Lecture Notes. About Fertility Levels, Trends and Fluctuations

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1 Lecture Notes About Fertility Levels, Trends and Fluctuations Larry E. Jones and Alice Schoonbroodt April Introduction Fertility is negatively correlated with economic development. It is very high, as high as seven children per women, in developing countries; and extremely low in many rich countries, often well below the replacement rate. Fertility rates are often discussed by policy-makers, who argue, for example, that lowering fertility in poor countries could lead to increased economic growth. At the same time, many OECD governments have recently become concerned about low fertility rates in relation to the viability of pensions systems in the future. In response, several countries have instituted policies that subsidize child-rearing in an effort to increase the fertility rate. While Economists view the world as populated by many individual actors making choices under constraints. Aggregate measures, like Gross Domestic Product or employment, are the outcomes of the interactions among these individuals. In neo-classical theory, the household is considered to be such a decision-making actor. Gary Becker in the 1960s initiated a new line of research in which parents and children, old and young, men and women are separate economic decision makers. This allows economic tools to be used to study topics such as fertility decisions, the division of labor between husband and wife, and transfers between parents and children. A lot of research has been done since to understand intra-household decision making. More recently, macroeconomists have assessed from a quantitative point of view various reasons for and implications of changing fertility patterns. While the trend from high to low fertility has been thoroughly investigated, very little has been done to University of Minnesota University of Southampton 1

2 understand temporary but large fluctuations in fertility. Furthermore, some policies aiming at controlling fertility have been addressed by various economists. In these notes simple models of fertility choice are described. In here we study the qualitative and quantitative properties of the Becker and Barro (1988) and Barro and Becker (1989) (hereafter B&B) models of fertility along with one based on the ideas of Caldwell (1978) on the demand for children as a source of old age support, see Boldrin and Jones (2002) (B&J henceforth) for the model formalization. The B&B model of fertility is based on the assumption that parents get direct utility from the consumption of their children, their grandchildren, etc. That is, parents have forward looking altruism, and, thus, they view their fertility decisions as extensions of their own dynastic family. This forms the basis for their demand for children in the B&B model. In contrast, much of the literature in demography (see Caldwell (1978) and Alter (1992) for excellent surveys), focuses on a demand for children arising out of a need for support in old age. Various authors have implemented this idea (e.g., parents are automatically entitled to a given fraction of the income of their children (Ehrlich and Lui (1991)), etc.), but, a natural candidate is to model this as a situation in which children care about the welfare of their parents, i.e., as reverse altruism. (See Laitner (2002) for uses of this in models of savings behavior.) That is, parents demand for children arises because they know that children care about them, and will, because of this, help support them in old age. This is the focus of the B&J implementation of the Caldwell approach. Of course, in reality, both of these sources of demand for children are operative with the B&B motive potentially more important in developed countries and the B&J one potentially more important on the path to development. Still, as will be seen, it is useful to study the properties of the two versions in isolation. Since in both approaches, fertility is an intertemporal decision, the methodology of modern dynamic general equilibrium theory is a natural tool to study the properties of the models. As is typical with this type of model, representative agent implementations are useful first approximations and this is what we will focus on. The models also have implications for the data in cross section, and these are pointed out below. Thus, this work complements a growing and interesting literature using dynamic general equilibrium techniques to study decisions about fertility and other choices made by families (e.g., marriage and female labor supply) in quantitative settings. This approach, using macro-dynamic tools to study family economics questions, complements the existing literature in micro/labor economics on the same topics (see Heckman and Macurdy (1980), Eckstein and Wolpin (1989), etc.) 2

3 2 The Barro-Becker Model of Fertility In this Section, we revisit versions of the standard Barro-Becker model in which each person goes through two stages of life, childhood and adulthood. In childhood, no decisions are made. In subsequent period(s) they are adults and make consumption and fertility choices. 2.1 Utility functions in Barro-Becker models The standard presentation of the Barro-Becker model usually begins with a description of the preferences of a period t adult. It is assumed that parents care about three separate objects: i) their own consumption in the period, c t, ii) the number of children they have, n t, and, iii) the average utility of their children, U t+1. This is usually specialized further. It is assumed that utility of the typical time t household is of the form: U t = u(c t ) + βg(n t )U t+1 = u(c t ) + βg(n t ) n t i=1 1 n t U it+1 where U it+1 is the utility of the i-th child of the parent. Assuming that U it+1 = U it+1 = U t+1 for all i, i this simplifies to: U t = u(c t ) + βg(n t )U t+1. Intuitively, it makes sense to assume: 1.) Parents like the consumption good: Utility is increasing and concave in own consumption; 2.) Parents are altruistic: Holding n t fixed and increasing U t+1 increases (strictly) the utility of the parent, U t ; 3.) Parents like having children: Holding U t+1 fixed and increasing n t increases (strictly) the utility of the parent. It is also natural to assume: 4.) The increase described in 3.) is subject to diminishing returns. 3

