## 2. Implications of Linear and Nonlinear Models of Family Planning Diffusion with Social Interactions

In this section we formally identify the direct program impact versus total effect (i.e., the direct effect modulated by social interactions, see below) of increases in family planning program efforts in both linear and nonlinear models. We then compare the implications of linear and nonlinear models in situations in which program efforts are increased and in situations in which social interactions are intensified.

Linear probability model: We begin with the linear model because it is simpler and more transparent despite its well-known limitations. Let the probability that a woman adopts modern family planning (y = 1) be:

 P(y=1 | x, yc) =*(-.5 + yc) + *x + {1}

The term *(-.5 + yc) represents the influence of social interaction on a woman's probability to use family planning and is chosen to match our subsequent specification of the nonlinear model. The parameter reflects the `strength' or relevance of social interaction and determines the extent to which the adoption probability is affected by the contraceptive behavior in the village or reference group (yc). For = 0 there is no effect of social interaction, and increasing levels of tend to increase the relevance of social interaction for a woman's family planning decision. As shown below, 0 < < 1 is necessary for there to be an interior equilibrium in which some women in a community, but not every woman in the community, uses contraceptives. The term 0.5 represents a critical level that determines the direction of the social influence on a woman's contraceptive decision. If the contraceptive prevalence in the reference group (yc) is above 0.5, then social interaction increases the probability of using family planning as compared to the situation when no social interaction is present, and otherwise it decreases the probability. In a situation with yc = .5, i.e., a situation where half of the population uses and half does not use family planning, social interactions has no effect on a woman's decision to adopt contraception. The coefficient is the direct effect of program efforts (x), and larger program efforts increase the probability of using contraception when > 0. The final term is the constant. For simplicity, in our discussion of this theoretical model in this section (but not in our estimates in Section 4) we consider only women who are identical with respect to individual characteristics, which permits us to combine the effect of these characteristics into the constant term. As in the nonlinear model below, the constant term is assumed only to represent these individual characteristics and is assumed not to adjust to offset changes in the first term that determines the relevance of social influences on women's fertility decisions.

The solid line in Figure 1 plots the curve implied by equation {1}: the vertical axis gives an individual's probability of using contraception as related to the average contraceptive use for the individual's reference group (yc, on the horizontal axis) given the program effort x (e.g., proportion of other villagers who "heard a family planning message on the radio"). The slope of the solid line indicates how the probability of individual use changes when there is a discrepancy between the probability of an individual's use and the average contraceptive use of other women in her village.

It is important to note that the lines in Figure 1 specify the dependence of individual behavior on the family planning prevalence in the social environment of that individual. The figure therefore represents a micro-macro relation between individual and population behavior, and not an intertemporal relation. In particular, the linear model in Figure 1 is consistent with the typical S-shaped diffusion curve reflecting the increased adoption of an innovation over time in aggregate data [e.g., see Rogers 1995].

The linear model in Figure 1 exhibits only one equilibrium, the point at which each individual's behavior mirrors the village average -- where the solid line intersects the 45o ray from the origin in Figure 1 [Note 3]. This equilibrium therefore satisfies P(y=1 | x, ye) = ye, where ye is the equilibrium level of contraceptive use. In linear models, this equilibrium level can be calculated directly from the model parameters as ye=x/(1-) + ( - .5)/(1-). If individuals prefer to behave somewhat like others in their reference group (so that is positive but less than one), this equilibrium is stable. To the left of it the individual probability of use is above the village average use; therefore the average village use increases because the individual is in the reference group for others in the village, which causes movement to the right towards the equilibrium (and vice versa to the right of the equilibrium).

What happens when there is an increase in program effort, for example a new media campaign? We depict this changed relation between the program and social interaction as a shift from the solid to the dashed line in Figure 1. The direct effect on the probability of the individual's use of changing program efforts is the vertical distance indicated as the "direct program effect" in Figure 1 (the result of changing program effort by one unit while holding constant village average use). This direct program effect is not modulated by social interactions. If, however, the individual adjusts to her reference group, we get a social multiplier [Montgomery and Casterline 1993]. The social multiplier leads to a new and higher equilibrium level of contraceptive use, i.e., where the dashed line intersects the 45o ray. The total increase in the probability of contraceptive use is thus the total program effect, consisting of a direct program effect plus its multiplication by social interaction.

