifically, it suggests a number of factors likely to be important in determining the time an individual allocates to inflicting harm on other family members. Further, for some factors it either provides expected signs for estimated coefficients, or at least suggests possible interpretations for those coefficients. The model developed suggests that the time allocated to harm, leisure, and work are determined jointly. These time allocations will depend upon the dictator’s wage, the level of income from other family members, the level of income from family assets and transfer payments, the level of disposable income available to the victim outside the marriage, the probability of outside intervention, the level of both time and money sanctions when outside intervention occurs, and the tastes of the dictator and the victim. The model further suggests that time allocation in one use does not directly affect the allocation of time in the others, but rather only affects the other time allocations indirectly through a correlation of the error terms in the equations for the time allocations, and through the necessity for the sum of all three time allocations to equal the total time available. One can incorporate the correlation in the error terms in the time allocation equations by using an estimation procedure such as Zellner’s seemingly unrelated regression technique (Zellner, 1962). The fact that the total time allocated to the three uses cannot exceed total time available can be incorporated directly into the system of equations or one may estimate only two of the three time allocation equations and solve for

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the third time allocation. The former approach will provide more efficient estimates. ” Before our model is ready for estimation, we must do two things: (1) specify a mathematical form for the time allocation equations, and (2) suggest empirical measures for the variables in our theoretical model. The appropriate mathematical form for the time allocation equations depends on the form of the dictator’s utility function. When selecting mathematical forms for the utility function, economists choose those which conform to commonly observed characteristics of individual tastes (e.g., diminishing marginal utility). The simplest form for the utility function which economists commonly find acceptable is the multiplicative, simple Cobb-Douglas form.14 This form of the utility function implies supply and demand equations which are linear in the logarithms of all variables.15 Our final task is to suggest empirical measures for the theoretical variables in our model. Table 2 summarizes our suggestions for empirical measures for the theoretical variables in our model. The measures which we have suggested are not intended to be definitive, but rather are designed to stimulate discussion and additional development. Although the model we have developed is a time allocation model, it may be necessary to proxy the time spent in violent activities by the frequency and severity of harm. The frequency and severity of harm is probably best proxied by self-reports although we are aware of the difficulties involved in using such measures.16 The accuracy of self-reports could be checked by comparing such reports with civil and criminal records. If self-reports are not available, police reports and court data (particularly civil court) would provide possible alternative measures. Measures for the expected probability of sanction imposition are probably best obtained from an interview with the individual inflicting harm. If I’ See Deaton and Muellbauer (1980) for a discussion of the “adding-up” restrictions. Economic theory suggests other parameter restrictions as well. For example, this theory indicates that own substitution effects should be negative for non-Giffen goods. Generally these restrictions are not imposed, but rather parameter estimates are checked for conformity with the restrictions after estimation. For example, see Abbott and Ashenfelter (1976). I4 See Varian (1978) or Deaton and Muellbauer (1980) for discussions of this and other functional forms. Is A potential empirical difficulty when using the double log form is how one treats Lero or negative values for variables. Researchers have traditionally handled this situation by adding an arbitrary constant to variables with zero or negative values. Statisticians have developed methods for discerning the best constant to use in such situations. For example, see Aitchinson and Brown (1976). I6 See Hindelang, Hirschi, and Weis (1981) for a recent survey of the use and difficulties with this type of data. Locander, Sudman, and Bradburn (1976) and Elliott and Ageton (1980) provide examples of recent developments in methods designed to provide more accurate self-reports.

Suggested Empirical

TABLE 2 Measure for Theoretical

Variables

Suggested empirical measure

Expected sign of coefficient

Family income from sources other than dictator’s earnings

Income of other family members and from family assets and transfer payments


Subjective estimates of the probability that external interventions will occur

Estimates of this probability from the general experience in the community under consideration

Ambiguous unless individual is risk neutral in which case
Subjective estimate of the time required to cope with external interventions

Estimate of the average time required by various types of external interventions

>O if violence is an inferior good GO if violence is a normal good

Subjective estimate of the money costs of external intervention

Estimate of the average money cost of external interventions; statutory fines for family violence might be a useful estimate


Dictator’s

The market wage rate received by the dictator

>O as long as the total income effect outweighs the partial income effect and violence is a normal good

Actual or potential wage of victim; educational level of victim; labor market experience of victim

Ambiguous if sanctions depend on the level of violence
Theoretical

variable

wage

Victim’s disposable income outside the marriage

Number of children and dependent adults for whom the victim would be responsible if he/she left the family Distance from supportive friends and relatives; number of shelter and other public support facilities available Victim and dictator’s taste (e.g., dictator’s internal sanctions against violence)

