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Explaining International Fertility Diﬀerences Rodolfo E. Manuelli and Ananth Seshadri Department of Economics University of Wisconsin-Madison February 2006 Very Preliminary - Please no not cite Abstract Why do fertility rates vary so much across countries? Why are European fertility rates so much lower than American fertility rates? To answer these questions we extend the Barro-Becker framework to incorporate the full life-cycle of the individual’s problem. The model is rich enough to generate predictions on years of schooling (a proxy for the quality of children) and the number of children. We arrive at two conclusions. First, diﬀerences across countries in TFP and life expectancies are able to capture the wide variation in fertility rates across the world income distribution. Second, the diﬀerential evolution of labor income taxes between the US and Europe over the last few decades can account for the lower fertility rates in Europe. 1 9 8 7 Total Fertility Rate 6 5 4 3 2 1 0 0 5000 10000 15000 20000 25000 30000 35000 40000 GDP per capita Figure 1: Income and Fertility, 2000 1 Introduction The Question: Fertility rates vary considerably across countries. While the average family in the U.S. has 2.1 children, • the average family in Niger has 7.4 children and, • the average European family (check) has 1.6 children. Our main objective in this study is to understand what role economic forces play in the decisions of the typical Niger and European families 2 The Motivation: Why study international fertility diﬀerences? Diﬀer- ences in fertility rates have a very large impact on output per worker. Our previous work (Manuelli & Seshadri, 2005) suggests that if countries in the bottom decile of the world income distribution were ‘endowed’ with the US demographics, output per worker in these poor countries would more than double. Consequently understanding these diﬀerences in fertility can further our understanding of the international income diﬀerences. Moreover, this work complements the large literature that attempts to explain the demo- graphic transition. The Methodology: There are two general features shared by most papers that endogenize fertility. First, it is common to assume a two or three period overlapping generations set-up in which parents care about the quantity and quality of children. We depart from this way of modeling by incorporating the full life-cycle of the individual’s utility maximization. As we will make clear, this helps to get a rich set of predictions and makes it possible to analyze the eﬀects of changes in life expectancies on the fertility rate. More important, we will argue that the explicit consideration of the life cycle elements are crucial in trying to understand these dramatic diﬀerences in birth rates across countries. Furthermore, adding the full life-cycle problem of the individual also allows us to look at the joint predictions of the model for fertility rates and years of schooling (which proxies for the quality of a child). This permits a direct test of the quantity quality trade-oﬀ. Second, we depart from the literature in the way we model human cap- ital. The seminal contribution by Becker, Murphy and Tamura (1990) uses a human capital production function in which the return to human capi- 3 9 8 7 Total Fertility Rate 6 5 4 3 2 1 0 40 45 50 55 60 65 70 75 80 Life Expectancy at age 1 Figure 2: Life Expectancy and Fertility, 2000 4 14 12 10 Years of Schooling 8 6 4 2 0 0 2 4 6 8 10 Total Fertility Rate Figure 3: The Quantity-Quality Trade-oﬀ, 2000 tal increases with the stock of human capital. Lucas (2002) takes a similar approach and assigns an important role to endogenous human capital ac- cumulation. Both papers view shifts in the production function for human capital as the underlying reason for the long term fertility decline. Finally, Galor (2005) uses a human capital production function wherein the growth rate of human capital is an input in the production of human capital. In con- trast to these approaches we follow Ben-Porath (1967) and Mincer (1974) in formulating the human capital accumulation decision over the course of the life-cycle in order to explain the joint behavior of the quantity and quality of children across countries. 5 The Mechanism: The exogenous sources of variation are three-fold –life expectancies, TFP and taxes. It is instructive to analyze the impact of each of these in the context of our model. a. Changing Life Expectancies: When life expectancy rises –holding re- tirement age constant– years in retirement also go up. This leads individuals to save more for retirement. Hence, the capital stock rises relative to GDP, thereby leading to a decline in the real interest rate and consequently the fertility rate. Thus, all else equal, increasing the number of years in retire- ment, reduces fertility. Further, even if increases in life expectancy increase the eﬀective working life span of the individual, we ﬁnd that the aggregate stock of physical capital rises by more than the aggregate stock of human capital. Hence fertility falls. b. Changing TFP: When total factor productivity goes up, wages rise. This leads parents to invest more in the human capital of their progeny (and in their own human capital). This increases the