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4 Learning and Lags in Mortality Perceptions Mark R. Montgomery Each culture has experts, people of unusual acumen or specialized knowledge, who detect covariations and report them to the culture at large. Nisbett and Ross (1980: 1 1 1~. Every demographer can sketch in two curves the broad outlines of the demo- graphic transition. One curve depicts high and then falling mortality; the other shows how fertility, also high at the outset, follows a similar downward course after a lag. Fertility change depends on many factors other than mortality per se, but it is reasonable to believe that, as survival prospects improve, the need for high fertility must be reduced. Although much empirical research supports the view that mortality decline is a causal factor in fertility decline (Schultz, 1981; Wolpin, in this volume), the literature also contains puzzling cases in which considerable mortality decline fails to evoke any response in fertility, as well as examples in which the usual sequence is reversed, with fertility decline preceding the decline in mortality (Matthiessen and McCann, 1978~. In spite of the fundamental role assigned to mortality change, no theory has yet been formulated to explain the lag in fertility responses or to show why long lags are to be expected in some circumstances and rapid response in others. In assembling the elements of such a theory, it is worthwhile to pose to ourselves this question: How, in fact, do people learn that mortality has de- clined? Imagine oneself semiliterate, situated in a rural area of a developing country, and isolated to a great degree from the media and from the formal health-care system. Suppose that even in such an environment, a decline in child mortality is under way. Are the improvements in child survival simply self- evident? Is it a trivial matter to learn the facts? How would one know of mortality decline without having been informed of the facts by health personnel or by those who are better educated? 112

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MARK R. MONTGOMERY 113 These questions came to mind in the course of a series of focus groups that Olukunle Adegbola and I conducted in Lagos, Nigeria, in 1989. Lagos is of course at the opposite end of the spectrum from the situation just described: It is perhaps Africa's largest urban community, saturated by information from radio, television, and newspapers, with a population that contains numerous health personnel and a mixture of the well-educated and the thoroughly uneducated. Without really expecting much from the question, we asked our focus group participants in Lagos whether they thought children were more likely to survive these days than a generation ago. We were somewhat taken aback by the range and variety of their replies. No consensus existed about mortality decline. The better-educated expressed confidence that a decline was under way and pointed to the many medicines and modern treatments that were unknown in their fathers' time. Some of the less-educated, by contrast, exhibited confusion about the direction of change, some arguing that children were more likely to die now than in the past, while others cited improvements in health care. Several participants observed that currently in Lagos, one sees many more funerals than used to be the case; to them, this was evidence of rising mortality. They were clearly thinking of the numerator of a mortality rate, rather than of the rate itself. Evidently, it is not a simple matter to deduce mortality decline from observa- tion alone. It may be that few people think naturally in terms of rates or prob- abilities, and that some training is required to appreciate the distinction between the number of deaths and the death rate. Perhaps educated Nigerians know of mortality decline because they have learned about it in school or because they are more attentive to information presented in the media or by health personnel. In addition, the mortality decline in Nigeria may have been uneven (Hill, 1993), with a clearer downward trend for socioeconomically advantaged groups and a slower change, perhaps too subtle to reach the threshold of perception (McKenzie 1994), among groups that are less well-off. The aim of this chapter is to explore such issues with the aid of formal models of learning and decision making under uncertainty. I draw on a diverse literature that includes studies in economics, demography, and social and cogni- tive psychology. The chapter is frankly speculative. I know of only three empiri- cal studies that have directly measured mortality perceptions in developing coun- tries: Pebley et al. (1979) for Guatemala, Heer and Wu (1978) for Taiwan and urban Morocco, and Cleland et al. (1992) for five Central and West African countries. The Pebley et al. study focused on perceived child survival probabili- ties in four villages in the Institute of Central America and Panama (INCAP) study of Guatemala. Likewise, Heer and Wu (1978) considered child mortality perceptions in two Taiwanese townships and for Morocco they gathered data on perceptions of change. Cleland et al. explored the perceived threat of AIDS to adult survival and behavior. In developed country settings, recent work by Hurd and McGarry (1995) for the United States is concerned with the perceptions of mature and elderly men and women of old age survival prospects. In view of the