4 The first requirement is straightforward. It is satisfied as long as u is increasing and concave. The second has implications for what g can be. Since (u(c) + βg(n)u)/ U = g(n), it follows that (b) implies that g(n) > 0 for all n. The third requirement is less straightforward. Although this requirement makes intuitive sense, there are, because of the special restrictions implicit on functional forms, some issues that arise. For example, suppose that U t+1 > 0. Then (3.) implies that g(n) must be increasing in n. On the other hand, if U t+1 < 0 (3.) implies that g(n) should be decreasing in n. It follows that if it is possible for U t+1 to be either positive OR negative, it is impossible to satisfy all of these requirements simultaneously. In sum, (1.)-(3.) are mutually inconsistent without some sort of restrictions on the possible values for U t+1. 1 Similar issues arise with respect to (4.). If U t+1 is restricted to be positive, (4.) requires g to be concave while if U t+1 is restricted to be negative, (4.) requires that g is convex. This is not to say that these conditions cannot be satisfied. We must simply assume that EITHER U t+1 > 0 always OR U t+1 < 0 always and then make the appropriate assumptions on g. Without an assumption like this, the natural monotonicity properties of utility cannot be guaranteed. Thus, we are left with two options: A. Assume that g(n) is non-negative and strictly increasing and U > 0, OR B. Assume that g(n) is non-negative and strictly decreasing and U < 0. This is complicated by the fact that U t+1 is not exogenous. Because of this, more work is needed to state the sets of sufficient conditions in terms of primitives. A second, related goal comes from noting that the utility formulation above is time consistent there is no inherent conflict in preferences between the agents in period t and period t + 1. Because of this, it is possible to reformulate sequential decision problems in terms of a one time, time zero choice. However, to do this, we must have the full preferences over sequences of the time zero decision-maker the dynasty head completely specified. Formally, assume symmetry, then, successive substitution leads to a formulation of dynastic utility at time 0 in terms of the basic choice variables: U 0 = t=0 Π t 1 s=1g(n s ) ] u(c t ). Because of the term Π t 1 s=1g(n s ), this utility function is typically not concave as written. However, as discussed in Alvarez (1999), under certain conditions, this can be rewritten 1 U t+1 > 0 would arise if, for example U t+1 = c1 σ t+1 1 σ and 0 σ 1, while U t+1 < 0 would arise if, for example U t+1 = c1 σ t+1 1 σ and σ > 1. It follows that U t+1 = log(c t+1 ) could not be allowed without some restrictions on the allowable values for c t+1. 4

5 as a concave problem in dynasty aggregate variables. Assume that g(n) = n η, and let N t = Π t 1 s=1n s, this is the total number of adult descendants alive during period t. Then Π t 1 s=1g(n s ) = g(π t 1 s=1n s ) = g(n t ), and so preferences for the dynasty head can be rewritten as: U 0 = ] t=0 βt C g(n t )u t N t, where C t = N t c t is total consumption in period t. Note that this assumes that consumption is the same for all adults in a period. U t for t > 0 is defined similarly. Following the discussion above, since g(n) = N η is always positive, there are two possible ways to satisfy conditions (a)-(d) above: A.) Assume that u(c) 0 for all c 0, that u is strictly increasing and strictly concave and that 0 < η < 1; B.) Assume that u(c) 0 for all c 0, that u is strictly increasing and strictly concave and that η < 0. 2 Either of these are consistent with the entire set of intuitive requirements laid out in the original Barro-Becker papers. Typically we will want more however. The extra requirements that we will want are that U 0 as written here is increasing and concave in (C, N). This is for standard reasons, we want the solution to maximization problems to be unique, etc. For this, we specialize further and assume that and u(c) = c1 σ 1 σ. Given this assumption, there are two sets of parameter restrictions that satisfy the natural monotonicity and concavity restrictions for this functional form, both in terms of the aggregate, or dynasty variables, (C, N), and in terms of per capita values, (N, c) = (N, C N ). The first is the standard assumption in the fertility literature: 0 η + σ 1 < 1, 0 < 1 σ < 1. In this case, 0 < η = η + σ σ < 1. In this case U > 0 for all (N, C) R 2 +. This corresponds to case A above. The second possibility is one which allows for intertemporal elasticities of substitution in line with the standard growth and business cycle literature: σ > 1, η + σ 1 0 (i.e., η 1 σ < 0). In this case, utility is negative and η < 0. 3 above. 4 This corresponds to case B 2 The condition 0 < η < 1 in A implies that N η is non-negative, increasing and concave. The condition that η < 0 in B implies that N η is non-negative, decreasing and convex. 3 The case with parameters in this second case is also discussed, briefly, in Alvarez (1999), footnote 2. 4 The possibility of σ > 1 and η < 0 is also discussed, briefly, in Alvarez (1999)), footnote 2. Mateos-Planas (2002) also uses this configuration. 5

6 In the case where, η = 1 σ (allowed under both configurations), utility becomes a function of aggregate consumption only. Hence, conditions for monotonicity and concavity of U involve U C and U CC only. This leads to the following proposition. Proposition 1. If either i) 0 < η < 1, 0 < σ < 1 and 0 η + σ 1 < 1, or ii) σ > 1 and η + σ 1 0, then U is strictly increasing in C, weakly (strictly, if η + σ 1) increasing in N and weakly quasi-concave (strictly, if η + σ 1). Furthermore, V is strictly increasing in c, weakly (strictly, if η + σ 1) increasing in N and weakly quasi-concave (strictly, if η + σ 1). Proof. See Appendix. From the above proposition, one can see that the original Barro-Becker parameter configuration excludes one side of the admissible parameter space. This results from the initial assumption that utility is positive. In what follows, we derive comparative statics and perform simple quantitative assessments. We show how the parameter configuration in ii) preserves desirable properties but accommodates additional historical observations since the demographic transition. 2.2 Budget and Feasibility Constraints We will revisit three types of feasibility/budget constraints. Let N bt denote the total number of births for the dynasty this period and X t dynasty investment in physical capital, K. 5 That is, for given initial conditions and the assumption of CRS production function when appropriate, the following sets are convex for any sequence of productivities, {A t } t=0, interest rates, {r t } t=0, and/or wages, {w t } t=0. General equilibrium (Barro and Becker (1989)): (N t, K t, C t ) t=0 t 0, C t + θ t N bt + X t A t F (N t, K t ) Λ(N 0, K 0 ) = N t+1 = πn t + N bt K t+1 = (1 δ)k t + X t Partial equilibrium (Becker and Barro (1988)): (N t, K t, C t ) t=0 t 0, C t + θ t N bt + X t w t N t + r t K t Λ(N 0, K 0 ) = N t+1 = πn t + N bt K t+1 = (1 δ)k t + X t 5 Throughout we only look at the goods costs case but this is without loss of generality, except for comparative statics with respect to income taxes and for cyclical properties of the model. Here, this goods costs is denoted by θ t and we assume that it grows at the rate of overall technological improvement. In the end, the only thing that ends up mattering is θt A t which is constant by assumption. If on the other hand, there was a time cost, say b fraction of the time available in a period, we would find that w t = A t and that b = θ A. 6