marginal cost of having children relative to consumption. Consequently fertility declines. There other eﬀects on the optimal investment, but we ﬁnd that those are quantitatively smaller than the wage eﬀect. Thus our framework is able to capture the quantity-quality trade-oﬀ despite having decreasing returns to scale in the production of human capital. c. Changing Taxes: When the tax rate on labor income goes up, individu- als decrease their investments in human capital. Thereby the capital-output ratio rises. Consequently, fertility declines. In comparing the rich and poor nations, TFP is higher in the US relative to the poor nations and so is life expectancy. These two eﬀects alone ac- 6 count for the large fertility diﬀerential between the United States and poorer nations. Furthermore, taxes on labor income in Europe are higher than in the United States. This leads to a lower fertility rate in Europe (as well as the lower schooling level). Quantitatively, we show that these three forces combined, explain international diﬀerences in fertility rates. 2 Economic Environment: In this section we describe the basic model. We present an economic environ- ment with imperfect altruism and we show that, under some conditions, the solution to the utility maximization problem is identical to the solution of an income maximization problem. We then compute the aggregate variables in this economy using the exogenously speciﬁed demographic structure. 2.1 The Individual Household Problem The representative household is formed at age I (age of independence). At age B, ef children are born. The period of ‘early childhood’ (deﬁned by the assumption that children are not productive during this period) corresponds to the (parent) age B to B +6. The children remain with the household (and as such make no decisions of their own) until they become independent at (parent) age B + I. The parent retires at age R, and dies at age T . Let a denote an individual’s age. Each parent chooses his own consump- tion, c(a), as well as consumption of each of his children, ck (a), during the years that they are part of his household, a ∈ [a, I), to maximize his utility. We adopt the standard Barro-Becker approach, and we specify that parent’s 7 utility depends on his own consumption, as well as the utility of his chil- dren. In addition to consumption, the parent chooses the amount of market goods to be used in the production of new human capital, x(a), and the fraction of the time allocated to the formation of human capital, n(a) (and, consequently, what fraction of the available time to allocate to working in the market, 1 − n(a)) for him and each of his children while they are still attached to his household. The parent also decides to make investments in early childhood, which we denote by xE (e.g. medical care, nutrition and development of learning skills), that determine the level of each child’s hu- man capital at age 6, hk (6), or hE for short. Finally, the parent chooses how much to bequeath to each children at the time they leave the household, bk . We assume that each parent has unrestricted access to capital markets, but that he cannot commit his children to honor his debts. Thus, we restrict bk to be non-negative. The utility function of a parent who has h units of human capital, and a bequest equal to b at age I is given by Z T Z I P −ρ(a−I) −α0 +α1 f V (h, b) = e u(c(a))da+ e e−ρ(a+B−I) u(ck (a))da I 0 +e−α0 +α1 f e−ρB V k (hk (B + I), bk ) Thus, the contribution to the parent’s utility of an a year old child still attached to him is e−α0 +α1 f e−ρ(a+B−I) u(ck (a)), since at that time the parent is a + B years old. In this formulation, e−α0 +α1 f captures the degree of altruism. If α0 = 0, and α1 = 1, this is a standard inﬁnitively-lived agent model. Positive values of α0 , and values of α1 less than 1 capture the degree of imperfect altruism. The term V k (hk (I), bk ) is the utility of a child at the time he becomes independent. 8 Each parent maximizes V P (h, b) subject to two types of constraints: the budget constraint, and the production function of human capital. The former is given by Z T Z I Z R −r(a−I) f −r(a+B−I) e c(a)da + e e ck (a)da + e−r(a−I) x(a)da + (1) I 0 I Z I f e e−r(a+B−I) xk (a)da + ef e−rB bk + ef e−r(B+6−I) xE 6 Z R Z I −r(a−I) f ≤ e wh(a)(1 − n(a))da + e e−r(a+B−I) [whk (a)(1 − nk (a))]da + b. I 6 We adopt Ben-Porath’s (1967) formulation of the human capital production technology, augmented with an early childhood period. We assume that ˙ h(a) = zh [n(a)h(a)]γ 1 x(a)γ 2 − δ h h(a), a ∈ [I, R) (2) ˙ hk (a) = zh [nk (a)hk (a)]γ 1 xk (a)γ 2 − δ h hk (a), a ∈ [6, I) (3) hk (6) = hB xυ , E (4) h(I) given, 0 < γ i < 1, γ = γ 1 + γ 2 < 1, The technology to produce human capital of each child at the beginning of the potential school years, hk (6) or hE is given by (4). Our formulation cap- tures the idea that nutrition and health care are important determinants of early levels of human capital, and those inputs are, basically, market goods.1 Equation (2) correspond to the standard human capital accumulation model initially developed by Ben-Porath (1967). There are two important features of our formulation. First, we assume that the technology for human capital 1 In section XXXX we extend this formulation to the case in which adult time enters the production function. We do not restrict this time to be the parent’s own time, as we allow for the possibility of hiring qualiﬁed individuals to take care of infants. 