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## 1. Classical Stable Population Theory

A population is in ‘demographic equilibrium’ if the fraction of individuals in each age-sex category remains constant over time and the population is growing (or declining) at a constant rate. Demographers call a population that is in demographic equilibrium a ‘stable population.’ This use of the phrase stable population blurs the distinction between questions involving the existence of a demographic equilibrium and those involving its dynamic stability; hence, I depart from standard demographic terminology and in its place use the phrase ‘demographic equilibrium.’ In deference to traditional usage, however, I refer to demography’s standard model as ‘classical stable population theory.’

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Nondemographers are often surprised to learn that the classical stable population theory operates with a one-sex model. The theory postulates an age-speciﬁc fertility schedule for women and an age-speciﬁc mortality schedule for women. These two schedules, in conjunction with the very strong assumptions that both of these schedules remain constant over time, imply an equilibrium growth rate and an equilibrium age distribution for the female population.

Demography’s two-sex problem is the problem of integrating men into this one-sex, female-based model. The two-sex problem is often motivated by discussing the eﬀect on fertility of a ‘marriage squeeze.’ A marriage squeeze is an imbalance between the female and male populations of marriageable age brought about, for example, by wars that disproportionately reduce the number of marriageable men. A marriage squeeze can also result from a baby boom: because women typically marry older men, a baby boom increases the number of marriageable women many years before it increases the number of marriageable men. Common sense suggests that a marriage squeeze is likely to reduce age-speciﬁc fertility rates for women, but the classical stable population theory, because it assumes that age-speciﬁc fertility rates remain constant, cannot accommodate this common sense conclusion.

## 2. The Birth Matrix-Mating Rule Model

Demography’s two-sex problem is solved, in a formal sense, by the ‘birth matrix-mating rule’ (BMMR) model introduced in Pollak (1986). (Feeney (1972), in an unpublished doctoral dissertation, presents a two-sex model with the same basic structure, but without a satisfactory proof of the existence of equilibrium.) The BMMR model has three primitives (i.e., components not derived from more fundamental components):

(a) a birth matrix, which indicates the number of oﬀspring born to each possible type of union (e.g., the union of a female of age i and a male of age j);

(b) a mating rule, which indicates the number of unions of each type as a function of the number of females in each age category and the number of males in each age category; and

(c) age-speciﬁc mortality schedules for males and females.

Under relatively weak assumptions, the BMMR model determines equilibrium growth rates and equilibrium age distributions for the female and male populations. It also determines who marries whom or, more precisely, the age pattern of assortative mating. The BMMR model assumes that all fertility is marital fertility and all unions are monogamous, although both of these assumptions are easy to relax. Because the BMMR model allows changes in the age structure of the population (e.g., shortages of marriageable men relative to marriageable women) to change the number of unions, it accommodates our common sense conclusion about the eﬀect of a marriage squeeze. Throughout a marriage squeeze, the birth matrix (i.e., the number of births to each type of union) remains constant, but the number of unions changes, and these changes in the number of unions imply changes in the number of births and, hence, changes in the implied age-speciﬁc female fertility rates.

### 2.1 Linear And Nonlinear Models

Because the classical stable population theory utilizes a linear model, the mathematics is straightforward. The age-speciﬁc fertility and mortality schedules for the female population together deﬁne a mapping of the female population vector (by age) in period t into the female population vector in t + 1. Because the implied mapping is linear, demographers use the simple and powerful tools of linear algebra to investigate the existence, uniqueness, and dynamic stability of the equilibrium. The matrix corresponding to the mapping of the population vector in period t into the population vector in period t + 1 is called the Leslie matrix. The equilibrium age distribution is an eigenvector of the Leslie matrix; the equilibrium growth rate is obtained by subtracting 1 from the corresponding eigenvalue. In linear demographic models, cases in which an equilibrium fails to exist, cases of multiple equilibria, and cases in which the dynamics do not exhibit global stability are generally regarded as pathological.