7 Labor as { only input: } (Nt, C t ) Λ(N 0 ) = t=0 t 0, C t + θ t N bt w t N t N t+1 = πn t + N bt Under any of these constraint sets, the time 0 maximization problem has a convex constraint set and a concave objective function. Thus, the problems have unique solutions, concave value functions, etc. 2.3 A Simple Version of the Barro-Becker Model In this section, we briefly describe a version of the Barro-Becker model (Becker and Barro (1988) and Barro and Becker (1989)) with three alterations. First, we will assume that the only source of income is labor and there is no possibility of bequests. Second, the usual formulation also assumes that there is only one period of adult life which makes comparisons to data difficult. To handle this we will assume that adults survive to the next period of life with probability π. We assume that all alive adults are otherwise identical, however. 6 Thus, death among adults is treated as in Blanchard (1985). Finally, we introduce a stochastic component which may affect productivity, youth mortality or longevity (conditional on reaching adulthood). Hence in this model, households are alive for one period as a child, where no decisions are made. In subsequent periods they are adults and make consumption and fertility choices. Adults care about consumption, the number of children and their (surviving) children s future utility. In every period they decide how many children to have and how much to consume given their income. All income is stochastic labor income. Children cost θ which may be a goods cost or a time cost. We will follow the approach in Alvarez (1999) in which a time zero dynastic head chooses the time paths of aggregate, dynasty level variables a discussion of the period by period formulation of the dynasty problem is presented in Jones and Schoonbroodt (2007). The problem solved is: max {Ct (s t ),N s,t (s t )} Subject to: ] t=0 βt g(n t (s t 1 ))u Ct(st) N t(s t) C t (s t ) + θ t (s t )N s,t (s t ) w t (s t )N t (s t 1 ), and N t+1 (s t ) π t (s t )N t (s t 1 ) + N s,t (s t 1 ), N 0 given, where C t is aggregate consumption in period t, N t is the number of adults alive, N s,t is the number of births in period t that survive to become adults in period t + 1, θ t is the cost 6 Note that π is independent of age. This is a short-cut to an explicit age-structure that keeps the state space small but allows longer than one period life-times. An explicit age-structure is addressed in Section ***. 7

8 of producing a surviving child for the next period (with θ t = (a t + bw t )/π y, with a t, the goods cost per birth, b the fraction of time spent per birth and π y, survival probability to adulthood), w t is the wage rate s t = (s 0,..., s t ) is the history of shocks which may affect parameters such as w t, θ t and π t. Note that if survival to adulthood, π y < 1, it takes 1 π y births for 1 surviving child. Hence, we can interpret a decrease in mortality as a decrease of the cost of producing a surviving child, θ. Below, we also assume that wages and goods costs of children grow at the exogenous rate, γ. Implicit in this formulation is the assumption that each adult has the same level of consumption C t N t 1 that the expected working lifetime of an adult is 1 π formulation has π = 0. = c t in any period. From this it can be seen periods. The original Barro-Becker Equilibrium populations in the deterministic case The first order condition for the stock of population in period t + 1 is given by: θ s,t N η+σ 1 t Ct σ = β w t+1 + θ s,t+1 π] N η+σ 1 t+1 Ct+1 σ (η + σ 1) + β }{{} (1 σ) A N η+σ 2 t+1 C 1 σ t+1 } {{ } B (1) This equation can be written in per capita variables. Dividing by N η 1 t, we get: θ s,t c σ t = β w t+1 + θ s,t+1 π] γ η 1 (η + σ 1) N,t c σ t+1 + β }{{} (1 σ) A γ η 1 N,t c1 σ t+1 } {{ } B (2) where γ N,t is the population growth rate between t and t + 1, or the number of children per adult in period t. The intuition for this is as follows. On the left is the marginal cost in terms of changed current utility of increasing N t+1 (i.e., of producing an extra child). This cost is just the direct cost of reduced consumption today (rescaled by the fact that it only takes θ s,t units of C to make one extra unit of N). On the right hand side are the two pieces of the marginal benefits next period from increasing N t+1. These are: A. the value of the extra output the dynasty will have next period; B. the marginal value of utility from having extra children. To gain some more insight, consider the special case in which η = 1 σ. In this case, N is exactly like a capital good since the two utility effects of increasing N exactly cancel out. These two effects are: 1. the direct benefit of having extra children g(n) = N η in the utility function; 2. the direct cost of having children by diluting per capita consumption C ] 1 σ N /(1 σ) in the utility function. As we can see in the first order condition, there 8

9 are two things that give simplification. The first is that (η + σ 1) = 0 and so the first term disappears entirely and second that N η+σ 1 t = N η+σ 1 t+1 = 1, i.e., the marginal value of increased total consumption by the dynasty in periods t and t + 1 no longer depend on the size of the dynasty in the period. Because of this, we get: ] σ Ct+1 wt+1 = β + θ ] s,t+1 π C t θ s,t θ s,t This is the standard Euler Equation from an Ak model in terms of aggregate consumption, modified for the case θ s potentially different from 1 with π corresponding to 1 δ (where δ denotes depreciation), and time varying costs and benefits, i.e., w t and θ s,t. The Euler equation in (1) or (3) together with the feasibility constraint (3) which can be rewritten as C t + θ s,t N t+1 πn t ] = w t N t ] C t Nt+1 + θ s,t π = w t. (4) N t N t and the initial condition N 0 completely describe the equilibrium path Balanced growth One advantage of the version of the model with labor income only is that it delivers simple, and qualitatively reasonable analytic comparative statics results across Balanced Growth Paths (BGPs). 7 These are summarized here for completeness. We will focus on three distinct changes based on the quantitative experiments that we will explore in the next section, namely the results of: i) changing γ, ii) changing θ s, and iii) changing π. These correspond to three commonly discussed quantitatively important changes in demographic patterns over the period from 1800 to 1990: i) the increased growth rate of labor productivity that came with industrialization, ii) the fall in the direct cost of creating a surviving child through the reduction in infant and child mortality rates, and iii) the dramatic increase in expected lifetime that has occurred in the last 100 years. As a first approximation, we study these changes by doing simple comparative statics with respect the parameters of the model. Assume that θ s,t = γ t θ s and w t = γ t w. Let γ C = C t+1 C t be the growth rate in aggregate consumption and γ c = C t+1/n t+1 C t /N t the growth rate in per capita consumption. On a BGP, the resource constraint in (4) becomes 1 γ t C t N t = w + θ s π θ s γ N (5) 7 Some, but not all of these results carry over to generalized versions of the model with capital, (see Barro and Becker (1988) for that case). In that case, the extra condition that must be satisfied that the rate of return on investing in children must equal the rate of return on investing in capital complicates matters. 9