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4 LEARNING AND LAGS IN MORTALITY PERCEPTIONS prominent causal role thought to be played by child mortality decline in fertility transitions, it is surprising that the empirical literature on perceptions of survival remains so thin. The chapter is organized as follows. In the first section I consider a model of Bayesian learning about child survival in which individual perceptions are deter- mined by a prior distribution that summarizes initial subjective beliefs about mortality probabilities, these beliefs then being updated by reference to a sample of information on mortality experience, yielding a posterior distribution that summarizes how beliefs change in the light of experience. Bayesian learning supplies a natural benchmark for learning models, in that the approach assumes that individuals process information in an optimal fashion, acting much as statis- ticians do. In view of the difficulties that individuals may face in thinking in probabilistic terms, however, to assume Bayesian learning may well be inappro- priate. An extensive literature in both psychology and economics, which makes use of experimental evidence on judgment and belief updating, provides strik- ingly little support for the Bayesian model. It seems that individual reasoning about uncertainty can and often does depart systematically from Bayesian predic- tions. In the second section some of the major findings from this experimental literature are reported, and I speculate about their implications for mortality perceptions. In the final section I present conclusions and offer suggestions for a demographic research agenda. BAYESIAN LEARNING ABOUT CHILD SURVIVAL The Bayesian approach to learning is perhaps viewed most usefully as an optimal benchmark, providing a standard against which less precisely formulated models can be judged. I focus on a type of learning that might be termed social learning (Montgomery and Casterline, 1996), whereby individuals gather infor- mation about child survival prospects by observing the experiences of their peers. Each family is assumed to be linked by way of its social networks to a sample of other families that provides N observations on child survival in each year. The true child survival probability will change from one year to the next, following an upward trend, and each year the family' s beliefs about survival will be updated in a Bayesian fashion, taking into account that year's sample of size N. To draw out the implications of survival probabilities for fertility, I rely on a simple utility maximization model (details are reported in the Appendix) that is an idealized representation of fertility decision making. Parents are assumed to choose the number of births they bear, denoted by B. so as to maximize an expected utility function U(S), where S < B is the number of surviving children. I explore a specification in which U is a quadratic function of S. that is, U(S) = aS- bS2,

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MARK R. MONTGOMERY 115 with parameters a, b > 0. In this specification there is an ideal numbers of surviving children S* that corresponds to the maximum of U(S). Because child survival is uncertain, parents are not assured of having S* children and will adjust their fertility B to protect themselves against the possibility of child loss. The number of surviving children S is assumed to be binomially distributed given the number of births B and the child survival probability O. In developing the ex- amples below, I take ~ to be the probability of surviving to age 5, that is, as the complement of the child mortality probability.2 Known Survival Probabilities To begin, suppose that the survival probability ~ is known with certainty. Figures 4-1 and 4-2 illustrate the optimal fertility distributions for a range of values of the survival probability O. In developing these figures, I assumed that the desired number of surviving children is fours and that ~ lies in the range that has been characteristic of the postwar experience in West Africa. To motivate the example, I take from Hill (1993:Table 5.2) a time series of child mortality esti- mates for Ghana, a West African country whose mortality decline is well docu- mented. In the immediate postwar period (circa 1948) the probability of child mortality was about 0.301 in Ghana. A period of mortality decline then set in, bringing child mortality to a level of 0.199 by 1967. In the 1970s and 1980s little further improvement took place, and by 1985 Ghanaian child mortality had fallen only to 0.163. Figure 4-1 presents illustrative fertility estimates derived from values that range from 0.699, a number that corresponds to the immediate post- war years, to a maximum of 0.95, the latter being some 10 points above what Ghana had achieved as of the mid-1980s. As can be seen in Figure 4-1, improvement in child survival brings about a reduction in mean desired fertility. The mean value of fertility declines from 5.5 to 4.2 over the range for 0, a difference of slightly more than one child. The net reproduction rate, however, remains essentially constant, this being a common 1 For simplicity, I choose parameter values to ensure that there is only one maximum of u(s), associated with an integer value for s. Alternatively, one could consider survival to adulthood rather than to age 5. In most developing country settings in which mortality rates are very low between age 5 and adulthood, there would be little empirical difference between the two survival probabilities. Where perceptions are concerned, however, the death an older child or young adult might exert a powerful influence out of proportion to its empirical likelihood. 3The optimal number of surviving children is produced by making assumptions on the ratio of utility parameters a and b such that 4 = al(2b) or a = 8b. we can then consider different values for b. The greater the value for b, the sharper the fall off in utility as the number of surviving children deviates from four.

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6 5.6 5.4 5.2 t .~ . ~ 4.6 4.8 4.4 4.2 4.0 LEARNING AND LAGS IN MORTALITY PERCEPTIONS \ 0.70 0.75 0.80 0.85 O.gO 0.95 Child Survival Probability FIGURE 4-1 Mean fertility by child survival probability. 0.7 0.6 0.5 .o 0.4 o 2 0.3 0.2 0.1 \ ~ I .g , - l . . . .8 I / -' / ~ 2. ; J . \ ~ \ .7 _ ~ ~1 ~-~-e ~,1 1 0.0 0 1 2 3 4 5 6 7 8 9 10 Parity FIGURE 4-2 Distributions of fertility given different survival values.

