Proposition 1 (Socialization Investments). In cultural heterogeneous societies, qi ∈
(0, 1), for i, j∈ {a, b}, in equilibrium:
i. in homogamous married families (i = j), the parents socialization efforts are strictly positive, τ({i, j}, 0, qi) > 0,∀p ∈ {m, f}and the probability of successful socializa-
tion of children is greater than the horizontal socialization rate, Pi({i, j}, 0, qi) > qi;
the parents socialization efforts, τ({i, j}, 0, qi) > 0,∀p ∈ {m, f} are monotonically increasing in cultural intolerance preferences,∆Vij and decreasing in the population distribution, qi;
ii. in heterogamous married families (i 6= j), the socialization effort of the parent with higher cultural intolerance is strictly positive, while the other parent does not invest in socialization. Let’s assume∆Vji > ∆Vij, then τjf({i, j}, 0, qi) > 0; τmi({i, j}, 0, qi) = 0, the probability of successful socialization of children to trait j is greater than the hor- izontal socialization rate, Pj({i, j}, 0, qi) > (1−qi) while Pi({i, j}, 0, qi)) < qi. Par-
ent j socialization effort, τfj({i, j}, 0, qi) is monotonically increasing in his own cul- tural intolerance preference,∆Vji; decreasing in his spouse cultural intolerance pref- erence, ∆Vij; and finally increasing in the population distribution qi. Under per- fect symmetry between parents’ cultural intolerance preferences, the direct social- ization technology of heterogamous families nullifies, τ({i, j}, 0, qi) = 0 and social- ization probabilities equate to horizontal socialization rates, Pi({i, j}, 0, qi) = qi and
Pj({i, j}, 0, qi) = (1−qi);
iii. in homogamous divorced families (i = j), the mother socialization effort is strictly positive: τfj({i, j}, 1, qi) >0. Divorced families underinvest in socialization compared to married ones, τ({i, j}, 1, qi) ≤τ({i, j}, 0, qi);
iv. in heterogamous divorced families (i 6= j), the mother socialization effort is strictly positive: τjf({i, j}, 1, qi) > 0. Divorced families overinvest in socialization compared to married ones, τ({i, j}, 1, qi) ≥τ({i, j}, 0, qi);
Parents make costly investments in order to socialize their children, both in homoga- mous and heterogamous families. Socialization investments in homogamous families ben-
efit from coordinate incentives. Conversely, a positive socialization investment in heterog- amous families hinges on cultural intolerance asymmetries. Indeed, in case of perfect sym- metry between parents preferences, the model goes back to the Bisin and Verdier(2000) initial assumption of zero vertical transmission.
Parents have the incentive to positively socialize their children, both under marriage and divorce states, if socialization is available11. Conditioning on being married, parents’ socialization investments are Pareto efficient, resulting from marital cooperative behaviour. Conversely, under divorce, mothers’ socialization investments are socially inefficient. Posi- tive or negative externalities in socialization between homogamous and heterogamous mar- ried families, respectively, translate in underinvestment or overinvestment effort provision under divorce.
In addition, homogamous families hold a more efficient socialization technology, com- pared to heterogamous ones, τ({i, j}, 0, qi | i=j) >τ({i, j}, 0, qi | i6=j)and Pi({i, j}, 0, qi |
i= j) > Pi({i, j}, 0, qi | i6= j) >qi in case of marriage, while under divorce, both families
are equal in their socialization technologies, τfj({i, j}, 1, qi | i= j) =τjf({i, j}, 1, qi | i6=j).
Interestingly, Proposition 1 outlines an asymmetry in household behaviours between the minority ethnic group as opposed to the majoritarian one, because parents socialization choices interact with their social environment of reference. In particular, the direct efforts of parents in the intergenerational transmission of cultural values substitute the horizon- tal socialization channel, which implies that the direct efforts are larger for the minority group, ceteris paribus. Finally, any increase in cultural intolerance of parent of type i,∆Vij translated into an higher socialization quality of children successfully socialized to trait i. This positively affects parent i socialization investment, τmi({i, j}, d, qi). Notice in particular that, while socialization efforts in homogamous families,{i, j}with i = j, respond only to cultural intolerance preferences ∆Vij, heterogamous families socialization efforts respond to both parents cultural preferences∆Vijand∆Vji, in opposite directions.