It is obvious that if one evaluates changes in family planning programs without taking the indirect effects of social interaction into account, the total effect provides an overestimate of the direct influence of program interventions. Whereas the direct effect is immediately linked to the intervention, the total effect results in addition from feedbacks in which initial program interventions are augmented through interaction. Formally, the total change in contraceptive use due to a unit increase in the program effect is ye/x = /(1 - ), which is greater than the direct impact of because of social interactions if 0 < < 1. The factor 1/(1 - ) represents the social multiplier effect ML in the linear model, and this multiplier effect equals the ratio of the total program effect to the direct impact of the program. Since ML in the linear model exceeds one for 0 < < 1, the total program effect is always larger than the direct effect of the program. Moreover, the social multiplier has the same value for all levels of contraceptive use, and its value is greater the greater is . That is, intensifying social interaction - in the sense of making the reference-group behavior more relevant for a woman's contraceptive decision by increasing the parameter - increases the social multiplier effect. Formally, this positive effect of strengthening social interaction on the multiplier effect is seen in the derivative ML/ = 1/(1 - )2 > 0. It is important to note that this increase of the social multiplier due to more intensive interaction is independent of the sign of P(y=1 | x, yc)/, i.e., the effect of the change in on the individual probability to use modern contraception.

While a fully dynamic model of social interaction is desirable [e.g., see Montgomery and Casterline 1998 or Montgomery and Zhao 1998 for simulation-based dynamic models of social interactions and fertility decisions], the distinction between direct and total effects provides a first approximation for interpretation of estimates within a dynamic framework. The direct effects of program changes can be interpreted to be short-term effects of program intervention, while the total effects incorporate the long-term implications that also include the eventual indirect consequences of family planning programs on knowledge about contraception and on norms of reproductive behavior.

A nonlinear model: We use the logistic form of a nonlinear model, a specification that is frequently used in theoretical models of social interactions [Brock and Durlauf 1995, Kohler 2000a,b, Manski 1993] and for empirical estimates [Arends-Kuenning 1997, Entwisle and Godley 1998, Kohler, Behrman, and Watkins 2001, Montgomery and Chung 1994, Munshi and Myaux 1997]. In addition, and importantly for our discussion below, this logistic model can be motivated from individual utility maximization [McFadden 1981]. While the standard motivation is individualistic--behavior depends only on individual characteristics (and prices)--we also include dependence of behavior on the social environment. In particular, we represent social interactions via a social utility term in which women experience disutility if their behavior deviates from the average level of contraceptive use in their reference group. This assumption captures our findings in the ethnographic interviews in the Kenya Diffusion and Ideational Change Project (KDICP) that in Nyanza, where family planning use was still quite low, there was substantial uncertainty about the appropriateness and the safety of modern methods [Rutenberg and Watkins 1997]. Alternatively, small families may be socially stigmatized if a large family norm is prevalent. Thus, we expect that increases in the prevalence of family planning in a village lead to a reduction in uncertainty about the consequences of small families and that these increases also make the use of family planning methods socially more acceptable.

We assume that the disutility from deviating from the average behavior of woman's reference group is related linearly to the difference between an individual's decision to use or not to use and the average reference group behavior yc. More specifically, we assume that the social utility term takes the form of a*(-.5 + yc), where .5 is the critical level above which the prevalence of contraceptive use in a woman's village or reference group has a positive influence on the adoption of family planning, and a is the `strength' or relevance of this social interaction effect. The standard derivation leads to the probability that a woman uses a modern method of family planning given by

 P(y=1|x, yc) = F(a*(-.5 + yc) + b*x + d), {2}

where d is a constant including the effect of the individual characteristics and F is the cumulative logistic distribution.