10 Since on a BGP (5) has to hold for every t, we must have C t γ t N t = C t+1 γ t+1 N t+1 or γ = C t+1/n t+1 C t/n t = γ C γn = γ c Then, using this and (5) in (1) after dividing both sides by γ t+1 N η+σ 1 t+1 C σ t+1, we get This can be rewritten as: ] (η + σ 1) θ s γ 1 η N γσ 1 = β w + θ s π θ s γ (1 σ) N ] + w + θ s π] 1 β γ1 η N γσ 1 + (η + σ 1) γ (1 σ) N = ] η w + π (1 σ) θ s (6) Comparative statics of population growth The only endogenous variable in equation (6) is the population growth rate, γ N, which only enters on the left-hand side. Moreover, π (expected working life) and θ s (costs of children or youth mortality) only enter on the right-hand side while the productivity growth rate, γ, only enters on the left-hand side. That is, holding (σ, β, η, w) fixed, ] this equation is of the form: LHS(γ N ; γ) = D(θ s, π), where D(θ s, π) = η w (1 σ) θ s + π (see Figure 1). Note that LHS(γ N ; γ) is increasing in γ N, for all values of γ, since our parameter restrictions imply that η, 1) and (η+σ 1) η > 0. Similarly, D > 0 since > 0. (1 σ) (1 σ) Because of this, it follows that D is increasing in π and decreasing in θ s, and hence so is γ N. Similarly, LHS(γ N ; γ) is increasing in γ σ 1 holding σ fixed. However, whether an increase in γ causes γ σ 1 to increase or decrease depends on whether σ > 1 or σ < 1. If σ < 1, an increase in γ causes γ σ 1 to fall while if σ > 1, an increase in γ causes γ σ 1 to rise. From this it follows that γ N is increasing in γ if σ < 1 and decreasing in γ if σ > 1. This has important implications for studying trends in fertility using this Barro-Becker type model. Since the rate of growth of productivity has increased over the period describing the Demographic Transition, it follows from this that we would expect population growth rates to fall as a result as long as σ > 1. Further, the above equation simplifies considerably when 1 σ = η. In fact, ] w γ σ N = βγ 1 σ + π. (7) θ s In this case, we can also derive results about the size of effects of changes in parameter values on γ N for different values of σ. To see this, note that the left-hand side of equation (7), γ σ N is increasing in γ N, and is 1 when γ N = 1. Further, it is concave if σ < 1, and convex in γ N if σ > 1 (see Figure 2). Because of this, it follows that if γ N > 1, a given size change in γ, θ s, or π will have larger effects on γ N if σ < 1. E.g., a reduction in θ s gives rise to a larger 10

11 Figure 1: Comparative statics of γ N 10 8 LHS D LHS increase γ D decrease θ s /increase π LHS,D γ N increase in γ N if σ < 1 than if σ > 1. This will play an important role in our quantitative findings below. Summarizing the discussion to here we see: Proposition 2. The following comparative statics results across BGP s hold for population growth, γ N : 1. An increase in γ causes γ N to rise if σ < 1 and to fall if σ > 1; 2. An increase in the cost of children, θ s, causes the population growth rate, γ N to fall; 3. An increase in adult survival, π, causes γ N to increase; ] Moreover, if η = 1 σ and βγ 1 σ w θ s + π > 1 (i.e., γ N > 1), a change in θ s or π will have larger effects on the BGP level of γ N if σ < 1 than if σ > 1. It also follows that γ N > 0 and γ N > 0. We do not emphasize these because they will β w play no role in the quantitative discussion we focus on below. 8 Note that a decrease in youth 8 If all child costs are goods costs, increases in labor income taxes are equivalent to reductions in w. Thus, it follows that increasing the labor income tax rates will decrease both population growth rates and fertility on the BGP, cf. Manuelli and Seshadri (2005). At the other extreme, if all costs are in terms of time (θ = bw), fertility and population growth rates are independent of labor income taxes. 11

12 Figure 2: Increase in γ N is larger for σ < 1 than σ > RHS LHS σ>1 RHS change LHS σ<1 LHS,RHS γ N mortality will be interpreted as a decrease in the cost of surviving children, θ s, while increases in expected working life lengths will be interpreted as increases in π. From the beginning of the 19th century to the end of the 20th century, Crude Birth Rates (CBR) fell substantially, while population growth rates (net of immigration) decreased only slightly. Hence, smaller increases in population growth in response to changes in mortality declines (through θ s or π) may be a desirable prediction. From the proposition, this is the case with lower IES (i.e., σ > 1). In the next two subsections, we derive a model analogue for the Crude Birth Rate and introduce survival probabilities to adulthood to discuss relevant comparative statics Comparative statics of the Crude Birth Rate, surviving children In this version of the model, the variable of choice is the total number of surviving births in the dynasty in a given period, N s,t. This does not map naturally to the usual measures of fertility used by demographers such as Total Fertility Rate (TFR) or Cohort Total Fertility Rate (CTFR). This is complicated by the fact that the model is unisex and monoparental i.e., each agent has children and is on its own and this is true for everyone, not just women. 12