9 accumulation is the same during the schooling and the training periods. We resisted the temptation to use a more complicated parameterization so as to force the model to use the same factors to account for the length of the schooling period and the shape of the age-earnings proﬁle. Second, we assume that the market inputs used in the production of human capital –x(a) and xk (a)– are privately purchased. In the case of the post-schooling period, this is not controversial. However, this is less so for the schooling period. Here, we take the ‘purely private’ approach as a ﬁrst pass. In an extension of the basic model we explore the role of public education2 , even though all that is needed for our assumption is that, at the margin, individuals pay for the last unit of market goods allocated to the formation of human capital during the schooling period. In the steady state, it is possible to separate the optimal consumption de- cision from the optimal human capital accumulation decision. In particular, given the discount factor equals the interest rate e−rB = e−[α0 +(1−α1 )f ] e−ρB , or, r = ρ + [α0 + (1 − α1 )f ]/B. (5) it follows that Proposition 1 Assume that r = ρ + [α0 + (1 − α1 )f ]/B, then the solution to the optimal human capital accumulation corresponding to the maximization 2 An alternative explanation is that Tiebout like arguments eﬀectively imply that pub- lic expenditures on education play the same role as private expenditures. The truth is probably somewhere in between. 10 of (??) subject to (1)-(4) is identical to the solution of the following income maximization problem Z R max e−r(a−6) [wh(a)(1 − n(a)) − x(a)]da − xE (6) 6 subject to ˙ h(a) = zh [n(a)h(a)]γ 1 x(a)γ 2 − δh h(a), a ∈ [6, R), (7) and h(6) = hE = hB xυ E (8) with hB given. Proof. : See Appendix B An intuitive (and heuristic) argument that shows the correspondence be- tween the utility maximization and the income maximization problem is as follows: Suppose that parents (who make human capital accumulation deci- sions for their children until age I) do not choose the maximize the present value of income of their children (only part of which they keep). In this case, and since bk > 0, the parent could increase the utility of each child by adopting the income maximizing human capital policy and adjusting the transfer to ﬁnance this change. It follows that the cost to the parent is the same and the child is made better oﬀ. Since the parent appropriates the income generate by child labor, one might wonder if it is not in the best interest of the parent to take the child out of school early and send him to work. However, this cannot be optimal as the parent can choose the optimal –from the point of view of the child– human capital policy and change the 11 bequest as necessary. Since the parent’s income is unchanged and the child is better oﬀ, this results in an increase in the utility of the parent. As our informal discussion suggests, the key ingredient is that the inter- generational no borrowing constraint is not binding. Since this option is, eﬀectively, another technology that the parent can use to transfer wealth to his children, standard arguments show that there will be no distortions. In related work we show that, when the non-negative bequest is binding, this is no longer true. In that case, which requires r > ρ + [α0 + (1 − α1 )f ]/B, the equilibrium human capital choices no longer maximize the present value of income. (See Manuelli and Seshadri (2004).) In the unconstrained case, it is possible to fully characterize the solution to the income maximization problem. The main features of the solution are summarized in Proposition 2 There exists a unique solution to the income maximization problem. The number of years of schooling, s, satisﬁes 1. µ ¶(1−γ)υ Ã γ (1−γ ) !−(1−υ) h1−γ υ γ22 γ1 2 F (s) = 1−υ γB−υ(1−γ ) , (9) zh w 2 1 r + δh r + δh where F (s) ≡ m(6 + s)1−υ(2−γ) e(1−γ)(δh +rυ)s ⎡ γ r+δ h (1−γ ) ⎤ (1−γ)(1−υ(1−γ1 )) 1−γ 1 − 2 (1−γ ) 1 s ⎣1 − r + δ h (1 − γ 1 )(1 − γ 2 ) 1 − e 2 ⎦ , γ 1 γ 2 r + δ h (1 − γ 1 ) m(s + 6) and m(a) = 1 − e−(r+δh )(R−a) , 12 provided that µ ¶(1−γ)υ Ã γ (1−γ ) !−(1−υ) h1−γ υ γ22 γ1 2 m(6)1−υ(2−γ) > 1−υ γB−υ(1−γ ) . zh w 2 1 r + δh r + δh Otherwise the privately optimal level of schooling is 0. 2. The level of human capital at the age at which the individual ﬁnishes his formal schooling is given by ∙ γ2 γ1 ¸ 1 γ 2 γ 1 zh wγ 2 1−γ γ 1 1 h(s + 6) = m(6 + s) 1−γ (10) (r + δh )γ r + δh Proof. : See Appendix B To obtain a more precise characterization of the solution, we assume that the utility function is isoelastic and given by, c1−θ u(c) = . 