The above model is interesting because with slight modifications it can capture two important processes that are frequently used to motivate the relevance of social interactions: social learning and social influence [Montgomery and Casterline 1993]. The former stresses that contraceptive adoption decisions are subject to substantial uncertainty, for example about the medical side effects of modern contraception and/or the benefits of choosing a smaller number of children. Learning about the experiences of other women through social interactions may reduce this uncertainty and thus change the probability that a woman adopts contraception or reduced fertility herself. The second aspect, social influence, emphasizes normative influences on behavior. Social influence captures the fact that preferences regarding modern contraception and/or the number of children are affected by the fertility-related opinions and behaviors that prevail in an individual's social environment. These two aspects can be incorporated in {2} by replacing the term .5 in the social utility term with a more general critical level . Kohler et al. [2001] have shown that the above framework can then be used to empirically and theoretically distinguish between social learning and social influence based on data that include information about the structure of women's social networks. If social influence is most relevant, then we expect to be relatively large since represents a critical level of contraceptive use among her network partners that needs to be exceeded before networks have a positive effect on the adoption of modern contraception. If social learning dominates, then is small because even a small proportion of users in a network can provide useful information about modern contraception that reduces the respondent's uncertainty about this innovation.

In the context of this paper, however, the critical level is only of secondary importance. In particular, we focus here on the strength of social interaction - as reflected in the parameter a - since this aspect is most relevant for the comparison between linear and nonlinear models and for the existence of multiple equilibria. In our analyses of model {2} we therefore assume that the critical level equals 0.5 because this level constitutes in our opinion a plausible specification and most data-sources will not allow the estimation of an alternative value.

The total effect of family planning programs in the presence of social interactions can be characterized in the above nonlinear model, as in the linear case, by equilibria in which an individual's choice probability mirrors the cluster or village average. That is, an equilibrium is a level of contraceptive use that satisfies P(y=1|x,ye ) = ye, or equivalently, an equilibrium is a fixed point at which ye = F(a*(-.5 + ye) + b*x + d). These equilibria are thus at intersections of the "s-shaped" curve F(.) with the diagonal.

The solid line in Figure 2 displays a case in which only one such equilibrium exists. The solid line in Figure 3, on the contrary, shows a case with three intersections. Multiple equilibria are a possibility, though not a necessity, with nonlinear models; they arise when the effect of social interaction (the social utility term in {2}) is sufficiently strong, or alternatively, if the coefficient a is sufficiently large (A necessary but not sufficient condition for the existence of multiple equilibria is that the largest derivative {2} with respect to yc exceeds one, which implies that a needs to be larger than four). The equilibria at low and high levels of contraceptive use are stable for reasons parallel to those discussed with regard to Figure 1. The same reasoning, however, indicates that the center equilibrium always is unstable. A population converges to one of the two stable equilibria depending on whether it is to the left or right of the unstable equilibrium.

The use of the logistic distribution F(.) in the nonlinear model {2} is convenient choice, but it is not essential for the implications of our model. The most important aspect of `nonlinearity,' which is implied by the logistic function F(.) but also many other functional forms, is the convex-concave property, or the s-shape, of the curve in Figure 3. Multiple equilibria can arise if the functional form in model {2} exhibits a s-shape for sufficiently strong social interaction effects, i.e., for a sufficiently large value of a. Analytically less convenient, but otherwise very similar functional forms that can be used instead of the logistic function F(.) are the cumulative normal distribution or the arctan function (i.e., the inverse of the tangent function).

The above nonlinear model points to three relevant aspects of social interactions for program effects.

First, social interactions increase the impact of program effects beyond the direct program effect similarly to the linear model in Figure 1 by changing the location of equilibria. Social interaction leads again to a social multiplier effect that amplifies and enhances the direct effect. If the nonlinear model exhibits only one equilibria, the total effect is measured by the shift in the equilibrium level in Figure 2, similar to our earlier discussion of the linear case in Figure 1. If there are multiple equilibria, as in Figure 3, small and large program changes lead to qualitatively different implications. On one hand, large program changes, which are further discussed below, can displace an equilibrium and lead to transitions between the high and low fertility equilibria. On the other hand, small program changes lead to shifts in the level of each equilibria and to social multiplier effects in populations that are close to the high or low fertility equilibrium. The relation between the direct and the total program effects, which we developed for the single equilibrium case above, therefore applies `locally' to both the high and low fertility equilibria as long as the program changes are small.