13 These assumptions are made for simplicity. 9 There is a natural relationship, however, between N s,t and a common measure of fertility, the Crude Birth Rate CBR. In the data, the latter is calculated as the number of children in a period divided by the population in that period. In the model, we have: CBR s,t = N s,t N t + N s,t = N t+1 πn t N t + N t+1 πn t = γ Nt π 1 + γ Nt π, where γ Nt = N t+1 N t is the growth rate of the adult population between periods t and t + 1. This expression for CBR s corresponds to the number of births during a period divided by the end of period population. Here, we are assuming that all children survive giving the expression in the text. Below, we also introduce infant and child mortalities and differentiate between CBR measured in births vs. CBR measured in surviving children. Thus, this is one obvious identification between data and model quantities that can be used. On the BGP, CBR s is constant and is given by: CBR s = γ N π 1 + γ N π = γ N π This is of the form 1 and because of this it follows that the comparative statics results 1+1/x given above for γ N for changes in γ and θ s will also hold for CBR s. The one exception to this concerns the effects of changes in expected life length. Even in this case, CBR s is a monotonically increasing function of γ N π, and so the sign of CBRs is the same as the π sign of γ N 1. Thus, while an increase in π will always increase population growth rates, π whether or not it increases CBR s depends on the size of γ N. If it is less then 1, CBR s < 0 π π while the opposite holds if γ N > 1. When η = 1 σ, we find that: π dγ N dπ = β (γγ N) 1 σ. σ Note that γγ N = γ C, the growth rate of aggregate consumption by the dynasty. As is standard in growth models, β (γγ N ) 1 σ is the growth rate in utility. This must be less than one for a well defined (finite) solution to exist for the dynasty s maximization problem. If σ > 1, that condition will hold if β < 1, γ > 1 and γ N 1. Thus, it follows that in this case, CBR s π < 0 as long as γ N 1. Summarizing, we have: Proposition 3. The following comparative statics results hold across BGP s for surviving children, CBR s : 9 see Section?? for further discussion. 13

14 1. An increase in γ causes CBR s to rise if σ < 1 and fall if σ > 1; 2. An increase in the cost of children, θ s, causes the fertility rate, CBR s to fall; ] 3. If η = 1 σ < 0, and βγ 1 σ w θ s + π > 1 (i.e., γ N > 1), an increase in π causes CBR s to fall. ] Moreover, if η = 1 σ and βγ 1 σ w θ s + π > 1 (i.e., γ N > 1), a change θ s will have larger effects on the BGP level of CBR s if σ < 1 than if σ > 1. For the quantitative assessments below, it is interesting to note that by analogy to a growth model, although a decrease in the depreciation rate causes the rate of growth of the capital stock (i.e., population growth in this interpretation) to increase, this does not necessarily imply that investment s share in output (i.e., fertility) increases. Indeed, when σ > 1, population growth, γ N, increases while fertility, CBR s, falls in response to an increase in π The effects of changes in survival to adulthood: births vs. survivors Going back to the original Barro-Becker work one common exercise has been to examine the implications for the model quantities when survival rates of children (e.g., Infant Mortality Rates (IMR) and Child Mortality Rates (CMR)) change. This has been done by reinterpreting N s,t as the number of surviving children and using a certainty equivalent version of the costs for producing a surviving child. 10 above as follows: max {Ct,N b,t,n s,t,n t } subject to: C t + θ b,t N b,t w t N t, and N t+1 πn t + N s,t, N s,t π s N b,t, N 0 given. t=0 βt N η t The simplest version of this is to adapt the model C t N t ] 1 σ /(1 σ) where θ b,t is the cost of a birth, N b,t is the total number of births in period t, N s,t is the number of children that survive to working age and π s is the unconditional probability of a child that is born surviving to working age. This problem can be rewritten by eliminating N b,t to obtain: 10 This was used in the original Barro and Becker (1988) and Becker and Barro (1989) as well as in Boldrin and Jones (2002), Schoonbroodt (2004), Doepke (2005), etc. 14

15 max {Ct,N s,t,n t} t=0 βt N η t C t N t ] 1 σ /(1 σ) Subject to: C t + θ s,t N s,t w t N t, and N t+1 πn t + N s,t, N 0 given, where θ s,t = θ b,t π s is the cost of producing a surviving child. This is formally equivalent to the problem analyzed above but where, now, the cost of raising a child to working age depends on the survival probability an increase in π s decreases θ s,t. Because of this equivalence, the comparative statics results given above apply. For example, an increase in π s lowers the cost of children and hence, by the argument above increases γ N and CBR s. Note that this interpretation here has one additional layer of subtlety because it is γ N and CBR s calculated in terms of surviving children that increase and this does not necessarily imply that, for example, CBR calculated in terms of births will go up. To make this distinction clear we will introduce one new piece of notation CBR t = N b,t N t +N s,t = CBRs,t π s. Thus, even though CBR s is increasing in π s it need not be true that CBR is. Additional layers of complexity can also be added (cf. Schoonbroodt (2004) and Doepke (2005) for examples). For example suppose that there are 3 stages that children must pass through to become adults, say, infancy, childhood and youth. Suppose i) π i is the survival probability of infants which cost θ i to raise ii) π ic is the conditional survival probability to be a youth given survival through infancy and θ c is the cost borne by parents during childhood, and iii) π cy is the probability of surviving to adulthood conditional on surviving to be a youth and θ y is the cost to parents of children in youth. Then the total child-rearing cost to a parent that has N b births is θ i N b + θ c π i N b + θ y π ic π i N b while the total number of surviving children produced is N s = π s N b, where π s = π i π ic π cy. Thus, to produce N s surviving children, N b must be N s π s and hence, the total cost is: θ s N s (θ i + π i θ c + θ y π ic π i ) N b = θ i+π i θ c +θ y π ic π i π s N s ; or, the per surviving child cost is: θ s = θ i π i π ic π cy + θ c π ic π cy + θ y π cy. Again, we see that increasing survival rates (π i or π ic or π cy or any subset) lowers the cost of producing a surviving child and hence from the results above both the population 15