Proposition 2 (Divorce Choice). In cultural heterogeneous societies, qi ∈ (0, 1), for
i, j∈ {a, b}and assuming positive fertility n>0, at the equilibrium:
i. in homogamous families (i = j), the difference between the expected marital utility under divorce and marriage is strictly negative, n ˜W({i, j}, 1, qi) −n ˜W({i, j}, 0, qi) <0; ii. in heterogamous families (i 6= j), the difference between the expected marital utility under divorce and marriage is strictly negative, n ˜W({i, j}, 1, qi) −n ˜W({i, j}, 0, qi) <0; iii. for given realizations of marital stability shock,θ and assuming ∆Vji > ∆Vij, the di-
11Potential extensions to non-negative probability of father custody assignment after divorce, will not change
vorce probability of heterogamous families is higher compared to homogamous fam- ilies: π({i, j}, n, qi | i6=j) >π({i, j}, n, qi | i= j);
iv. for given realizations of marital stability shock, θ, and assuming ∆Vij > ∆Vji, the divorce probability of heterogamous families is higher with respect to homogamous minorities families, π({i, j}, n, qi | i 6= j) > π({i, j}, n, qi | i = j)if qi < 1/2 and
viceversa for qi >1/2,
Two different and opposed mechanisms contribute to explain divorce choices: the dif- ferential in socialization technologies across marital states and the free riding incentives driven by different attribution of socialization costs. Proposition 2 outlines that, for given socialization preferences and technologies, only negative surprises on marital quality trig- ger divorce decisions. Indeed, for a neutral realization, θ =0, the families remain married, because of the marital gains arising from the presence of public good. This implies that the loss in socialization technologies associated to divorce decisions is not counterbalanced by a comparable reduction in socialization costs, both for homogamous and heterogamous families. This is quite intuitive for homogamous families, as they largely lose from mar- ital deviations to divorce, because of less efficient cultural transmission technologies. In heterogamous families, instead, the divorce guarantees a more efficient socialization tech- nology to the mother at a cost of the father socialization. The composite effect is negative.
Moreover, for a given realization of marital quality shock, when the mothers have larger cultural intolerance preferences compared to the fathers, the divorce probability of heterog- amous families is higher with respect to homogamous families, independently of the distri- bution of cultural traits in the population. Thus, divorce choices for heterogamous families might be interpreted as a strategic deviation from marriage for mothers who have a pref- erence to socialize children and expect to have a higher probability of children custody attainment. This perspective is pointed out byDohmen et al. (2012). Indeed, they make the point that children are more strongly socialized by a divorced mother rather than in heterogamous families, while there is no significant difference in socialization compared to homogamous families. Finally, Proposition 2.iv states that the distribution of cultural traits in the population matters in the comparison of divorce decisions among families, when in heterogamous families the fathers have larger cultural intolerance preferences compared to the mothers. The result comes from the fact that being i the majoritarian group, qi > 1/2, the loss in socialization technologies of homogamous families is counterbalanced by the increase in the favourable horizontal socialization and by free riding incentives on social- ization costs.
Proposition 3.(Comparative Statics on Divorce Choices). In cultural heterogeneous so-
i. the divorce probabilities in homogamous and heterogamous families, π({i, j}, n, qi),∀i, j, are decreasing in the number of children n;
ii. the divorce probability in homogamous families (i = j), π({i, j}, n, qi | i = j), is decreasing in the level of cultural intolerance of parents ∆Vij and it is increasing in the proportion of the society with similar cultural trait qi;
iii. the divorce probability in heterogamous families (i 6= j), π({i, j}, n, qi | i 6= j), is decreasing in the level of cultural intolerance of parent with trait i, ∆Vij, and it is
decreasing in the proportion of the society with opposite cultural trait qi;
Divorce choices depends on the socialization quality of children, but also on the quantity of the fertility investment. Interpreting children as a marital specific public good, Proposi- tion 3.i uncovers a negative relationship between the probability of divorce of a family and the number of children in that family.
The probability of divorce rests on the distribution of cultural traits in the population. Other things equal, the marital instability of homogamous families belonging to the ma- joritarian group i is higher with respect to the instability of homogamous minority couples. Indeed, marital deviations bring a socialization risk that is lower the higher the proportion of the population with a similar cultural trait i, i.e. the more favourable the horizontal so- cialization channel qi. A symmetric result holds for heterogamous families. Finally, any increase in cultural intolerance of parent of type i,∆Vijtranslates into lower rates of marital
dissolution.
Proposition 4. In cultural heterogeneous societies, qi ∈ (0, 1), for i, j∈ {a, b}, at equilib-
rium:
i. the fertility rates in homogamous and heterogamous families, n({i, j}, qi), are strictly positive;