1−θ The optimal choice of consumption is given by r−ρ (a−I) c(a) = c(I)e θ , a≥I (11a) r−ρ (r−ρ)B−(α0 +(1−α1 )f ) (a−I) ck (a) = c(I)e θ e θ , a ≤ I. (11b) Note that, at the steady state interest rate, the equilibrium consumption r−ρ (a−I) function is just c(a) = c(I)e θ , for all a ∈ [0, T ]. The ﬁrst order condition for the equilibrium choice of fertility is Z I Z I −r(a+B−I) e ck (a)da + e−r(a+B−I) xk (a)da + e−r(B+6−I) xE (12) 0 6 Z I −rB +e bk − e−r(a+B−I) whk (a)(1 − nk (a))da 6 Z I −(α0 +(1−α1 )f ) = α1 e [ e−ρ(a+B−I) u(ck (a))da + e−ρB V k (hk (B + I), bk )]/Φ 0 13 The left hand side represents the cost of having a child and it is given by the sum of its components: consumption, expenditure on goods required to produce human capital and bequests, net of the income produced by each child. The right hand side is simply the utility of each child multiplied by its marginal contribution to the parent’s utility. In this formula, Φ is the Lagrange multiplier corresponding to the budget constraint. Thus, the term on the right is measured in goods. In Manuelli and Seshadri (2005) we showed that the solution to the in- vestment in human capital problem is completely described by µ ¶ γ2w 1 r+δ h (1−γ 1 ) (a−s−6) x(a) = Ch (zh , w, r)m(6 + s) 1−γ e (1−γ2 ) (13a) , a ∈ [6, 6 + s), r + δh µ ¶ γ2w 1 x(a) = Ch (zh , w, r)m(a) 1−γ , a ∈ [6 + s, R). (13b) r + δh " # 1 (1−γ 2 ) γ (1−γ 2 ) γ 1 γ 2 γ 1 (1−γ 1 )(1−γ 2 ) 1−γ γ11 γ 2 zh w m(6 + s) 1−γ xE = υ (13c) (r + δ h )(1−γ 2 ) e(r+δh (1−γ 1 ))s where ∙ ¸ 1 γ γ γ 2 2 γ 1 1 zh wγ 2 1−γ Ch (zh , w, r) = . (r + δ h )γ For a ≥ 6+s net income deﬁned as y(a) = wh(a)(1−n(a))−x(a) satisﬁes 1 −δ h (a−6−s) m(6 + s) 1−γ y(a) = Ch (zh , w, r)w{γ 1 e (14) r + δh 1 Z δ (a−R) m(a) 1−γ e−δh (a−R) e h r+δ h γ −(γ 1 + γ 2 ) + [(1 − x δh ] 1−γ dx}. r + δh δh eδh (6+s−R) while the supply of human capital to the market by an individual of age a is 1 −δ h (a−6−s) m(6 + s) 1−γ h(a)(1 − n(a)) = Ch (zh , w, r)w{γ 1 e (15) r + δh 1 Z eδh (a−R) m(a) 1−γ e−δh (a−R) r+δ h γ −γ 1 + [(1 − x δh ] 1−γ dx} r + δh δh eδh (6+s−R) 14 2.2 Equilibrium Given the individual decision on human capital accumulation and investment as a function of age, all we need is to compute the age structure of the population to determine aggregate human capital. Since the capital-human capital ratio is pinned down by the condition that the marginal product of capital equal the cost of capital, this suﬃces to determine output per worker. Demographics Since we consider only steady states, we need to derive the stationary age distribution of this economy. Let N(a, t) be the number of people of age a at time t. Thus, our assumptions imply N(a, t) = ef N(B, t − a) and N(T, t) = 0. It is easy to check that in the steady state N(a, t) = φ(a)eηt , (16) where e−ηa φ(a) = η , (17) 1 − e−ηT and η = f/B is the growth rate of population. ¯ Aggregation Let h(r, w) be the average (per person) level of human capital ¯ as a function of r and w. Thus, h(r, w) is given by Z R ¯ h= h(a)(1 − n(a))φ(a)da. 6+s 15 Similarly, let x(r, w) be the average (per person) investment in human ¯ capital as a function of r and w. Thus, x(r, w) is given by ¯ Z R x(a)φ(a)da 6+s Equilibrium From (5) it follows that if the bequest constraint is not bind- ing, the interest rate is given by α0 f r =ρ+ + (1 − α1 ) . (18) B B Optimization on the part of ﬁrms implies that pk (r + δ k ) = zFk (κ, 1), (19) where κ is the physical capital - human capital ratio. The wage rate per unit of human capital, w, is, w = zFh (κ, 1). (20) Aggregate output and consumption per person satisfy Z T ¯ c(a)φ(a)da = [zF (κ, 1) − (δ k + η)κpk ]h(r, w) − xE φ(6) − x(r, w). (21) ¯ 0 For this to be an equilibrium, we need to verify that, at the candidate solution, b > 0. In order to better characterize the equilibrium conditions deﬁne, for any functions g and K (a kernel) the operator P as Z y P (g; x, y; K) ≡ g(a)K(a)da. (22) x For our purposes, it suﬃces to use two diﬀerent kernels: K = r = e−r(a−I) ˆ and K = φ(a). The ﬁrst one corresponds to the present discounted value 16 using the interest rate, while the second aggregates values using population weights. Using this notation, and imposing bk = b in the budget constraint (1), it follows that b = (1 − e−rB+f )−1 {P (c; I, T ; r) − P (y; I, R; r) + ˆ ˆ (23) e−rB+f [P (c; 0, I; r) + P (x; 6, 6 + s; r) + e−r(6−I) xE − P (y; 6 + s, I; r)]}. ˆ ˆ ˆ Since the marginal cost of an additional child (the left side of (12)) is given by e−rB {P (c; 0, I; r) + P (x; 6, 6 + s; r) + e−r(6−I) xE − P (y; 6 + s, I; r) + b}, ˆ ˆ ˆ it follows that the marginal cost of a child is e−rB {P (c; 0, I; r) + P (x; 6, 6 + s; r) + e−r(6−I) xE − P (y; 6 + s, I; r) ˆ ˆ ˆ 1 − e−rB+f +P (c; I, T ; r) − P (y; I, R; r)} ˆ ˆ which simpliﬁes to e−rB {P (c; 0, T ; r) + P (x; 6, 6 + s; r) + e−r(6−I) xE − P (y; 6 + s, R; r)}. ˆ ˆ ˆ 1 − e−rB+f (24) Thus, the equilibrium marginal cost of a child is proportional to the life- time diﬀerence between the present discounted value of expenditures (con- sumption and expenditures related to human capital) minus income. The right hand side of (12) gives the beneﬁt of an additional child. At the steady state (ignoring the dependence of V on h and b) we obtain that U P + e−α0 +α1 f U k V = , 1 − e−ρB e−α0 +α1 f 17 where Z T P U = e−ρ(a−I) u(c(a))da I Z I k U = e−ρ(a+B−I) u(ck (a))da. 0 Thus, the beneﬁt from another child is e−(α0 +(1−α1 )f ) α1 [U k + e−ρB U P ]/Φ. 1 − e−ρB e−α0 +α1 f The term [U k + e−ρB U P ]/Φ measures the lifetime utility of the child in consumption units (i.e. divided by the marginal utility of consumption). Using the speciﬁc utility function and the equilibrium function c(a) the marginal beneﬁt is e−ρB e−(α0 +(1−α1 )f ) θ c(I)1−θ [(r−ρ)B−(α0 +(1−α1 )f )] 1−θ α1 {e θ 1 − e−ρB e−α0 +α1 f ρ − (1 − θ)r 1 − θ ρ−(1−θ)r ρ−(1−θ)r ρ−(1−θ)r I e θ (1 − e− θ I ) + 1 − e− θ (T −I) }/Φ. Since