16 growth rate and the CBR s increase. As above, this need not be true of CBR, however. We use this formulation in our time series experiment for the U.S. in Section?? Two special cases are worthy of mention. The first is one in which π i = π ic = 1. In this case we are back to the case described in the beginning of this section where π s = π cy and θ b = θ i +θ c +θ y. This is the case we will consider in our first set of quantitative experiments. The second one is where only survivors are costly, while births that die before a certain age are costless. For example, if π ic = π cy = 1 (i.e., children die in infancy or survive to adulthood) and θ i = 0. In this case, N s and CBR s are independent of π s = π i. Thus, N b = N s π i and CBR = CBR s π i are decreasing (and hyperbolic) in π i (see Doepke (2005) for a full exposition). In general, however, whether the number of births, CBR is increasing or decreasing in π s depends on which of the two percentage increase is larger, the one in CBR s or the one in π s. Again, we summarize in a proposition: Proposition 4. The following comparative statics results hold across BGP s for total births, CBR: 1. An increase in γ causes CBR to rise if σ < 1 and fall if σ > 1; ] 2. If η = 1 σ < 0, and βγ 1 σ w θ s + π > 1 (i.e., γ N > 1), an increase in π causes CBR to fall. CBR s Moreover, in both cases, percentage changes in CBR are equal to percentage changes in As noted in the previous section, it is important for what is coming to remember that although increases in the components of π s (and hence decreases in θ s ) will increase CBR s, this increase will be less when σ > The US experience Jones and Schoonbroodt (2007) simulate the U.S. experience, taking the timing of events and all three changes in economic environment into account, namely changes in productivity growth rates, youth mortality and longevity. They find that overall, the model predicts that CBR would fall from 36.3 to 17.2 births per 1000 population. This corresponds to about twothirds of the observed change in the U.S. (and about 100 percent of the change seen in the U.K.). In terms of population growth rates, the model predicts a fall from 1.4 to 0.65 percent per year which captures about one half of observed changes in the U.S.. Interestingly, about 90 percent of changes in fertility before 1880 are accounted for by changes in productivity growth rates, while changes in mortality account for about 90 percent of the fertility decrease thereafter. This finding speaks to the debate about what the root cause of the fertility decline 16

17 was was it reductions in infant mortality rates or was it accelerating economic development? What we find is: it was a combination of both with the latter being the most important factor early in the transition and mortality being the most important factor later on, a key assumption being low intertemporal elasticity of substitution in consumption. 2.4 Stochastic Version: Productivity Shocks We can formulate the dynastic head planner s problem as follows. 11 The dynasty head at time zero solves the maximization problem: Max {Ct,N t } subject to: E 0 ( t=0 βt g(n t (s t 1 ))u C t (s t ) + θ t (s t )N bt (s t ) N t (s t 1 )w t (s t ), N t+1 (s t ) πn t (s t 1 ) + N bt (s t ), N 0 fixed. ]) Ct(s t ) N t(s t 1 ) Where s t = (s 0, s 1,..., s t ) is the history of shocks up to and including period t, w t is the stochastic process for wages (assumed to be a function of the shocks), θ t is the cost of raising a child born in period t, C t is aggregate consumption for the dynasty in the period (assumed split across all individuals of working age), N bt is the number of new children born in the dynasty in period t and N t is the number of dynasty members of working age alive in the period. Note that each individual alive in period t is assumed to have a survival probability to period t + 1 of π. This is unusual for a Barro-Becker fertility model, where it is usually assumed that individuals only have one period of active decision making. This corresponds to the assumption that π = 0, one of the special cases we will discuss below. However, since some of the TFP movements that we want to discuss are at frequencies higher than a generation (in the fertility sense), it will be convenient to also consider cases in which π > 0. Because of this choice of functional form (admittedly a gross simplification of actual dynastic survival processes), note that the model is equivalent to one in which the stock, N, depreciates at rate δ = 1 π over each period. This will allow us to use the analogy to a stochastic Ak model with less than full depreciation below. We have assumed that the flow utility function is of the form U(C, N) = g(n)u( C ), i.e., utility depends on both the size of N 11 Alvarez (1999) shows the equivalence between the equilibrium allocations from a sequence of individual problems and the dynastic head s problem in this type of model. 17

18 the dynasty and per capita consumption. Assuming that g(n) = N η and u(c) = c1 σ 1 σ, we can rewrite this problem as: P1 Max {Ct,N t } subject to: t=0 βt N η t C t N t ] 1 σ /(1 σ) ] = t=0 βt N η+σ 1 t C 1 σ t 1 σ C t + θ t (N t+1 πn t ) N t w t. There are two sets of parameter restrictions that satisfy the natural monotonicity and concavity restrictions for this functional form, both in terms of the aggregate, or dynasty variables, (C, N), and in terms of per capita values, (N, c) = (N, C ). These are: i) 0 < η < 1, N 0 < σ < 1 and 0 η + σ 1 < 1, and ii) σ > 1 and η + σ 1 0. We will explore both of these options below. Notice that if η + σ 1 = 0, then N does not enter the period utility function except in aggregate consumption, and hence, N plays exactly the same role in this model as K does in a stochastic Ak model. 12 There is one twist however. That is θ t is also stochastic, at least in the case where child-rearing is modelled as a time cost. In that case, θ t = bw t and hence, if w t is stochastic, so are child-rearing costs. That is, the periods when productivity is high are also those when capital is expensive. Also, in this case aggregate consumption, C, grows at the same rate as N, but per capita consumption is constant (without shocks). Other than that, the analogy is very close. Let us introduce one additional piece of notation. Let N ft be the fertile part of the working population, i.e. N ft = λ t N t. Then children per fertile person - the fertility rate, n bt can be expressed as n bt = N bt N ft. This is the model quantity that we will identify with the Total Fertility Rate (TFR) in the Data Homogeneity properties of the planner s problem Recall that we want to study solutions to maximization problems of the form: P (N 0, s 0 ) Max {Ct,N t } U 0 ({C t, N t }) = E 0 t=0 βt N η+σ 1 t subject to: ] Ct 1 σ 1 σ C t + θ t N t+1 N t (w t + θ t π), N 0 given. 12 See Jones and Manuelli (1990) and Rebelo (1991), seminal papers on this model. 18