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MARK R. MONTGOMERY 117 feature of models that take births themselves to be costless (see the Appendix). The effects of child survival are perhaps clearer in Figure 4-2, which shows the distribution of desired fertility for several ~ values. Uncertain Survival Probabilities Having sketched the main features of the case in which ~ is known, I now ask how fertility decisions might differ if the survival probability ~ is an unknown quantity about which individuals have more or less certain prior beliefs. Two questions must be addressed. First, how does one represent such prior beliefs? Second, how does experience cause these beliefs to be revised? As they enter adulthood, most women and men in developing countries will have had some exposure, whether direct or indirect, to the risks of child mortality. Many will have experienced the death of a sibling, an event that must leave a deep and vivid impression, and the experiences of relatives might figure in as well.4 Those who have attended school would have acquired some general information about health and survival, which could also shape initial perceptions of risk. But for many young adults, childbearing and the risks of child death are matters that belong to a stage of life that they have not yet entered and about which they know little. We would therefore expect perceptions of the survival probability ~ to be hazy and subject to considerable uncertainty. Imagine two young Ghanaian adults who begin their family building in 1948. In each year, they will have access, through social networks of relatives, friends, and peers, to a sample of N observations on child survival. Each year the family assesses the information contained in this sample and revises its subjective beliefs about ~ accordingly. The size N of this sample will depend on the extent of social networks in general, the level of fertility in the population, and the degree of privacy that shrouds matters of birth and death. For example, it may be that the educated have access to larger social networks than do the less educated and so possess a larger sample of information. For simplicity, however, I take N to be a constant and set N= 10 for purposes of illustration. Returning to the household's fertility choice, the problem is to choose the number of births B that maximizes E U(S) = E (aS- bit. As noted in the Appendix, if the value of the survival probability ~ were known, this problem could be restated as 4As John Casterline (personal communication, 1996) notes, this supposition forms the basis of the sisterhood method for estimating adult mortality.

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118 LEARNING AND LAGS IN MORTALITY PERCEPTIONS max B(a - b) ~ - bB(B - 1)02. With ~ unknown, however, the values of ~ and 02 must be replaced in the expres- sion by their subjective expected values, E(~) and E(02), which in turn depend on prior beliefs and the available sample of child survival observations. Illustrative Results 11- we take the (]hanaian experience as a guide, Figure 4-3 shows the se- quence over time of the actual survival probability as compared with the evolu- tion of subjective beliefs about survivorship. Figure 4-3 shows the subjective mean E(~) and the 25th and 75th percentiles of the subjective distnbution. It illustrates the evolution of perceptions for a family beginning in 1948 with a prior distribution whose mean is identical to the true survival probability (0.699), but with considerable uncertainty about 0, as evident in the 25th to 75th percentile range that stretches from 0.57 to 0.86. In 1948 and in each year thereafter, the family updates its beliefs on the basis of ten new observations, each such sample being drawn from a binomial distribution characterized by that year's (true) survival probability. The learning shown here is therefore cumulative; that is, beliefs in 1980 are influenced to some degree by the initial prior distribution in 1948 as well as by the intervening sequence of social network samples. As can be seen in the figure, although the uncertainty evident in the initial ~ .0 . 0~9 ._ ~ 0.8 - .> 0.7 A} 0.6 0.5 true mean - - subjective mean 1950 1960 - 1970 1980 1990 Calendar Year FIGURE 4-3 Actual trend versus subjective distribution.

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MARK R. MONTGOMERY 119 prior distribution is quickly reduced, and even though the initial subjective mean and the true mean were selected to exactly coincide, the family's perception of child survival fails to keep pace with the empirical realities. There is an upward trend in the subjective mean, to be sure, but the year-to-year gain is smaller than the true improvement in survivorship. By the early 1960s even the 75th percen- tile of the subjective ~ distribution has fallen below the true value for 0, and by the end of the period the gap between the subjective mean and the true ~ value has grown to some eight percentage points. One might ask whether larger social networks, which contain more annual observations on children's deaths and survival, would bring the subjective mean into line with the true survival probability. A change in the sample size N does have some effect, but the gap shown in Figure 4-3 persists even if N is doubled to 20 annual observations. The persistent lag in expectations, as compared with the true level of child survival, is the product of two factors. The first is that the true values for ~ follow a trend, whereas in the simple Bayesian model outlined above, beliefs are up- dated without there being a recognition that a trend in survivorship exists. If a more complex Bayesian model were employed (see Zellner and Rossi, 1984; Koop and Poirier, 1993; Poirier, 1994) in which both level and trend components for ~ were updated from sample experience, the subjective values would adhere more closely to the actual trend. Such Bayesian trend models require greater sophistication on the part of decision makers, who must now make mental com- parisons of their social network samples over time. The second factor is that even with relatively small social network samples (recall that N = 10), only a few rounds of experience are required to substantially reduce subjective uncertainty about O. As the distribution of beliefs becomes ever more concentrated about the subjective mean, upward revision of that mean is made ever more grudgingly. With time and experience, individuals become more and more set in their beliefs and begin to resist new evidence of improve- ments in survival. Implications for Fertility How much difference does learning and uncertainty make to fertility deci- sions? One way to assess the net effect of learning is to compare the mean level of desired fertility with the survival probability ~ set equal to its true end-of- period value (see Figure 4-3) to the mean fertility level produced by the subjec- tive distribution of ~ for that year. The difference in fertility is on the order of 0.6 children. In other words, an objectively correct evaluation of survival prospects would be predicted to lead to 0.6 fewer desired births than in the case in which subjective mortality impressions are employed. The projected difference of 0.6 desired births is in no way an empirical finding, being simply the outcome of a particular model and set of assumptions.

