# Online Appendix I: A Model of Household Bargaining with Violence. In this appendix I develop a simple model of household bargaining that

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1 Online Appendix I: A Model of Household Bargaining ith Violence In this appendix I develop a siple odel of household bargaining that incorporates violence and shos under hat assuptions an increase in oen s relative incoe leads to a decline in violence. Let U (C, S) be a oan s utility hich is increasing in her on consuption (C ) and increasing in safety (S) and let U (C, V) be a an s utility hich is increasing in his on consuption (C ) and in violence (V). Assue that there is an upper bound to violence (death) V and S= V V. Both utility functions are assued to be strictly concave, onotonically increasing, differentiable and hoothetic. I is total household incoe and α is the share of incoe a oan ould get if she ere not in the partnership (not necessarily hat she earns inside the relationship). To focus on partnerships that ill experience soe violence, assue that the arginal rate of substitution beteen consuption and violence is greater for oen than en here V=0 (MRS ((1- α)i, 0) < MRS (αi, V )). We denote as T the set of feasible utility pairs (U, U ) that the partners ay obtain if they reach an agreeent and the utility each gets fro dissolution of the partnership (the disagreeent payoff or single state utility) as (d, d ) = (U((1- α)i, 0), U (αi, V )). I first sho that the doestic violence proble constitutes a Nash bargaining proble and that a Nash bargaining solution provides a unique solution to the proble. Lea 1: Under the above assuptions, the proble is a Nash bargaining proble ith a Nash bargaining solution (Nash, 1950). The folloing necessary and sufficient conditions are et:

2 1) The set T is copact. This is satisfied since the set of feasible allocations is copact and U and U are continuous. 2) The set T is convex. This is satisfied since the set of feasible allocations is convex and U and U are continuous. 3) d= (U ((1- α)i, 0), U (αi, V )) is a eber of T. 4) For soe eber (U, U ) of T it is the case that U >d and U >d. This holds under the assuption that MRS ((1- α)i, 0) < MRS (αi, V ). To allo for different degrees of bargaining poer beteen partners, e ill use Kalai s asyetric Nash bargaining solution. The solution to the asyetric bargaining proble (U*, U* ) axiizes the folloing expression as shon by Kalai (1983): (U -d ) τ (U -d ) 1-τ subject to (U, U ) (d, d ) and (U, U ) feasible, here τ is a easure of bargaining poer of the an. I study next ho an increase in the relative age of oen affects utilities of both en and oen. Lea 2: An increase in α results in an increase in U* and a decrease in U*. Take α and α such that α'> α. Then d (α )> d (α) and d (α )< d (α). Therefore, τ ( U d ( α' )) ( 1 τ ) ( U d ( α' )) < τ ( U d ( α )) ( 1 τ ) ( U d ( α )) and by convexity of T e have that U* (α ) U* (α).

3 This alone does not guarantee that an increase in α results in a decline in violence. This depends on the shape of the contract curve. Lea 3: the contract curve has a positive slope. Let (C, V) and (C, V ) be points on the contract curve such that C > C and V < V. By definition, MRS (C, V) = MRS (C, V V ) since (C,V) is a point on the contract curve. This iplies that C' C > and V ' V V C' V ' < V C V By hootheticity and strict concavity of utility functions, MRS (C, V ) < MRS (C, V) and MRS (C, MRS (C, curve. V V V V ) > MRS (C,V). This iplies that ) > MRS (C, V ) and the point (C, V ) cannot be on the contract Theore 1: An increase in α results in a decrease in violence. Take α and α such that α'> α. Then U* (α ) > U* (α), by lea 2, hich iplies that V(α ) < V(α), since the contract curve has a positive slope by lea 3. The above siple odel of household bargaining shos that under certain reasonable assuptions (naely strict concavity, differentiability and hootheticity of utility

4 functions) an increase in a oan s relative incoe leads to a decline in violence against her. 1 1 Incorporating the arriage arket should not affect this coparative static result. Rather, the arriage arket siply influences the disagreeent payoff (d) hich ill becoe the axiu of an individual s single state utility and the expected utility fro another atch. In this paper I do not focus on the arriage arket because there is no data for California on arriages and divorces.

5 Online Appendix II: Deterinants of the Wage Gap I estiate the ipact of changes in underlying coposition of the labor force on ale and feale ages as an additional test of the exogeneity of the age easures. In coluns 1 and 2 of Appendix Table 1 I present estiates of the ipact of feale college enrollent, iigration, incarceration flos and changes in the feale population aged on feale ages. In colun 1 all covariates are entered together and in colun 2 I present estiates in hich the covariates are entered separately (each ro represents a separate regression.) I do the latter because the lack of precision hen all covariates are entered together ay reflect collinarity. For each specification I also include county and year fixed effects and county and race specific linear tie trends and the natural log of the relevant population age (these controls are included in the ain regression specifications presented in Table 2). In coluns 3 and 4 I present the sae results for ales. It should be noted that these results exclude the lags of doestic violence (hospitalization for assault or arrests for doestic violence) that I include in soe of the regressions of the ipact of ages on violence hich ould, if anything, likely further decrease the estiated effects of changes in underlying characteristics of orkers on ages. As is evident fro the table, these variables appear to have no significant ipact on either feale or ale ages as constructed according to equation (1). The point estiates are all sall and iprecise. In coluns 5-8 I present the sae set of estiates but on ages that are constructed based on changes in the underlying industrial structure of the county not changes in state-ide ages in industries doinant in the county in While the

6 results are soehat siilar to those in coluns 1-4, the point estiates for education (college enrollent) are slightly larger and ore precisely estiated for ales, though still relatively sall and insignificant. This is consistent ith the possibility of selective, endogenous, changes in industrial structure over tie.

7 Appendix Table 1: Predicting Wage Changes (1) (2) (3) (4) (5) (6) (7) (8) feale feale ale ale feale feale ale ale Ln(feale college students) [0.041] [0.041] [0.050] [0.055] Ln(ale college students) [0.052] [0.050] [0.040] [0.045] Ln(non-intiate hoicides) [0.004] [0.004] [0.004] [0.003] [0.007] [0.007] [0.008] [0.008] Ln(iigration) [0.004] [0.004] [0.005] [0.005] [0.002] [0.002] [0.004] [0.002] Incarceration flos per 1000 ales [0.012] [0.011] [0.013] [0.011] [0.027] [0.031] [0.020] [0.027] Ln(feale pop 15-44) [0.038] [0.050] Ln(ale pop 15-44) [0.048] [0.044] Observations R-squared Robust standard errors in brackets Wages used state state state state industry industry industry industry Covariates entered together Y Y Y Y Covariates entered separately - each ro a separate regression Y Y Y Y

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