Subjective measure of individual attitudes to violence, work, leisure, risk, and the family

379

Will depend on the measure used

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this is not possible, estimates could be obtained from the victim or could be proxied by civil and criminal records. ” The dictator’s level of internal sanctions can probably be best measured by interviews with the dictator which use psychometric scales designed to measure attitudes toward violence and marriage. If this is not possible, this variable could be proxied by the dictator’s past record of violent activity. The nonmonetary costs (time costs) and monetary costs of sanctions for violence can be measured from the dictator’s subjective beliefs concerning these costs if it is possible to interview the individual inflicting harm. If this is not possible, nonmonetary costs could be proxied by time spent “cooling off” at the station house, or the length of civil restraining orders. Monetary costs could be proxied by the amount of fines and court costs. The victim’s attitude toward violence, like the dictator’s, is probably best proxied by psychometric scales administered in individual interviews. It would be useful if one of these scales were to measure the degree of submissiveness of the victim. As we have noted in the previous section, we cannot observe the victim’s level of disposable income outside the marriage; however, we can proxy these opportunities by measures of the victim’s level of labor market skills, the number of individuals for whom the victim would be responsible outside the marriage, and the social support system available to the victim outside the marriage. It should be quite easy to determine the number of children and other dependents in the family. The existing literature suggests that it will be useful to measure the extent of dependency. This might be proxied by the age of children (perhaps a preschool age child binary) or the extent of physical impairment. It is extremely important to estimate the opportunities available to the victim outside the marriage. The nature of these opportunities are quite complex as families serve numerous functions (e.g., provide physical and emotional sustenance). A major function of many families is to provide physical sustenance. The victim’s ability to provide for such needs outside the family will depend on his or her marketability (i.e., ability to earn a living) and the availability of alternative sources for physical needs (e.g., food, housing). For victims who are working, measuring potential earnings outside the family should not prove too difficult although earnings may be different if the victim remains in the marriage than if he or she leaves. For victims who have not been in the labor market for some time, measurement of this variable will prove more difficult although the labor market literature provides a number of possible ” See Witte (1980) for additional this variable.

suggestions concerning possible methods of estimating

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measures. For example, Heckman (1976, 1979) has developed a method of adjusting estimated wages by incorporating a measure of the probability of working in a wage estimating equation. If the victim is unable or unwilling to work, the level of transfer payments available should directly affect the decision to leave the family. These transfer payments may either come from family, friends, or public bodies. The level and availability of such payments might be measured subjectively from the victim’s perceptions or objectively from public records or by the number of family and friends supportive of the victim. Perhaps, the most difficult time for the victim is the time immediately after departure from the family. The availability of physical and emotional support at this time may be crucial in the victim’s decisions. The availability of such support might be measured by the availability of shelters or the proximity of supportive family and friends. 5. SUMMARY

AND CONCLUSIONS

In this paper, we have developed a model of violence between adult family members which draws from the sociological theories of family violence and social exchange, and from the economic theories of crime and the family. The theoretical base of the model is that of an individual family member weighing the costs and gains of alternative choices in maximizing his or her level of well-being. Since the family consists of more than one individual, it is necessary to make assumptions concerning the way in which the family as a unit makes decisions. We have assumed that the individual inflicting harm, who we have designated as the dictator, makes all the decisions regarding family consumption and his or her own time allocations. In making these decisions, the dictator is constrained by two factors: (1) the income available for consumption spending is limited by the level of family income, and (2) the marriage will only remain intact so long as the other adult family member, referred to as the victim, is as well off in the marriage as he or she would be outside the marriage. In addition, the dictator’s decisions are affected by the possibility of costly outside intervention occurring as a result of the violence. The dictator chooses to allocate his or her time between three uses-leisure, work, and violence-in order to maximize his or her level of well-being. The dictator’s level of well-being will increase with an increase in family income, time spent in violence, and time spent in leisure. Using the conditions necessary for the attainment of a maximum for the dictator’s optimization problem, the implicit function theorem, and Cramer’s rule, we have determined the way in which factors beyond the dictator’s control affect his or her time allocation choice. We find, as we would expect, that an improvement in the victim’s best alternative situation outside the marriage will improve the well-being of the individual within the marriage, regardless of whether the level of violence changes.