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120 LEARNING AND LAGS IN MORTALITY PERCEPTIONS The model itself could be elaborated in any number of dimensions, for example by incorporating both trend and level components in learning and by adding a dynamic structure that would permit child replacement effects to be analyzed. These are worthwhile extensions; they will inevitably lead to a more sophisti- cated Bayesian model. Before we undertake to construct such an elaborate model, we should assess the experimental evidence on learning to determine whether any support exists for the type of optimal learning being considered here. DEPARTURES FROM BAYESIAN LEARNING Camerer (1995) has thoroughly reviewed the literature in cognitive psychol- ogy, social psychology, and economics and found evidence that indicates a de- parture from Bayesian learning in two principal areas: Individuals employ learn- ing strategies that are cognitively much simpler than the Bayesian rules; and in some circumstances, individuals consistently make fundamental mistakes in probabilistic reasoning, tending to assign too much weight to certain types of evidence and not enough to others. Nonexperimental economists have been inclined to dismiss such findings, arguing that, as individuals gather experience and are repeatedly subjected to the discipline of the market or other social institutions, their judgments will come to resemble optimal Bayesian judgments. Such conditions are unlikely to obtain in respect to mortality perceptions and fertility decision making. In developing transitional societies, individuals are often called upon to make decisions in novel situations and in environments that are in flux, where the old rules no longer quite apply but new rules and institutions have not yet evolved to take their place. Mistakes and misperceptions are to be expected as individuals traverse such new terrain. To begin, it should come as no surprise that individuals prefer to use simple rules of thumb in making their judgments and updating beliefs, rather than to engage in complex Bayesian calculations. Such common decision rules usually involve a simple averaging of prior beliefs and new sample evidence (Lopes, 1985, 1987; McKenzie, 1994~. This need not in itself put the Bayesian perspec- tive in doubt, as one could argue that individuals often behave as if they were Bayesians. In the case of updating in mortality perceptions, the main features of Figure 4-3 above would probably not change greatly if individuals simply split the difference between their prior beliefs and the latest sample evidence. However, there is considerable doubt that individuals consistently behave as if they were Bayesians. As Camerer (1995) and Conlisk (1996) demonstrate, when economic experiments are designed to reward optimal decisions and punish mistakes in judgment, as a market would, these incentives do not consistently lead participants to adopt Bayesian decision rules. A new area of research in economics, exemplified by El-Gamal and Grether (1995) and Cox et al. (1995), attempts to discover by statistical means the types of learning rules participants