the marginal utility of wealth satisﬁes Φ = c(I)−θ , (12) is equivalent to e−rB {P (c; 0, T ; r) + P (x; 6, 6 + s; r) + e−r(6−I) xE − P (y; 6 + s, R; r)} ˆ ˆ (25) ˆ 1 − e−rB+f e−ρB e−(α0 +(1−α1 )f ) θ c(I) [(r−ρ)B−(α0 +(1−α1 )f )] 1−θ = α1 {e θ 1 − e−ρB e−α0 +α1 f ρ − (1 − θ)r 1 − θ ρ−(1−θ)r ρ−(1−θ)r ρ−(1−θ)r I e θ (1 − e− θ I ) + 1 − e− θ (T −I) }. In this expression we have not imposed (18) so as to be able to separate the impact of changes in fertility on costs and beneﬁts from the point of view of the individual, and the general equilibrium eﬀects.3 3 This is not quite right, as the derivation of V assumes that a steady state exists and this, in turn, requires that the steady state be given by (18) 18 The feasibility constraint can be written as P (c; 0, T ; φ) + P (x; 6, 6 + s; φ) + φ(6)xE (26) µ ¶ 1−α 1 αz = P (y; 6 + s, R, φ) + (r − η) P (h(1 − n); 6 + s, R; φ). r + δk The expressions (25) and (26) depend on (c(I), f, s, r, w). If one uses (18), (19), (20) to eliminate (r, w), from (9), (25) and (26), we are left with three equations and three unknowns. It is possible to get a better picture of what is going on by (selectively) imposing the steady state condition r = ρ + α0 + (1 − α1 )η. This implies that B −rB + f = −ρB − α0 + α1 f , and (25) simpliﬁes to P (c; 0, T ; r) + P (x; 6, 6 + s; r) + e−r(6−I) xE − P (y; 6 + s, R; r) (27) ˆ ˆ ˆ θ α1 ρ−(1−θ)r I ρ−(1−θ)r = c(I) e θ [1 − e− θ T ]. ρ − (1 − θ)r 1 − θ The left hand side of (27) is the cost of an additional child. After this simpliﬁcation it is given precisely by the diﬀerence between the present dis- counted value of expenditure minus income over the lifetime of an individual. The right side gives the beneﬁt which is proportional to initial consumption at independence (c(I)). [Add more analysis of how f inﬂuences both sides). At the steady state θ ρ−(1−θ)r ρ−(1−θ)r P (c; 0, T ; r) = c(I) ˆ e θ I [1 − e− θ T ]. ρ − (1 − θ)r and the condition for the optimal choice of the number of children simpliﬁes to P (x; 6, 6 + s; r) + e−r(6−I) xE − P (y; 6 + s, R; r) ˆ ˆ (28) α1 + θ − 1 θ ρ−(1−θ)r ρ−(1−θ)r = c(I) e θ I [1 − e− θ T ]. 1 − θ ρ − (1 − θ)r 19 This last equation shows the role played by preferences over consumption and children. If individuals care suﬃciently about their oﬀspring (α1 + θ − 1 > 0), their human capital choices will be such that the net expenditure on children (early childhood and schooling) exceeds what they receive in the form of child labor. If, on the other hand, α1 + θ − 1 < 0 then the present value of what parents receive from their children exceeds the out of pocket costs of producing human capital. [Add some discussion. The term P (x; 6, 6 + s; r) + e−r(6−I) xE − P (y; 6 + s, R; r) is messy. At the steady state ρ−r ηe θ I θ ρ−(r−θη) P (c; 0, T ; φ) = c(I) −ηT ρ − (r − θη) [1 − e− θ T ]. 1−e Then, feasibility implies that ρ−r ηe θ I θ ρ−(r−θη) c(I) −ηT ρ − (r − θη) [1 − e− θ T ] (29) 1−e µ ¶ 1−α 1 αz = P (y; 6 + s, R, φ) + (r − η) P (h(1 − n); 6 + s, R; φ) r + δk −P (x; 6, 6 + s; φ) + φ(6)xE . 3 Calibration We use standard functional forms for the utility function and the ﬁnal goods production function. The utility function is assumed to be of the CRRA variety c1−θ u(c) = , 0 < θ < 1. 1−θ The production function is assumed to be Cobb-Douglas F (k, h) = zK α H 1−α . 20 Our calibration strategy involves choosing the parameters so that the steady state implications of the model economy presented above is consistent with observations for the United States (circa 2000). Thus, we calibrate the model to account for contemporaneous observations in the U.S. We then vary the exogenous demographic variables and choose the level of TFP for other coun- tries so that the model’s predictions for output per worker match that for the chosen country. Consequently, while output per worker for other coun- tries are chosen so as to match output per worker by construction, the model makes predictions on years of schooling, age earnings proﬁles and the amount of goods inputs used in the production of human capital. There are some parameters that are standard in the macro literature. Thus, following Cooley and Prescott (1995), the discount factor is set at ρ = 0.96 and the depreciation rate is set at δ k = .06. Capital’s share of income is set at 0.33. Less information is available on the fraction of job training expenditures that are not reﬂected in wages. There are many reasons why earnings ought not to be equated with wh(1 − n) − x. First, some part of the training is oﬀ the job and directly paid for by the individual. Second, ﬁrms typically obtain a tax break on the expenditures incurred on training. Consequently, the government (and indirectly, the individual through higher taxes) pays for the training and this component is not reﬂected in wages. Third, some of the training may be ﬁrm speciﬁc, in which case the employer is likely to bear the cost of the training, since the employer beneﬁts more than the individual does through the incidence of such training. Finally, there is probably some smoothing of wage receipts in the data and consequently, the individual’s marginal productivity proﬁle is likely to be steeper than the 21 individual’s wage proﬁle. For all these reasons, we equate earnings with wh(1 − n) − πx and set π = 0.5. The parameter α1 determines the degree of curvature in the altruism function of the individual. Note that this also determines the real interest rate. We proceed by choosing the level of α1 in the United States so as to match a fertility rate of 2.1. Finally, we assume that B = 25. Our theory implies that it is only the ratio h1−γ /(zh wγ 2 −υ(1−γ 1 ) ) that B 1−υ matters for the moments of interest. Consequently, we can choose z, pk (which determine w) and hB arbitrarily and calibrate zh to match a desired moment. The calibrated value of zh is common to all countries. Thus, the model does not assume any cross-country diﬀerences in an individual’s ‘abil- ity to learn.’ This leaves us with 8 parameters, α0 , δ h , zh , γ 1 , γ 2 , υ, α1 and θ. The moments we seek in order to pin down these parameters are: 1. Earnings at age R/Earnings at age 55 of 0.8. Source: SSA 2. Earnings at age 50/Earnings at age 25 of 2.17. Source: SSA 3. Years of schooling of 12.08. Source: Barro and Lee 4. Schooling expenditures per pupil (primary and secondary) relative to GDP per capita of 0.214. Source: OECD, Education at a Glance, 2003. 5. Pre-primary expenditures per pupil relative to GDP per capita of 0.14. Source: OECD, Education at a Glance, 2003. 6. Fertility Rate of 2.1. Source: UNDP 22 7. Lifetime Intergenerational Transfers/GDP of 4.5%. Gale and Scholz, 1994 8. Capital output ratio of 2.52. Source: NIPA Theory implies that when bequests are in the interior, the human capital allocations that result from the solution to the parents problem correspond to the allocations that result from the simpler income maximization problem. Consequently, proceed in two steps since the 8 equations in 8 unknowns are ‘block-separable’. For a given real interest rate and the wage rate, we calibrate the parameters δ h , zh , γ 1 , γ 2 and ν so as to match moments 1, 2, 3, 4 and 5. Thus, we use the properties of the age-earnings proﬁle to identify the parameters of the production function of human capital. This, of course, follows a standard tradition in labor economics. We then choose the other three parameters so as to match moments 6,7 and 8. The calibration requires us to solve a system of 8 equations in 8 unknowns and we obtained a perfect match. The resulting parameter values are Parameter α0 δh zh γ1 γ2 ν α1 θ Value 0.23 0.018 0.361 0.63 0.3 0.55 0.52 0.62 Of some interest are our estimates of α0 and γ i . Since the ﬁrst one is positive, it implies that agents are imperfectly altruistic. Our estimate of γ 2 is fairly large, and indicates that, in order for the model to be consistent with both average schooling in the U.S. as well as the pattern of the age-earnings in the data, market goods have to enter in the production function of human capital. 23 4 Results Before turning to the results, we ﬁrst describe the data so as to get a feel for the observations of interest. We start with the countries in the PWT 6.1 and put them in deciles according to their output per worker, y. Next, we combine them with observations on years of schooling (s), expenditures per pupil relative to output per worker (xs ), life expectancy (T ), and the total fertility rate (f ) for each of these deciles. The population values are displayed in the following table. Table 1: World Distribution Relative output Schooling Life Exp Fertility Years Expenditures TFR*(1-inf) y Decile yU S s xs T f 90-100 0.921 10.93 3.8 78 1.74 80-90 0.852 9.94 4.0 76 2.1 70-80 0.756 9.72 4.3 73 2.28 60-70 0.660 8.70 3.8 71 2.50 50-60 0.537 8.12 3.1 69 2.82 40-50 0.437 7.54 2.9 64 3.37 30-40 0.354 5.88 3.1 57 3.92 20-30 0.244 5.18 2.7 54 4.76 10-20 0.146 4.64 2.5 51 5.32 0-10 0.052 2.45 2.8 46 5.66 Table 1 illustrates the wide disparities in incomes across countries. The United States possesses an output per worker that is about 20 times as high 24 as countries in the bottom decile. Further notice that years of schooling also varies systematically with the level of income – from about 2 years at the bottom deciles to about 11 at the top. The quality of education as proxied by expenditures on primary and secondary schooling as a fraction of GDP also seems to increase with the level of development. This measure should be viewed with a little caution as it includes only public inputs and not private inputs (including the time and resources that parents invest in their kids). Next, notice that demographic variables also vary systematically with the level of development - higher income countries enjoy greater life expectancies and lower fertility rates. More important, while demographics vary substantially at the lower half of the income distribution, they do not move much in the top half. 4.1 Accounting for International Diﬀerences in Fertil- ity We now examine the ability of the model to simultaneously match the cross country variation in output per capita and years of schooling. To be clear, we change R (retirement age) and T (life expectancy) across countries and choose the level of TFP in a particular country so as to match output per worker.4 We then see if the predictions for the fertility rate are in accordance with the data. 4 We assume that R = min{64, T }. 