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If we assume that monetary sanctions (i.e., fines, court costs) are not dependent on the level of violence, then we can say unambiguously that an improvement in the victim’s outside opportunities reduces the level of violence within the marriage. We also find that an increase in the dictator’s level of internal sanctions against violence will result in a lower time allocation to violence. In determining the effect of other changes on the level of violence we must make an assumption concerning the way in which the dictator’s desire to inflict violence changes when family income changes. If we assume that time in violence is a normal good (i.e., the dictator desires greater time in violence as family income rises) then we can obtain the following results. The total income effect, which reflects the change in time in violence with an exogenous and equal change in family income in all states, will, by definition, be positive. The partial income effect due to the decrease in family income which results from an increase in the time sanction for violence will be negative. That is, an increase in the time costs of external intervention will decrease the level of violence. Such an increase would occur if police increased the “cooling-off” period imposed in cases of domestic violence. This result is referred to as a “partial” income effect since family income is changed only in the event that outside intervention does occur. An increase in the monetary sanction for violence will also result in a decrease in the level of violence. Thus, an increase in the size of fines imposed for family violence would be expected to decrease the level of violence. In determining the impact of a change in the dictator’s wage on time in violence we must make the additional assumption that the total income effect outweighs the partial income effect. With this assumption, an increase in the dictator’s wage will increase the dictator’s time allocated to violence. In addition to these results we have examined the effect of a change in the probability of external intervention on the time allocated to violence. While we cannot unambiguously sign this term in the general case, we can say that an increase in this probability will reduce the level of violence if both the dictator and victim are risk neutral as defined in the body of the paper. In the previous section we suggest some insights which our model provides for empirical work. First and most importantly, the model suggests factors which are likely to be important in determining the nature and extent of family violence. In addition, it provides expected signs or interpretations of estimated coefficients for a number of those variables. In conjunction with other economic work on family decisions we suggest a possible mathematical form for the time allocation equations and a stochastic error structure for the empirical model. Finally, we present possible empirical measures of the theoretical variables in our model.

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This paper represents a first attempt to formally model violence within the family in a structured framework. While we believe there are many insights to be gained from this model, we see several areas in which further efforts would be very beneficial. First and foremost, the concept of violence as a time allocation choice which gives the dictator direct utility is much too simple. Consideration of the severity, frequency, and duration of violence as well as the use of violence as a control mechanism by the dictator would provide interesting extensions of the model. Further effort might also be devoted to incorporating uncertainty in the victim’s responses to the dictator’s use of violence. Interviews of individuals involved in violent relationships suggest that violent relations remain intact to varying degrees and that over time either or both partners may choose to leave and return to the relationship many times. The intact marriage constraint as defined in this model would only be applicable to those violent relationships which were continuously maintained. One possibility for incorporating the intermittent nature of some violent relationships would be to develop a dynamic model of family violence. APPENDIX

The problem facing the dictator is to maximize his or her expected utility by choosing the amount of time to spend in leisure and in violence. Formally, the dictator’s optimization problem is max EU = (1 - p)U(Z’, t,,, t,; a) + pU(Z”, t,, t,; a)

(Al)

subject to the intact marriage constraint EV - V” = (1 - p)V(Z”, t,) + pV(Z”, t,) - VO(Y) 3 0

(A21

where T = to + t, + t, + ty,

(A3)

I” = A + wto = A + w(T - t, - t,),

(A4)

I” = A + w(t, - ty) - ,S(t,; /3)

(A3

and

= A + w(T - tv - t, - tY) - S(t,;p).

For ease of exposition, we indicate that a function is evaluated at I” by superscripting the function with an s and that a function is evaluated at Z” by subscripting the function by u. In addition, we refer to a partial derivative by superscripting the relevant function with the number of

384

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the argument. For example, in the case of the dictator’s utility function, aU(Z”, ZV, f,; a>/aZ’ = US since income is the first argument in the utility function and we are evaluating the derivative at I‘. We assume that the partial derivatives of the dictator’s and victim’s utility functions have the following signs. r/F, WY > 0,

i = 1, 2, 3,

v;, vy

> 0,

Uyi, Uz < 0,

i = 1, 2, 3,

v;, v;

< 0,

i = 2, 3,

V:;, Vs < 0,

Usi, U;lj > 0,

v;,, vl;, < 0,

G, 3 w, < 0, U:a, UL = 0,

i = 1, 2,

i = I,3

VXY)

> 0,

u;,, u;, -=I0. The signs on the derivatives of the monetary sanction function are assumed to be Si>O,

i=

1,/3,

s,, > 0. The interesting case in this model occurs when the intact marriage constraint is binding. If the constraint is not binding, then any changes in the parameters and/or variables in the intact marriage constraint do not affect the dictator’s actions and the problem becomes simply an unconstrained allocation of time as is often used in “economics of crime” studies. Since we wish to examine the response of the dictator to the intact marriage constraint we will assume throughout the analysis that the constraint is binding. The Lagrangian equation for the optimization problem is L = (1 - p)U(A + w(T - I, - t,), t,, t,; a) -t pU(A + w(T - fv - t, - q) - Sk; p), t,, tl; a) + A[(1 - p)V(A + w(T - t, - t,), t,) -t pV(A + w(T - t, - 4 - $1 - SC&; PI, f”) - VO(Y)].