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MARK R. MONTGOMERY 121 employ in experimental games. Bayesian decision making emerges as one in the set of rules employed, but other rules are equally prominent and substantial heterogeneity in learning evidently exists. Although this literature casts doubt on Bayesian learning, it has not yet succeeded in identifying a compelling and domi- nant alternative. Rather, it seems that the type of non-Bayesian reasoning that individuals employ is situation-specific, depending in a complicated fashion on the nature of the problem. Where mortality perceptions are concerned, the experimental literature sug- gests that, on the whole, perceptions may be even more resistant to change than was illustrated in the simple Bayesian model above. Not every aspect of the literature supports such a conclusion, but the bulk of the evidence does. It may be useful to preface the following discussion by reference to recent findings on the neurological basis of human learning and memory. The experi- ments of Knowlton et al. (1996), and the insightful comments on them by Robbins (1996), provide evidence that memory and learning in tasks related to probability classification involve a different section of the brain than do other tasks involving recognition and recall. (Their probabilistic experiments had to do with a weather forecasting game in which an arrangement of four cards imperfectly predicted the weather.) Moreover, as Knowlton et al. (1996:1400) observes, "the probabilistic structure of the task appears to defeat the normal tendency to try to memorize a solution, and individuals can learn without being aware of the information they have acquired." Because child survival and death are likewise imperfectly pre- dictable chance events, one wonders whether humans process information about them in ways that are fundamentally distinct from other types of learning. Might individuals over time come to appreciate trends and covariations in child survival without ever being able to say quite why they know what they do? The anomalies in probabilistic learning to be discussed below may well have common neuro- logical roots. Null Events Where mortality is concerned, the definition of an event is itself worth con- sidering. A statistician would note both the occurrence of a child death and the occurrence of a child survival; both are "events." But for the lay person, only the former may be noteworthy. There is no news in a child's survival; nothing has happened; it is a "null event." It may therefore be difficult for the lay person to give recollections of child survival their fully appropriate subjective weight. The experimental literature (Estes, 1976) contains numerous examples in which sub- jects demonstrate their difficulty in retaining information about such null events. If an event X is always described in an experiment as being "not Y." this trivial semantic difference appears to have real implications for recall.

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22 LEARNING AND LAGS IN MORTALITY PERCEPTIONS Use of Base Rates It is often observed in experimental contexts that when presented with a statistical portrait of a population, termed by psychologists a "base rate," and a sample of evidence from that population, individuals tend to give too much emphasis to the sample evidence and not enough to the base rate. For example, when told that these days, 90 percent of children survive to adulthood, and when given a small sample in which only six children in a group of ten survived, individuals might tend to give the sample evidence too much weight in their thinking (Grether,1980~. When asked to compare two situations having different base rates, such as two different mortality levels, individuals will often ignore these base rates entirely in favor of sample evidence, unless their attention is specifically directed to the difference in the base rates (Bar-Hillel, 1980; Bar- Hillel and Fischhoff, 1981; Argote et al., 1986~. It seems that the base rates are somehow viewed as rather abstract and pallid by comparison with the vividness and individuality of sample data (Bar-Hillel, 1980~. Perhaps for a developing country villager, general media presentations on the likelihood of child survival, or similar presentations by health personnel, lack the immediacy and persuasive power inherent in a tiny sample of recent village experience. One source of data is "merely statistical," whereas the other commands attention and demands inter- pretation. As Nisbett and Ross (1980:57) remind us, this bias in favor of sample evi- dence is reinforced by many folk sayings. Statements such as "Seeing is believ- ing," or "You can prove anything with statistics," which stress the reliability of one's own experience, are not counterbalanced by other maxims that warn against the danger of inferring too much from a small sample. The experimental evi- dence (see Nisbett and Ross, 1980:56-60; Tversky and Kahneman, 1974) shows that people prefer to rely on sample evidence even when they are made aware that such evidence has very little diagnostic content. Strength Versus Weight Griffin and Tversky (1992) argue that in making judgments about the likeli- hood of an event, people focus first on the apparent "strength" of an effect (such as the size of a sample mean) and then make an adjustment, but typically an insufficient adjustment, for the "weight" of the sample evidence (the sample size or the standard error of the sample mean). To put this differently, ". . . people are highly sensitive to variations in the extremeness of evidence and not sufficiently sensitive to variations in its credence or predictive validity. . ." (Griffin and Tversky,1992:413~. This bias might cause certain unusual events, such as a case in which a single family in a rural village lost all of its children, to acquire a disproportionate influence in mortality perceptions.

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MARK R. MONTGOMERY 127 have been described, this does not necessarily occur. Rather, groups are often more extreme in their judgments than individuals, sometimes (although not al- ways) amplifying individual errors into an erroneous group consensus (Argote et al., 1986, 1990~. Thus, groups as well as individuals may violate Bayes' rule. Cultural Differences A small literature explores cultural differences in how uncertainty itself is viewed and how degrees of uncertainty are expressed. This literature centers on the concept of probabilistic thinking, which Wright et al. (1978:285) define as "the tendency to adopt a probabilistic set, discrimination of uncertainty, and the ability to express the uncertainty meaningfully as a numerical probability." Re- views and experiments regarding probabilistic thinking are presented in Wright et al. (1978) and Kleinhesselink and Rosa (1991~. The major conclusions appear to be as follows. In a study of factual knowledge about geography and other matters that included students from Hong Kong, Malaysia, Indonesia, and Great Britain, Wright et al. (1978) found that, as seems to be typical (Fischhoff et al., 1977), all students tended to be overconfident about their judgments in relation to true performance (that is, to be wrong too often when confident they are right). Given this, the Asian students tended to express their evaluations and degrees of confidence in absolute terms (e.g., O percent or 100 percent confidence in a judgment) more often than did the British students, who adopted gradations in language and used nuance to describe their confidence levels. These tendencies apparently could not be attributed to different levels of substantive knowledge (for similar findings see Wright and Phillips, 1980~. It may be, therefore, that the Asian students are less prone to think in proba- bilistic terms. An alternative explanation is that, however the Asian students may think, they are less likely to express themselves in terms of degrees of certainty. This literature hints that cultural differences may exist, but leaves uncertain their extent and depth. Summary If the literature described above can be taken as a guide, I would argue that on the whole, improvements in child mortality are likely to be perceived with a greater lag than suggested by optimal Bayesian calculations. There are clearly many subtleties here, and it may be that recency effects and different types of adaptive learning could reverse this conclusion. Nevertheless, it seems far from clear that when mortality improves, individuals will possess the cognitive equip- ment to correctly and rapidly perceive the trend.