25 Table 2: Fertility and Schooling - Data and Model y Decile yU S TFP s xs f Data Model Data Model Data Model 90-100 0.921 0.99 10.93 11 3.8 3.7 1.74 2.10 80-90 0.852 0.98 9.94 10 4.0 3.8 2.1 2.32 70-80 0.756 0.96 9.72 10 4.3 3.9 2.28 2.46 60-70 0.660 0.94 8.70 8 3.8 4.1 2.50 2.73 50-60 0.537 0.91 8.12 7 3.1 4.5 2.82 2.96 40-50 0.437 0.88 7.54 6 2.9 4.1 3.37 3.51 30-40 0.354 0.85 5.88 5 3.1 3.5 3.92 3.86 20-30 0.244 0.82 5.18 4 2.7 3.1 4.76 4.38 10-20 0.146 0.79 4.64 2 2.5 2.4 5.32 5.12 0-10 0.052 0.68 2.45 1 2.8 1.8 5.66 5.76 26 Table 2 presents the predictions of the model and the data. The model is able to capture reasonably well the variation across countries in the quantity of children as captured by the fertility rate and the quality of children, as captured by years of schooling. As we move from the bottom to the top decile of the world income distribution, fertility in the model rises from 6.76 to 2.14 which compares very favorably with that observed in the data. Furthermore, the model also captures the variation in schooling quantity and quality across countries. To understand the ability of the model to match these observations, it is instructive to examine the eﬀect of each of the driving forces in isolation. Changes in TFP: A rise in TFP increases marginal beneﬁts and marginal costs. With suﬃcient curvature in the altruism function, beneﬁts at the margin rise by less than the costs thereby leading to a decline in fertility. This is perhaps the single most important eﬀect quantitatively. Notice that unlike Becker, Murphy and Tamura, the argument here does not rely on increasing returns to scale in the human capital production function. Changes in Life Expectancy: An increase in T , holding the retirement age R constant, also results in a decrease in fertility. Even though changes in T aﬀect several terms, we ﬁnd that, quantitatively, the positive impact on savings for retirement dominates. This in turn, results in an increase in the capital output ratio, thereby decreasing the interest rate and consequently the fertility rate. Changes in the Retirement Age: An increase in R holding ﬁxed life expectancy (whenever possible) in- 27 creases fertility. While this may seem backwards at a ﬁrst pass, the intuition is identical to the previous case. A rise in R, holding ﬁxed T , reduces the number of years in retirement and consequently reduces the capital output ratio. This increases the interest rate and thereby the fertility rate. Changes in Life Expectancy & the Retirement Age: An increase in T with a corresponding increase in retirement age de- creases fertility. This eﬀect is in parallel to a rise in TFP - the eﬀect of an increase in T and R is to promote human capital accumulation, but physical capital accumulation rises by even more. There are two eﬀects at play - ﬁrst, steeper earnings proﬁles lead to more life cycle savings on the part of the par- ent and second, the increase in the stock of human capital also prompts the parent to invest more in the complementary physical capital. For our para- meterization we ﬁnd that the capital output ratio rises and fertility declines. Thus, increases in life expectancy reduce fertility regardless of the elasticity of retirement age with respect to life expectancy. 4.2 Contribution of changing life expectancies to in- ternational fertility diﬀerences One of the main features of the model is the explicit consideration of the life-cycle of the individual. Here we attempt to demonstrate that changing life expectancies have a substantial eﬀect on fertility rates. Recall that in our cross-country analysis presented above there are only two sources of variation - life expectancies and TFP. Now imagine that all countries had the same TFP as the countries in the bottom decile of the world income distribution and consider the eﬀects of changing life expectancies alone. Table 28 3 displays the results in the columns labeled "Life Exp.". Notice when only T varies, the model generates the variation from 6.76 in the lowest decile to about 4.04 - a large eﬀect. Thus the consideration of the full life-cycle of the individual’s decision problem contributes signiﬁcantly to explaining international diﬀerences in fertility rates. Table 3: The Eﬀect of Life Expectancy on Fertility - Model Decile T s f Baseline Life Exp. Baseline Life Exp. 90-100 78 11 7 2.10 4.03 80-90 76 10 6 2.32 4.12 70-80 73 10 6 2.46 4.26 60-70 71 8 5 2.73 4.42 50-60 69 7 4 2.96 4.58 40-50 64 6 4 3.51 3.96 30-40 57 5 3 3.86 4.34 20-30 54 4 2 4.38 4.96 10-20 51 2 1 5.12 5.48 0-10 46 1 1 5.76 5.76 29 5 Accounting for US-Europe Fertility Diﬀer- ences The previous section demonstrated the ability of the model to capture the variation in fertility rates across the diﬀerent stages of development. Despite the model’s ability to capture the variation in fertility rates across the world distribution of income, there is one glaring failure - the inability to capture the low fertility rate observed in many European countries. Indeed this feature of the data has been puzzling - why would the US and European nations, which presumably are at similar stages of economic development, have dramatically diﬀerent fertility rates? In this section we examine the ability of the model to generate such diﬀerential behavior in fertility rates using diﬀerences in tax rates on labor income as a way of explaining these diﬀerences. Figure 4 shows the marked divergence in the fertility rates of the United States and the European nations starting around 1976.5 While