19 Let V (N 0, s 0 ) denote the maximized value of the objective at a solution (assuming one exists) and let {(Ct (N 0, s 0 ), Nt (N 0, s 0 )} t=0, denote the solution itself. Proposition 5. V (λn 0, s 0 ) = λ η V (N 0, s 0 ) and {(Ct (λn 0, s 0 ), Nt (λn 0, s 0 )} t=0 = λ{(ct (N 0, s 0 ), Nt (N 0, s 0 )} t=0. Proof. Step 1: {(C t, N t } t=0 is feasible for P (N 0, s 0 ) if and only if {(λc t, λn t } t=0 is feasible for P (λn 0, s 0 ). Step 2: U 0 (λ{c t, N t }) = λ η U 0 ({C t, N t }). Step 3: If the claim is false, then there is something better for P (λn 0, s 0 ) than λ{(ct (N 0, s 0 ), Nt (N 0, s 0 )} t=0. Call this feasible plan {(C t, N t)} t=0. From Step 1, the plan 1 {(C λ t, N t)} t=0 is feasible for P (N 0, s 0 ). From Step 2, 1 {(C λ t, N t)} t=0, gives higher utility than {(Ct (N 0, s 0 ), Nt (N 0, s 0 )} t=0, a contradiction. This is a standard homogeneous/homothetic type argument Recursive formulation Because of this result, and because the problem is stationary if (w(s), θ(s)) is assumed to be a first order Markov process, we can characterize the solution through Bellman s Equation: (BE) V (N, s) sup (C,N ) N η+σ 1 C 1 σ /(1 σ) + βe V (N, s ) s] s.t. C + θ(s)n (w(s) + θ(s)π) N. Because of Proposition 5, this can be rewritten as: (BE) V (N, s) sup (C,N ) N η+σ 1 C 1 σ /(1 σ) + βe (N ) η V (1, s ) s] s.t. C + θ(s)n (w(s) + θ(s)π) N, or, (BE) V (N, s) sup (C,N ) N η+σ 1 C 1 σ /(1 σ) + β (N ) η E V (1, s ) s] s.t. C + θ(s)n (w(s) + θ(s)π) N. Where the last step follows since N is a function of s alone (and not s ). Now, let D(s) E V (1, s ) s] to obtain: (BE) V (N, s) sup (C,N ) N η+σ 1 C 1 σ /(1 σ) + β (N ) η D(s) s.t. C + θ(s)n (w(s) + θ(s)π) N. 19

20 2.4.3 No catching up fertility From this, it follows that the FOC s for the problem are: (F OC1) N η+σ 1 C σ 1 = βη(n ) η 1 D(s) θ(s), (F OC2) C + θ(s)n = (w(s) + θ(s)π) N. These equations can be simplified to some extent. We have: (F OC1) C N ] σ = βηd(s) θ(s) N N ] η 1, so that, (F OC1) n = N N = ] σ 1/(η 1) C 1 η θ(s) N βηd(s)]. Substituting into F OC2 we get: (F OC2) C + θ(s)n = (w(s) + θ(s)π) N, C (F OC2) + θ(s) N = (w(s) + θ(s)π), N N C (F OC2) + θ(s) ] σ 1/(η 1) C 1 η θ(s) N N βηd(s)] = (w(s) + θ(s)π). It follows that C and N are functions of the current shock only (although C and N N N are not) and NOT the current level of the stock, N. Since this property plays a role in the ability of this type of model to exhibit a catching up of fertility after a low shock, we state this as a formal proposition: Proposition 6. The solution to the Planner s Problem is to find C and N N N ] σ 1/(η 1) 1 η θ(s) βηd(s)] = (w(s) + θ(s)π) C + θ(s) C N N and, N = ] σ C 1 η θ(s) 1/(η 1). N N βηd(s)] Thus, the growth rate in population, N N the size of the current stock, N. which solve: is a function of the current shock only, and not Furthermore, if the fraction of fertile people in the population, λ t, is independent of time, i.e. λ t = λ, then the fertility rate also only depends on the current shock, s t, and not the current stock, N t 20

21 Proposition 7. Let N f = λn. If λ is independent of time, then ( n b = N b N f = N πn N f = 1 N π). λ N That is, the fertility rate is independent of the size of the current stock, but, through N, N depends on this period s productivity shock. Demographers have argued that baby booms are a catching-up phenomenon partly due to previously low fertility and hence a low current population stock. This is an important qualitative property of the model. That is, the central idea behind the notion that the baby boom is catching up is that the size of the dynasty is too small (relative to trend) at the end of WWII because of the baby bust. Thus, fertility is increased so as to bring the size of the stock up to its desired level. However, due to this result, it follows that fertility in the model does not depend on the size of the dynasty, but only on the shock. Thus, this kind of model can never exhibit catch up fertility of this type. In formulations with land or capital this result may well be altered. Results may depend on whether additional inputs are accumulable or not. The idea, though, is different from the one advanced by most demographers. There will be catching up, not because of previously low fertility rates per se, but because of off balanced growth dynamics in land-to-labour or capital-to-people ratios. If the stock of population or labour is too low relative to the stock of the additional input, labour productivity and wages increase which increases investment in the stock of people that is, fertility An aside on trend growth in productivity In this section we add to the previous analysis trend growth in productivity. This is assumed to be exogenous. Thus, we will study solutions to maximization problems of the form: P (γ, β; N 0, s 0 ) Max {Ct,N t} U 0 ({C t, N t }) = E 0 t=0 βt N η+σ 1 t subject to: C t + γ t θ t N t+1 N t γ t (w t + θ t π), N 0 given. We assume that γ 1 is an exogenous constant. ] Ct 1 σ 1 σ Note that a sequence {C t, N t } is feasible for this problem if and only if it satisfies: C t + θ γ t t N t+1 N t (w t + θ t π) 21