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28 LEARNING AND LAGS IN MORTALITY PERCEPTIONS CONCLUSIONS My aim in this chapter has been to point to a gap in the theory of the demographic transition related to the factors that might induce lags or misper- ceptions of mortality change. If the wide-ranging literature cited above offers any basis for generalization, one might extract from it a few general themes. Social Learning and the Demographic Transition Economic development is associated with sweeping change in any number of social and economic arenas. Some of these changes require little effort to observe and interpret. Others, however, are difficult both to perceive and to understand. Among these society-wide changes, mortality is perhaps especially difficult in that some probabilistic thinking is required even to judge the direction of change. I have argued that individuals, if left to their own perceptual devices, are unlikely to infer that mortality prospects have improved; more precisely, they are unlikely to do so rapidly and without some interim period characterized by error, uncertainty, and debate. Social learning may also be important in other arenas of behavior, particu- larly in cases in which individuals are considering the adoption of innovations. Elsewhere (Montgomery and Casterline, 1996) it has been argued that social learning is a fundamental element in the diffusion perspective on fertility transi- tion. Diffusion models lay emphasis on factors affecting the demand for chil- dren, such as mortality, as well as those that affect the costs of fertility regulation. Some of the arguments made above with regard to mortality perceptions might also apply to learning about the properties of modern contraceptive methods or the risks and expected returns to be derived from educating children. The role played by exogenous variability and the mediating functions of social institutions also deserve consideration. High-mortality environments are likely to be environments in which mortality is highly variable. High variance, in turn, must surely add to the difficulties that individuals face in learning about the central tendencies of their environments, that is, in extracting signal from noise. Stable social institutions can act as buffers against extreme risks, or may play an insurancelike function in spreading otherwise local risks over wider populations. When institutions play such roles, they act to dampen variance and may thereby facilitate individual learning. Education and the Modern Health Sector It- mortality misperceptions are likely, it might now be asked how they come to be corrected. The argument developed above suggests a reconsideration of the special contribution of education and the provision of information. We do not yet

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MARK R. MONTGOMERY 129 understand how mortality perceptions are formed and whether they are influ- enced principally by direct experience, observed experience, or by the acquisition of learning in school, from the media, or from the health sector itself. Diffusion of information about mortality improvements from those who were taught the facts to those who might otherwise not have perceived them may be the funda- mental source of changes in perceptions. The roles that are played by national and international organizations charged with health interventions must surely be important. It is these agencies that communicate information about mortality and health change to the populations they serve. They can do so directly through presentations in the media and the discussions of health personnel. One wonders, in light of the discussion above, what factors determine the reception of these media messages. Are they typically judged persuasive, or abstract and irrelevant? The formal health organizations also communicate information indirectly by insertion of material related to health in national school curricula. LeVine et al. (1994) have argued that schooling itself equips individuals with the cognitive skills they need to translate the "decontextualized language" of the formal health sector into terms that are mean- ingful to individual experience and decisions. Perhaps it is in the schoolroom that individuals learn to be attentive to infor- mation provided by government and the formal health sector. Of course, few students will emerge from school with an understanding of the germ theory of disease and the workings of modern medicines. Many will come to know that such knowledge exists and that it rests in the hands of modern health care person- nel. They will also have been exposed to the knowledge that mortality is control- lable, at least to a degree, and this in itself will tend to heighten attention to information about health (Simons, 1989~. The Western experience in such matters may be instructive. As Preston and Haines (1991) argue, citing Dye and Smith (1986), the nineteenth century expe- rience in the United States was one of a gradually increased emphasis on the mother's role in protecting her children's health. There was an increasing faith in the controllability of mortality, although few effective means existed for preven- tion and even fewer for cure. Until the early twentieth century, the rising belief in controllability did little but increase anxiety and promote a sometimes frantic search for cures; when effective medicines finally emerged, however, the net effect was to improve child survivorship. Interestingly, in this era it was within the medical and public health professions that diffusion of information and the combat between traditional and modern health beliefs were important. Not until the second decade of the twentieth century did physicians fully relinquish nine- teenth century beliefs in bodily imbalances, innate racial constitutions, and vari- ous miasmas as explanations for disease.