(A61

FAMILY

385

VIOLENCE

The necessary first-order conditions for an internal maximum in which the constraint is binding are L1 = $

v

= (I - p)[U,(A

+ w(T - t, - t,), rv, t,; a)( -w)

+ UdA + WV - t, - 41,t, , 4; 41 + p[U,(A

+ w(T - t, - t, - r,)

- at, ; PI, t, , 4 ; a)( - w - S,) + w(T - t, - 4 - tY) - W, ; P), tv , fr ; 41

+ WA

+ MU - pW,(A + WV - 1, - t,), &I(-w) + V,(A + W - f, - 41,&)I + pW,(A + WV - t, - h - ‘?I - Sk; P), &N-w - S,) + V,(A + w(T - rv - t, - q) - Sk; L2 = $

I

= (1 - p)[U,(A + WA

+ W

+ pW,(A + &(A

/3), t,)]] = 0

647)

+ w(T - t, - t,), t,, t,; a)(-~) - t, - 41, t, , 4 ; 41

+ w(T - t, - t, - ‘J - Sk; PI, t,, tl; a)(- w) + w(T - t, - 4 - fv) - W,;

P), 2, , 6; a)]

+ MC1 - p)V,(A + w(T - t, - t,), t,)(-w) + pV,(A + w(T - t, - tl - fy) - Sk; Ls = $

,6), t,)(-w)]

= 0 (A8)

= (1 - p)V(A + w(T - t, - q), t,) + pV(A + w(T - t, - t, - fy) - S(t,; p), t,) - VO(Y) = 0.

649)

The second-order sufficient conditions for a maximum require that the determinant of the 3 x 3 bordered Hessian matrix, [m, be positive. The element of the Hessian in the ith row and jth column is denoted as h,.

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The cofactor of this element is denoted as IH;jl. The elements of the bordered Hessian are L,, = h,, = (1 - p)[w*u;,

- 2wu;,

+ cl;,]

+ PKW + W2Wl

- 2(w + s,)uy,

+ A(1 - p)[w*vq,

- 2wvi2 + Vi,]

+ b[(w

+ W2V~l - 2(w + S,)V;,

L*2 = h,, = (1 - p)[w*u;,

- wu;,

+ V;, - S,,v:]

- wu;,

+ L$]

- WV&]

+ Xp[w(w + S,)VY, - WV&] L,, = h,3 = (1 - p)[- WV; + v;1 + p[ -(w L2,

=

4512

(MO)

+ u;j]

- (w + S,)U$

+ P[W(W + s,w:, + X(1 - p)[w*v;,

- wu;,

+ r/y, - s,,uy]

(All) + S,)VY + v;]

6412)

G413)

= ha

L22 = h,, = (1 - p)[w’U:,

- 2wU;,

+ u;J

+ p[w2U’f, - 2uy3 + U,“,] + A(1 - p)w’V;, L23 = hZ3 = -(l

- p)wV;

+ Apw’V;,

- pwv;

(A14) (Al3

LI

= ~5, = h3,

(A161

-532

=

W7)

~523

= hx

Lj3 = hx3 = 0.

(A181

We assume that the second-order conditions for a maximum are satisfied. This implies that the determinant hit h 21 I h 31

42 hn h,, hz3 2 h,2 h33 I

which is the bordered Hessian Ia/, is positive. The first-order conditions for a maximum defined the endogenous variables as implicit functions of A, w, &, p, CX,p, and Y (i.e., the exogenous variables). We can use the implicit function theorem and Cramer’s rule to examine the effect of changes in the exogenous variables on each of the endogenous variables. We are interested in the effects of these variables