This effect of small changes in program efforts on the location of equilibria can be evaluated analytically. Assume that a population is currently at an equilibrium ye = ye(x,a,b,d) which satisfies ye = F(a*(-.5+ ye) + b*x + d). The direct effect of changing the program effort x then equals P(y=1|x,ye)/x = F(.)/x = b*ye*(1-ye), where the equilibrium value ye is held constant when taking the derivative with respect to x. The total-or long term-program effect is measured by the shift in the equilibrium level caused by program efforts and thus equals ye/x. Using the implicit function theorem we obtain ye/x = 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. Because at a stable equilibrium F(.)/yy=ye =a*ye*(1-ye) < 1, the total program effect exceeds the direct effect at a stable equilibrium. In particular, the ratio of the total to direct effect, or the social multiplier, equals 1/(1-a*ye*(1-ye)) and is larger than one.

Therefore, in the nonlinear case as well as in the linear case, the social multiplier effect implies that the total change in contraceptive use exceeds the direct program effect. Ignoring social interaction is therefore likely to lead to overestimates of direct program effects. Similarly, in the linear case the change in the equilibrium level of contraceptive use is maintained only if the change in program effort is permanent. But, in contrast to the linear model, in the nonlinear model the size of the social multiplier depends on the extent of average use in the reference group and the shape of the curves in Figures 2 and 3. If the curve is more sharply upwardly sloped around the initial equilibrium, for example, the social multiplier for a given increase in program effort is greater.

Second, whereas in the linear model there is only one steady state that a population reaches after the effects of a family planning innovation have worked their way through the population, nonlinear models may have multiple equilibria. If there are multiple equilibria, social interactions may reinforce large transitory program efforts and shift a population from a stable equilibrium with a low prevalence of contraceptive use (high level of desired fertility) to a stable equilibrium with high contraceptive use (low desired fertility) in which everybody may be better off (as in Figure 3). In such a case, small permanent changes in program effort may only affect the location of the equilibrium slightly, so that the population remains stuck in a Malthusian low-level equilibrium trap. But large changes in program effort, reinforced by social interactions, can shift the population from the low contraceptive use (i.e., high fertility) equilibrium to the high contraceptive use equilibrium, as illustrated by the shift from the solid to the dashed curve in Figure 3. Transitions between equilibria are often thought to occur at rapid pace, resulting in large changes of contraceptive prevalence within a relatively short time. Perhaps even more important for the financing of family planning programs is that because the high- and low-fertility equilibria are stable, a transitory increase in program effort, if sufficiently large, can yield this sustainable long-term changes in family planning usage if it results in the shift between equilibria. In contrast, if a change in program effort affects only the location of an equilibrium, the program effort needs to be maintained at the new level in order to result in permanent changes of family planning prevalence levels.

In the presence of these multiple equilibria, the onset of a fertility decline can constitute a coordination problem where a critical mass of fertility change can initiate a sustainable transition towards low fertility, and where expectations about future fertility levels are an important element in contemporary fertility decisions. Kohler [2000b] shows that in the context of such a coordination problem, social networks can be an important determinant of the onset and pace of fertility change in addition to program efforts and socioeconomic change. The specific effect depends on the structure and content of the interaction in social networks. In particular, `information networks' that provide information about the fertility intentions of other members of the village or reference groups have little effect on a population's ability to achieve a fertility decline as long as there is a stable high fertility equilibrium that inhibits the adoption of low fertility. However, if the population is out of equilibrium, for example, due to recent program efforts or socioeconomic changes, then information networks increase the speed of a change in fertility behavior that is already taking place. If social interaction is in the form of `coordination networks,' i.e., if the ties among individuals in a group are sufficiently strong to allow collective action among individuals, then social interactions can lead to an earlier onset and a faster pace of a fertility transition.