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130 LEARNING AND LAGS IN MORTALITY PERCEPTIONS A Research Agenda To know whether lags and learning are important in today's developing countries, there is no substitute for basic research aimed at eliciting perceptions of mortality levels, differentials, and change. This is a challenging area for research in that the populations whose perceptions are to be studied are not always fully literate and perhaps have a different vocabulary and understanding of uncertainty. In highly literate populations, several methods have been used successfully to elicit subjective probabilities (see Hurd and McGarry, 1995; Dominitz and Manski, 1994a,b; and Morgan and Henrion, 1990~. The early study of Pebley et al. (1979) tested simple probability scales in Guatemala, evi- dently with good results. These methods might be adapted profitably for use in developing country settings in which mortality and fertility transitions are now under way. APPENDIX In the simple Bayesian fertility decision model, parents are assumed to choose the number of their births B to maximize the expected value of the utility function U(S), where U(S) = as- bS2. The U(S) function is symmetric, a specification that imposes equal penalties for falling short of the target S* and exceeding it. This form of the utility function has been chosen principally for analytic convenience, and although I do not do so in what follows, other nonsymmetric specifications might also be considered. Note that the number of births B does not enter U(S) directly. The implicit assumption is that the costs of children that is, the factors that cause U(S) to slope downward beyond S* are largely the costs associated with surviving chil- dren. In high-mortality settings, where the probability of death after infancy is high, such an assumption is not entirely appropriate. An additional term involv- ing B could be added to the specification above, yielding a more general expected utility function U(S,B). This addition would certainly affect the fertility predic- tions derived from the model, but would not directly alter mortality perceptions and the process of learning. To simplify matters, therefore, I have not included B directly. In keeping with Sah (1991), the number of surviving children S is assumed to be binomially distributed given the number of births B and the child survival probability O. That is, P(S = s I B. B) = ( )0s (1 _ o)B-s The parameter ~ represents the probability of surviving to age 5.

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MARK R. MONTGOMERY 13 Known Survival Probabilities To begin, suppose that the survival probability ~ is known with certainty. In this case, parents confront the following maximization problem: max s=0 U(s) s (1 _ JIB s B (s) Although no analytic solution for B is available, the utility-maximizing B is easily found by numerical means for any given O. Indeed, from the binomial assumption and the simple form adopted for U(S), the problem reduces to finding the number of births B that maximizes E fU(s) I B,0] =B(a- big- bB(B- 1~02. Without additional structure, the steps taken to this point would lead to a single utility-maximizing value B* for fertility. It is perhaps more reasonable to allow for some heterogeneity in the population and to think of B* as the most likely value for fertility. Heterogeneity is introduced by attaching "disturbance terms" B specific to each level of fertility, giving E f U(s) I B. E, B] = B(a- big - bB(B- 1~02 + B . Provided that the disturbance terms B are mutually independent, extreme-value random variables, we are led to a distribution of the levels of optimal fertility of the form exp~B(`a - buff - bB(`B _ 1~02] Pr(`B I f9) = ~j=oeXptita - buff - bj(` j _ 1~2] The above is recognizable as the probability derived from a conditional-logit choice problem. Subjective Beliefs It is analytically convenient to summarize prior beliefs regarding ~ by means of the beta distribution, whose density is f(~3 1 0C,'(~) = 1 `3a-~`l_~3~-l where Be~a,b) is the beta function. This distribution is reasonably flexible and allows for a variety of representations of uncertainty (Lee, 1989; Pratt et al., 1995~. The mean and variance of the beta distribution are

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32 LEARNING AND LAGS IN MORTALITY PERCEPTIONS E(~) = var (a) = oc oc+D' aD (~+~)2(~+~+1) E(~) [1 - E(~) ] oc+~+1 When or = 13 = l, the distnbution of ~ is uniform over the interval from 0 to l; the case or = 13 = 2 yields a symmetric distnbution for ~ about the value ~ = 0.5; and with or > 13, or > 2 a skewed unimodal distnbution emerges. I illustrate this last case in Figure 4A-l, for which or and 13 have been chosen so that the subjective mean of the survival probability ~ = 0.699, a value equal to Hill's initial esti- mated probability of child survival in postwar Ghana. Social Learning When beta-distnbuted prior beliefs about ~ are updated by reference to a sample of N external observations on child survival, this sample being generated by the binomial distnbution, the posterior distnbution for ~ is also a beta. In other words, the beta distnbution is the conjugate prior for the binomial. We can see this as follows. Let pokey represent the prior distnbution for ~ and pap) represent the updated or posterior distnbution. We then have 2.2 2.0: 1.8 1.6 . ~ ].4 V, 1.2 0 a 1.0 0.8 0.6 0.4 0.2 l 0,0 ~-~ 1 1_ 1 1 1 1 0.0 0.1 0.2 0.3 0.4 FIGURE 4A-1 The beta density (mean = 0.699~. O.S 0.6 0.7 0.8 1 , 1 ' I ~1 1 0.9 1.0 Theta