FAMILY

on time spent in violence. that

or equivalently

387

VIOLENCE

By the implicit function

theorem

we know

we can rewrite this as

at,/ax [J] at,/ax = [D] [ ah/ax1 where x is any one of the exogenous variables (A, w, t,, p, a, p or Y). Using Cramer’s rule we have

where [Jjl is the determinant of the matrix which results when the column of [J] relating to the relevant endogenous variable has been replaced by the vector [II] (in this case j = 1). Since [J] is equivalent to [r;i] we know that the determinant of [J] is also positive. Using these relationships we can derive expressions for the effects of the exogenous variables on time spent by the dictator in violence. For simplicity, we present the results which occur under the assumption of additive separability of the dictator’s and victim’s utility functions in the endogenous variables. Under this assumption Uij = 0 and V, = 0, where i # j for all endogenous variables. The results are analogous but more complex if we do not make the separability assumption. First consider the effect of a change in A on t,: at,1 aA - $(I + Mw

- pbw,

+ p(w + s,>w,

+ A(1 - P)WVS,

+ S,)V’;,l I&II + [Cl - pbw,

+ pm;,

While we are unable to sign this result on a priori grounds, from economic theory that at,/aA is defined to be positive or is a normal good, and negative if t, is an inferior good. This the total income effect, the effect of an exogenous change which income the same amount in all states of the world.

(A191 we know zero if t, is simply increases

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Next consider the effect of a change in 1, on t,: EL at,

- - $w[P(w + s,w;, + Mw + S,)VY,lIW,,l - W[PWU’;l+ ApwVY,ll&l - W[PW l~31l~ =

- w 2 + $w[(l - PhW, + Ml - P)WV,l IW ( 1 + w[(l - phw, + Ml - p)wv;,l p41)- w[(l - p)Cl lfm.

This is a partial income effect. Changing ty alters income in the unsuccessful state but has no effect on income in the successful state. Although we cannot sign a&/at, a priori, we would reasonably expect, by analogy with the total income effect, that at,/&, s 0 if t, is a normal good and that at,/&,, > 0 if t, is an inferior good. The effect of a change in w on t, is

at,aw-(‘-t’-t’~(~)+$~)+~I(l-p)u: + PUI; + AU - p)V; + ApVlfl[I&II + I&d.

6421)

The third term of this expression is unambiguously positive since 1~~~1 is greater in absolute value than IH,, I. The first term is a total income effect and will be positive or zero when tv is a normal good and negative when t, is an inferior good. The second term is a partial income effect of the type discussed above. We would normally expect this term to be negative or zero when t, is a normal good and positive when t, is an inferior good. Thus, the effect of a change in w on t, will depend on (1) whether t, is a normal or inferior good, and (2) the relative magnitude of the total and partial income effects. The effect of a change in /3 on t, is

The second term of this expression is clearly negative and gives the impact of increasing the marginal price of violence, i.e., it is a substitution effect. The sign of the first term depends on the sign of at,/at,. which, as we have discussed above, we would expect to be negative or zero if t, is a normal good and positive if t, is an inferior good. Thus, unless t, is an inferior good and the weighted partial income effect outweighs the substitution effect we would expect at,/ap < 0.

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The effect of a change in (Y on t, is

The effect of a change on Y in t, is

ah-

1

aY - phV~(Y) I&,Il.

We are not able to sign this result a priori since lH3,1 cannot be signed as long as the monetary costs of external intervention depend on the level of violence. If the victim’s situation outside the marriage improves, the dictator must increase the victim’s expected utility within the marriage in order to maintain the relationship intact. The dictator can do this by working more hours, and thereby increasing income, by decreasing the level of violence, or by undertaking some combination of the two. If time in violence is a normal good, the impact of any combination of changes in work time and time in violence will lead to countervailing partial income effects on the amount of time allocated to violence. This occurs because the level of monetary sanctions and, thus, the level of family income when there is outside intervention, is a function of the dictator’s time in violence. This countervailing partial income effect will only be eliminated if the monetary sanctions do not depend upon the level of violence. When this is the case (i.e., S, = 0) then &/aY < 0. The effect of a change in p on t, is

s1 -[WV; - (w + s,yYy + hwvs - A(w+ s,>vy Ifb,I ap - m{ - [wu; - wu;l + hwv; - AwV’t] I&,1

- WV”, t,) - VU”, &)I I&II.

(A29

An increase in p increases the probability of the less desirable state of the world. In order to maintain the marriage intact the dictator must compensate the victim by either decreasing the level of violence, increasing family income, or some combination of the two. However, we cannot determine unambiguously the effect of an increase in p on the level of violence since the compensation could take the form of only an increase in income. We can unambiguously sign at,/@ if we assume U,i and VI1 equal zero. This is equivalent to the assumption of risk neutrality when income alone enters the utility function. With this assumption we have

at,/ap < 0. REFERENCES Abbott, M., and Ashenfelter, 0. (1976), “Labor supply, commodity demand and the allocation of time,” The Review of Economic Studies 43, 389-411.

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