The BMMR model, like the classical stable population theory, deﬁnes a mapping of the population vector in period t into the population vector in period t +1. The initial population vector, together with the mating rule, determines the number of unions of each type; the number of unions of each type, together with the birth matrix, determines the number of newborns in the next period. The initial population vector, together with the mortality schedules, determines the number of individuals in each of the other age-sex categories, just as in the classical stable population theory.

Nonlinearity is a necessary feature of any two-sex demographic model. The two-sex problem arises precisely because the classical stable population theory fails to recognize that the eﬀect on population in period t +1 of adding a female of age i to the population in period t depends on the number of males. In the BMMR model, this dependence is captured by the mating rule. Any mating rule for monogamous unions must satisfy two accounting requirements: a nonnegativity condition ensuring that the number of unions of each type is positive or zero, and an adding-up or balance condition ensuring that the number of mated females of each age does not exceed the total number of females of that age, with a similar requirement holding for males. This adding-up requirement is the fundamental source of nonlinearity in the BMMR model.

Because the BMMR model is inherently nonlinear, establishing the existence of equilibrium and analyzing dynamic behavior in the BMMR model is more complex than in the classical stable population theory. A ‘natural’ but unsuccessful strategy for establishing the existence of equilibrium is to apply a ﬁxed-point theorem to the mapping of the population vector in period t into a suitably normalized population vector in period t 1; a ﬁxed point of such a mapping—a population vector that maps into itself—is an equilibrium of the BMMR model. The diﬃculty with this strategy is that the mapping deﬁned by the BMMR model carries some population vectors into the 0 vector and, hence, this strategy fails to establish the existence of a nontrivial equilibrium. An alternative proof strategy, followed in Pollak (1986), avoids this diﬃculty by drastically limiting the domain of the mapping and reducing the problem to a single dimension.

### 2.2 Uniqueness And Dynamic Stability

Multiple nontrivial equilibria are not pathological in nonlinear models, and Pollak (1990a) demonstrates by example that the BMMR model can have multiple nontrivial equilibria. An overly strong apparently innocuous suﬃcient condition for the BMMR model to have a unique nontrivial equilibrium is that at every population vector an increase in the number of individuals in each age-sex category does not decrease the number of newborns in the next period.

Uniqueness and dynamic stability are intimately related. In a model with multiple equilibria, initial conditions determine which, if any, of several equilibria will be realized, and small diﬀerences in initial conditions can lead to large diﬀerences in long-run behavior. Pollak (1990a) discusses four sets of demographically meaningful conditions that imply local stability. Local stability rather than global stability is the relevant issue because, inter alia, some initial population vectors must lead to the trivial equilibrium. Further work—both theoretical and empirical—is required to determine whether any of the suﬃcient conditions for uniqueness and dynamic stability are satisﬁed and whether multiple equilibria or dynamic instability are realistic possibilities in two-sex demographic models.

Simulation is a possible approach to investigating these issues, but simulation techniques are much less informative in models that lack parametric speciﬁcations. In linear models, parametric speciﬁcation is a feature of the model itself, but to simulate the BMMR model, we must ﬁrst specify a functional form for the mating rule, and then specify the appropriate parameter values. Explicit functional forms for mating rules are discussed in Pollak (1990b).

## 3. Behavioral Models

Further work is also required to transform the BMMR model from a formal into a substantive model of population age structure and growth. The transformation requires importing behavioral theories from social science into the BMMR model to explain its three primitives—the birth matrix, the mating rule, and the mortality schedules. (In the generalized version of the model in which unions can persist for more than one period, there is a fourth primitive requiring a behavioral explanation: the schedule specifying the probabilities that unions of each type will end in desertion or divorce: see Pollak 1987.) From the standpoint of social science, however, these three primitives are diﬀerent kinds of analytical constructs. The mortality schedule is often regarded as a biological datum, although there is ample precedent (from Malthus to recent concern about excess female infant mortality rates in India) for regarding mortality as endogenous. The elements of the birth matrix reﬂect the decisions of individuals or of families facing economic and biological constraints. The analysis of these decisions is the subject matter of economic and sociological theories of fertility.