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MARK R. MONTGOMERY ) (I )Os (1 (3)N-s Bc~c`,0'(3 (1_(3)~ 1( )~35(1_(3)N-s = 1 (3a+5_1 (1 _ 0)~+N-s-l Be(oc+s,~+N-s) so that the posterior distribution is a beta with parameters oh = or + s, p~ = ~ + N - s. 133 Although flexible, the beta distribution is not an ideal choice for an applica- tion to child survival in that the distribution always assigns some subjective probability to values of ~ near 1 and 0. This gives a potentially unrealistic representation of subjective beliefs, as few individuals believe either that all children will survive or that none will. However, by selecting values for or and ~ with some care, one can confine most of the subjective distribution for ~ to a reasonable range. Note that the principal advantage of the beta distribution is that, when it is combined with the binomial, the posterior distribution for ~ has the same form as the prior distribution, both being betas. This is analytically convenient. By departing from the beta distribution one could represent subjective perceptions in a more reasonable way, perhaps by confining the perceived ~ to a range, but this advantage would be offset by greater difficulty entailed in analytic comparisons when the prior and posterior differ in form. Desired Fertility with ~ Uncertain Returning to the fertility choice facing households, the problem is to choose the number of births B that maximizes E U(S) = E (aS- by. As noted above, if the value of the survival probability ~ were known, this problem could be restated as max B(a - b) ~ - bB(B - 1~02 With ~ unknown, however, the values of ~ and 02 must be replaced in the expres- sion by their subjective expectations, E(~) and E(02), which in turn depend on prior beliefs and the available sample of child survival observations. The utility maximization problem can be stated as

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134 where E(~) = oc/(oc + p). LEARNING AND LAGS IN MORTALITY PERCEPTIONS max B(a - b) E(~) - bB(B - 1)E(~):E(~) + ~3( ~ ], ACKNOWLEDGMENTS This research was supported in part by a grant from the Rockefeller Founda- tion. I thank John Casterline, Barney Cohen, and two anonymous referees for their insightful comments. REFERENCES Argote, L., R. Devadas, and N. Melone 1990 The base-rate fallacy: Contrasting processes and outcomes of group and individual judg- ment. Organizational Behavior and Human Decision Processes 46:296-310. Argote, L., M. Seabright, and L. Dyer 1986 Individual versus group use of base-rate and individuating information. Organizational Behavior and Human Decision Processes 38:65-75. Bar-Hillel, M. 1980 The base-rate fallacy in probability judgments. Acta Psychologica 44:211-233. Bar-Hillel, M., and B. Fischhoff 1981 When do base rates affect predictions? Journal of Personality and Social Psychology 41(4):671-680. Caldwell, J., P. Reddy, and P. Caldwell 1983 The social component of mortality decline: An investigation in South India using alterna- tive methodologies. Population Studies 37(2):185-205. Camerer, C. 1995 Individual decision making. In J. Kagel and A. Roth, eds., The Handbook of Experimen- tal Economics. Princeton, N.J.: Princeton University Press. Cleland, J., M. Carael, J.-C. Deheneffe, and B. Ferry 1992 Sexual behavior in the face of risk: Preliminary results from first AIDS-related surveys. Health Transition Review 2(Suppl.):185-204. Conlisk, J. 1996 Why bounded rationality? Journal of Economic Literature 34:669-700. Cox, J., J. Shachat, and M. Walker 1995 An Experimental Test of Bayesian vs. Adaptive Learning in Normal Form Games. Un- published manuscript, Department of Economics, University of Arizona, Tucson. Dominitz, J., and C. Manski 1994a Eliciting Student Expectations of the Returns to Schooling. Unpublished manuscript, Institute for Social Research, University of Michigan, Ann Arbor. 1994b Using Expectations Data to Study Subjective Income Expectations. Unpublished manu- script, Institute for Social Research, University of Michigan, Ann Arbor. Dye, N., and D. Smith 1986 Mother love and infant death, 1750-1920. Journal of American History 73:329-353.

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