Social interactions therefore have a twofold implications for the dynamics of fertility change. First, if social effects are sufficiently strong and if the nonlinear model {2} describes a woman's decision to adopt family planning or low fertility, then the presence of social interaction can lead to multiple equilibria with a Malthusian `high fertility trap' and sustainable fertility decline that results from temporary interventions by family planning. Second, social interactions not can not only lead to these multiple equilibria, but also be an important determinant - in addition to program efforts and socioeconomic change - of whether a population can `escape' the high fertility equilibrium and initiate a sustainable transition to the low fertility equilibrium.

Third, the intensification of social interaction may have a negative effect on the size of the social multiplier; thus, intensified social interaction can lead to a reduction in the total change in contraceptive use that results from family planning programs. We define the intensification of social interaction as an increase of the parameter α or a in relations {1} or {2}, which implies a greater relevance of the contraceptive behavior yc of a woman's reference group for her own family planning decision. More intensive social interaction therefore implies that a women's decision to adopt family planning is relatively more influenced by her social environment, and relatively less influenced by her personal characteristics and socioeconomic incentives, than with less intense social interaction.

In the linear model in Figure 1, intensifying social interaction necessarily increases the social multiplier, and `more' social interaction always increases the ratio of total to direct program effects [Note 4]. In the nonlinear logistic model as in Figures 2 and 3, however, increasing the impact of social interaction may increase or reduce the social multiplier [Note 5]. Therefore, the ratio of total to direct changes in family planning use that result from program efforts may become larger or smaller when the intensity of social interaction is increased. More intensive interactions, therefore, can make it more difficult for program interventions to achieve a given amount of total-or long term-change in contraceptive use. We denote the fact that more intensive social interaction, i.e., a higher value of α or a in relations {1} or {2}, decreases the social multiplier effects as status-quo enforcement.

This result points to an important and intuitive, but in the literature on modeling social interactions and fertility not sufficiently emphasized and elaborated point [e.g., Bongaarts and Watkins 1996, Montgomery and Casterline 1993, Montgomery and Casterline 1996]. There are situations in which there are stable equilibria with positive but not universal contraceptive use in which social interactions are status-quo enforcing. For instance, the discomfort of deviating from a high-fertility social norm may increase the more a woman interacts with others in her village who behave according to the high-fertility norm. Increasing the strength of social interactions then tends to increase the disutility of a small family, and it can reduce the total behavioral change that results from program efforts. In such cases, social interactions are status-quo enforcing: the more intensive is social interaction, the more the multiplier effect of family planning programs decreases.

If there is only one equilibrium, then social interaction can be-- but need not be-- status-quo enforcing. On the other hand, if there are multiple equilibria then social interaction is always status-quo enforcing (see note 5 for the formal derivation of this property). Most importantly, therefore, an intensified social interaction at a Malthusian high fertility equilibrium reduces the multiplier effect of increased family planning efforts at this equilibrium and therefore makes it more difficult for program efforts and other socioeconomic changes to increase the level of contraceptive use in a population that is at such a high fertility equilibrium.

It is important to keep in mind that this status-quo enforcement of social interaction is a "local" property that reflects changes in the social multiplier due to small changes in program effort x in the neighborhood of a stable equilibrium ye. It does not affect the implication discussed above that transitory large changes in x can induce a transition from a high-fertility to a low-fertility equilibrium in cases with multiple equilibria. Moreover, status-quo enforcement in our definition depends only on the social multiplier effect, i.e., the ratio of total to direct changes in family planning use caused by program efforts. It is not necessarily associated with a specific level of contraceptive use. Status-quo enforcement thus characterizes situations where more intensive social interaction makes it more difficult for program efforts to change the level of contraceptive use in a population.

In summary, for distinguishing between the direct and indirect effects of small changes in family planning program interventions on contraceptive use, the linear model and the nonlinear model are similar, though the social multiplier in the nonlinear case depends on the nature of the initial equilibrium while in the linear case it is constant. However, only the nonlinear model allows multiple equilibria with a potential 'Malthusian Trap', and only the nonlinear model permits the possibility that an intensified social interaction has negative effects on the social multiplier such that social interaction can be status-quo enforcing.

 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