22 Thus, defining Ĉt = C t γ t, this Problem can be written equivalently as: P (γ, β; N 0, s 0 ) Max { Ĉ t,n t} U 0 ({Ĉt, N t }) = E 0 t=0 βt (γ 1 σ ) t N η+σ 1 t subject to: Ĉ t + θ t N t+1 N t (w t + θ t π), N 0 given. Note that this problem is the same as: P (1, ˆβ; N 0, s 0 ) Max { Ĉ t,n t} U 0 ({Ĉt, N t }) = E 0 t=0 ˆβ t N η+σ 1 t subject to: Ĉ t + θ t N t+1 N t (w t + θ t π), N 0 given. where ˆβ = βγ 1 σ. Note that this problem has no exogenous growth in it. Thus, we have: ] Ĉt 1 σ 1 σ ] Ĉt 1 σ 1 σ Claim 1. If {(Ĉ t, N t )} solves the problem P (1, ˆβ; N 0, s 0 ) for some (1, ˆβ; N 0, s 0 ), then {(γ t Ĉ t, N t )} is the solution to the problem P (γ, ˆβ/γ 1 σ ; N 0, s 0 ). Thus, to solve the problem with exogenously growing TFP, solve the one with γ = 1 and ˆβ = βγ 1 σ, and then multiply the C sequence by γ t. It follows that if C is constant (or N converges to a constant) in the solution to the problem with no growth, then, C grows at N rate γ in the one with exogenous growth. Because of this result, we will abstract from trend growth through most of the remainder of the paper. In those cases where the solution to the model depends on the discount factor, we will use this result to calibrate to the appropriate discount factor in the detrended model The stochastic Ak analogy In this section, we will specialize the model outlined above even further by assuming that η +σ = 1 and that the {s t } are i.i.d. There are several simplifications that occur under these assumptions. These are: 22

23 1. As noted above, in this case, the value function is homogeneous of degree 1 σ (since η = 1 σ) in N 0, V (λn 0, s 0 ) = λ 1 σ V (N 0, s 0 ). 2. Define D(s) E V (1, s ) s] for what comes below to simplify notation. Since the shocks are i.i.d., it follows that D(s) = E V (1, s ) s] = E V (1, s )] = D. 3. (F OC1) from Bellman s Equation (given above) simplifies to: ] 1/σ (F OC1) N βηd = θ(s) C. Furthermore, throughout this section, we will assume that: w(s) = A(s) = As, where the s are i.i.d. with E(s) = 1. Finally, we will consider two extreme cases for the form of θ(s). In the first, we assume that only goods are needed to raise children, with θ(s) a. In the second we assume that only time is used, θ(s) = bas. A. Goods cost only (θ t = a) Here we assume that all costs of raising children can be summarized as a time invariant cost stated in terms of the consumption good. In this case an analytic solution to the Planner s Problem can be given. It is summarized in: Proposition 8. Suppose θ t = a and assume that the shocks are i.i.d.. Then the problem has an analytical solution given by: { C = ϕ (As + aπ) N N = (1 ϕ) (As + aπ) N a where: ϕ = 1 1+a σ 1 σ (E(V (1,s ))β(1 σ)) σ 1 Both, the fertility rate and per capita consumption are pro-cyclical. Proof. See Appendix.. Note that ϕ and B depend on A, a and E (As + aπ) 1 σ]. Using the characterization in Proposition 8, it follows that fertility today (children per fertile person) is given by: n bt = N bt λ t N t = (1 ϕ)(as t+aπ) λ t π a λ t = (1 ϕ)a λ t s a t ϕπ λ t, and, c t = C t N t = ϕ (As t + aπ) = ϕas t + ϕaπ. 23

24 These depend on today s shock only. Thus, both the fertility rate and per capita consumption follow TFP movements procyclically in this case. B. Time cost only (θ t = ba(s t )) In this section, we switch to the other extreme, and assume that all child-rearing costs are in terms of time for the parents. That is, θ t = ba(s t ) = bas t. Under this assumption, the Planner s Problem has an analytic solution as summarized in the following proposition. Proposition 9. Assume that θ t = ba(s t ) = bas t and that the shocks are i.i.d.. Then the problem has an analytical solution given by: { C = ϕ(s) (1 + bπ) A(s)N N = (1 ϕ(s)) (1 + bπ) N b 1 where: ϕ(s) = 1+(bAs) 1 1/σ β(1 σ)ev (1,s )]] 1/σ ]. It follows that N is increasing in s if σ > 1 and decreasing in s if σ < 1. N Moreover C is increasing in s if N σ < 1 or σ > 1 and s > 1 1/σ and decreasing in s otherwise. Proof. See Appendix. Notice that if σ < 1, then E V (1, s )] > 0 and 1 1/σ < 0 and so s 1 1/σ is decreasing in s. It follows that in this case, ϕ(s) is increasing in s. In this case then, it follows immediately that N (N,s) N = 1 ϕ(s) b (1 + bπ) is decreasing in s, i.e. fertility is counter-cyclical, while C(N,s) N = ϕ(s) (1 + bπ) A(s) is increasing in s, i.e. per capita consumption is pro-cyclical. In the opposite case, if σ > 1, then E V (1, s )] < 0 and 1 1/σ > 0 and so s 1 1/σ is increasing in s. It follows that in this case, ϕ(s) is decreasing in s and hence N /N is increasing in s. Taking derivatives, one can show that C/N is increasing in s if s > 1 1/σ and decreasing in s otherwise. Hence, in the case where all costs of children are in terms of time, fertility and consumption move in opposite directions, except in the case where σ is relatively high (consumption smoothing is a priority) while the shock is particularly low. This is the only instance where consumption is not pro-cyclical. 24