Whether we interpret ‘unions’ narrowly or broadly, the mating rule of the BMMR model is more complex analytically than the mortality schedules or the birth matrix because it reﬂects not only individual behavior but also the interactions of individuals in the ‘marriage market.’ The marriage market metaphor is useful because it suggests notions of choice, competition, and equilibrium. That is, the mating rule is not a full-blown theory of marriage or union formation but a reduced form, presumably reﬂecting the requirements of equilibrium in the marriage market.

There are two approaches to modeling competition and equilibrium in marriage markets, one based on matching and the other based on search. Mortensen (1988) and Weiss (1997) provide accessible introductions to both matching and search approaches; Roth and Sotomayor (1990) provide the deﬁnitive treatment of matching models. Both matching and search models begin with the preferences of individuals and analyze their implications for marriage-market equilibrium. They diﬀer, however, in their assumptions about the information available to and the alternatives faced by marriage-market participants. Matching models are nonstochastic and timeless; they implicitly assume that complete and accurate information about potential spouses is instantaneously and costlessly available to all marriage-market participants. Search models, in contrast, are stochastic and dynamic; they recognize explicitly that individuals have less than complete and accurate information about potential spouses, and that acquiring more and better information takes time and, in some search models, requires resources. In a search model, when two individuals meet in the marriage market, they must decide whether to marry or to continue searching, balancing the attractiveness of marriage now to a particular individual against the expected value of remaining unmarried and continuing to search. Because search models do not assume that individuals have complete and accurate information, they may appear to be more ‘realistic’ than matching models. The price that search models pay for added realism is their sensitivity to speciﬁc assumptions about the rate at which potential spouses meet in the marriage market, the impatience or time preference of individuals, and individuals’ beliefs about the characteristics and beliefs of other marriage market participants.

Ecologists may ﬁnd models that allow the possibility of a positive equilibrium rate of population growth unsatisfactory because such models fail to recognize density dependence and other factors that prevent population growth from continuing indeﬁnitely.

## 4. Summary

To summarize: classical stable population theory is parsimonious both because it allows us to use well-known, powerful mathematical techniques to investigate existence and dynamic stability, and because it allows us to infer a population’s equilibrium age structure, growth rate, and dynamic behavior from very little data. The birth matrix-mating rule model, which allows fertility rates to depend on the population’s age-sex structure, is more complex analytically and more demanding in its data requirements. In return for these extravagances, the BMMR model solves demography’s two-sex problem and provides a framework for addressing the marriage squeeze and other important issues in demography that require a two-sex model.

**Bibliography:**

- Feeney G M 1972 Marriage rates and population growth: The two-sex problem in demography. Ph.D. thesis, University of California, Berkeley, CA
- Mortensen D T 1988 Matching: Finding a partner for life or otherwise. In: Winship C, Rosen S (eds.) Organizations and Institutions: Sociological and Economic Approaches to the Analysis of Social Structure. American Journal of Sociology, Supplement. University of Chicago Press, Chicago, pp. S215–40
- Pollak R A l986 A reformulation of the two-sex problem. Demography 23: 247–59
- Pollak R A 1987 The two-sex problem with persistent unions: A generalization of the birth matrix-mating rule model. Theoretical Population Biology 32: 176–87
- Pollak R A 1990a Two-sex demographic models. Journal of Political Economy 98: 399–420
- Pollak R A 1990b Two-sex population models and classical stable population theory. In: Adams J, Lam D A, Hermalin A I, Smouse P (eds.) Convergent Issues in Genetics and Demography. Oxford University Press, New York
- Roth A E, Sotomayor M A O 1990 Two-sided Matching: A Study in Game-theoretic Modeling and Analysis. Cambridge University Press, Cambridge, UK
- Weiss Y 1997 The formation and dissolution of families: Why marry? Who marries whom? And what happens upon divorce. In: Rosenzweig M R, Stark O (eds.) Handbook of Population and Family Economics, Vol. 1A. Elsevier, Amsterdam