## Notes

 1 In this paper we use the terms family planning, modern methods and contraception interchangeably to mean the methods promoted by family planning programs. These methods are distributed primarily through health services (clinics, hospitals) such as the pill, the injection and tubal ligation, as well as condoms. 2 If part of the previous program effort involved public subsidization of contraceptives, the elimination of these subsidies might increase the price of contraceptives, which in turn could induce some reduction in use from what use would have been in the absence of this increase. If this increase were small enough, it would cause a small downward movement in the same equilibrium. If it were large enough, it could cause a nonlocal movement to a lower equilibrium. 3 For this equilibrium to be in the [0,1] interval places constraints on the parameters in relation {1}, though if it were not the corner solution of 0 or 1 would prevail. We also note that an interior equilibrium value may seem peculiar in the sense that we have assumed identical individuals for this discussion, so in equilibrium all would seem to be either nonusers or users, not some mixture. Empirical estimates, from this perspective, combine different groups of individuals so that on the average the probability of use may be between 0 and 1. Alternatively, the equilibrium can be defined to be the point at which the probability of adoption of each individual (not whether they actually adopt or not) is identical (though this probability may be between 0 and 1 so that some but not everyone adopts at a point of time with stochastic differences determining who actually does and does not adopt). 4 Within the linear model above there may be status-quo reinforcement if α < 0, but only for corner solutions, not stable interior solutions. For interior solutions, increasing the impact of social interactions in the sense of increasing "α" in relation (1) unambiguously increases the social multiplier. 5 The total effect of program efforts in the nonlinear model is given by = b (ye*(1-ye))/ [1- a (ye*(1-ye))] = (1-a*ye*(1-ye))-1*P(y=1|x,ye)/x, where P(y=1|x,ye)/x is the direct program effect. The social multiplier MN in the nonlinear model therefore equals MN = 1/(1-a*ye*(1-ye)). Taking the derivative of MN with respect to a then yields the effect of intensifying social interaction on the social multiplier. Using the implicit function theorem, we obtain MN/a = (1-a*ye*(1-ye))-3* ye*(1-ye)*(1 - a/2 + a*ye*(1-ye)). The sign of this derivative can be positive or negative and depends on the final term in the expression. If the derivative is positive, then increases in the `strength' of social interaction, i.e, increases in the parameter a, result in larger multiplier effects and hence augment the program effort. If the derivative is negative, then increases in the parameter a reduce the multiplier effect. An important result is that, if the necessary condition for the existence of multiple equilibria, a > 4, holds, then at each stable equilibrium M/a < 0. In all situations with multiple equilibria, therefore, intensifying social interaction reduces the social multiplier effect and `more' social interaction is therefore not desirable from the standpoint of program effectiveness because they retard rather then reinforce the changes introduced by programs. The effect of intensifying social interaction on the total program effect can be analyzed similarly by taking the derivative 2(ye)/xa, and the sign of this derivative is also ambiguous. Similar to the effect of intensified social interaction on the multiplier effect, the total effect also decreases with 'more' social interaction at each stable equilibrium in a multiple-equilibria situation. 6 The transitions between the equilibria in our theoretical model are reversible, i.e., there can be a transition from the low to high contraceptive use equilibrium and also vice versa. The proportion of women who have ever used contraception in a population, however, cannot decline. Since S. Nyanza District is still at early stages in the fertility decline and has never experienced a high level of contraceptive use, only the former transition is empirically relevant. This transition from low to high contraceptive use can be measured with both dependent variables "ever used contraception" and "currently using contraception". We believe that the former provides a better measure for the adoption of modern family planning and we subsequently use it in our empirical analyses. The reverse transition from high to low levels of contraceptive use, which cannot be measured with a dependent variable like "ever use", is not empirically relevant in the present context. 7 Note that this variable is not confounded by the well-known problem of selective recall (i.e. that the respondent who is more disposed to using contraception is more likely than average in her community to recall hearing a family planning message on the radio) because the variable is constructed so that the respondent's response is not included in the measure used for that respondent. 8 That is, in none of the estimated models that underlie Table 2 do the estimates for the parameter a exceeded the critical level a > 4 which is necessary in order to have multiple equilibria [Table A1]. 9 That is, if the social multiplier is 175%, the proportion of the total effect due to social interaction is 75/175. 10 Related models of multiple equilibria and path dependency in the context of fertility decline are found in [e.g. Becker, Murphy and Tamura 1990, Galor and Weil 1996, Kohler 1997, Kohler 2000b].

 Empirical Assessments of Social Networks, Fertility and Family Planning Programs: Nonlinearities and their Implications Hans-Peter Kohler, Jere R. Behrman, Susan Cotts Watkins © 2000 Max-Planck-Gesellschaft ISSN 1435-9871 http://www.demographic-research.org/Volumes/Vol3/7