American fer- tility rates increased by more than 17% over the next two decades, European fertility rates fell by a little more than 11%. At the same time, while taxes on labor income in the United States virtually stayed constant, tax rates on labor income in most European countries as well as Japan and Canada went up. Prescott (2003) argues that these higher taxes explain the lower hours worked in Europe relative to the US. Davis and Henrekson (2004) present evidence in support of the negative eﬀect of taxes on labor supply. Table 4 5 The European nations included in Figure 4 are Finland, France, Germany, Hungary, Italy, Netherlands, Norway, Portugal, Sweden, Switzerland and the UK. 30 2.3 2.2 2.1 USA Total Fertility Rate 2 1.9 1.8 1.7 Europe 1.6 1.5 1971 1976 1981 1986 1991 1996 2001 Year Figure 4: US-Europe Fertility Diﬀerences, 1970-1995 31 presents data on tax rates (as reported in Prescott, 2003) and total fertil- ity rates. The remarkable aspect of the Table is the degree to which the % change in tax rates and the % change in fertility rates are negatively related. Table 4: Taxes and Fertility, G7 - Data Country Tax rate on labor income Total Fertility Rate GDP per worker 1975 1995 % change 1975 1995 % change 1975 1995 % change Germany 0.52 0.59 13 1.53 1.33 -13 0.76 1.07 41 France 0.49 0.59 20 2.28 1.68 -26 0.81 1.13 39 Italy 0.41 0.64 56 2.33 1.19 -49 0.81 1.27 57 Canada 0.44 0.52 18 1.97 1.64 -17 0.93 1.13 22 U.K 0.45 0.44 -2 1.79 1.73 -3 0.71 1.00 42 Japan 0.25 0.37 48 2.17 1.42 -35 0.53 0.92 75 U.S.A 0.40 0.40 0 1.92 2.04 6 1.00 1.41 41 To formalize the eﬀects of taxes on fertility, imagine adding a tax on labor income (τ h ) and capital income (τ k ) into the baseline model. The eﬀective e e prices that the consumer faces are r = r(1 − τ k ) and w = w(1 − τ h ). Further, assume that the revenues from these taxes are rebated back to individuals in a lump-sum fashion so that an individual receives exactly the same amount L regardless of his age. The government’s budget constraint then reads X R τ k rK + τ h wH = µi L. i=I Since the capital income tax rate doesn’t vary much across these countries, we hold τ k ﬁxed in what ensues. In order to re-calibrate the model to match the targets for the United States, we set τ k = 0.3 and τ h = 0.4.6 The parameter 6 Our quantitative results do not hinge on τ k = 0.3. 32 values change slightly. Now, imagine changing the tax rate on labor income and solving for the new steady state. Table 5: Eﬀect of Taxes of Fertility - Model τh f 0.40 2.10 0.45 1.83 0.50 1.65 0.55 1.49 0.60 1.31 Table 5 indicates that taxes have a powerful eﬀect on fertility. What happens when τ h rises? An increase in τ h reduces the eﬀective wage rate thereby leading to a reduction in human capital investment. Hence, the marginal cost of giving birth to children declines. However, the reduced wage rate, also implies lower consumption for the parent. A rise in the tax rate, increases marginal cost relative to consumption and consequently fertility declines. An alternative way to see this is to think about the impact of taxes on aggregate physical and human capital. When τ h rises, the stock of human capital falls. Furthermore, since the proceeds are rebated back to the consumer in a lump-sum fashion, the individual does not have much of a need to access the capital market in order to smooth the receipts across his life-cycle.7 Consequently, the stock of physical capital falls by less than that of human capital. This implies that the capital output rises and the real interest rate falls. Consequently, the fertility rate must also fall. 7 This is especially true since β(1 + r) which is the term that aﬀects the slope of the consumption proﬁle is pretty close to 1 in our baseline. 33 5.1 The Eﬀect of Social Security on Fertility The analysis above illustrates the rather powerful eﬀect that taxes have on fertility rates. Recall that the proceeds of the taxes were re-distributed back to consumers in a lump-sum fashion regardless of their age. In this section, we examine the implications of redistributing to retirees, i.e. examining the impact of payroll taxes that fund social security. Imagine starting from a situation wherein the tax rate on labor income is 40% (and the tax rate on capital income is 30%) and all the proceeds are redistributed in a lump- sum fashion to individuals between ages I and R. Now consider increasing the tax rate on labor income from τ h to τ 0h and redistributing the proceeds associated with (τ h 0 − τ h )wH to fund lump sum payments of equal amounts to individuals between the ages of R + 1 and T . Table 6 displays the results. Table 6: Eﬀect of Social Security on Fertility - Model τh f 0.40 2.10 0.45 1.94 0.50 1.82 0.55 1.71 0.60 1.64 Notice that social security has a negative eﬀect on fertility rates. This is driven by the way it is ﬁnanced –using labor income taxes– and not by the timing of payments. The eﬀects at play are the same as in the previous section. When the tax rate on human capital rises, the stock of human capital declines since the incentive to accumulate human capital declines. Since the proceeds of the higher tax rate are re-distributed only to retirees, this reduces 34 savings for retirement and consequently the stock of physical capital. The results indicate that the negative eﬀect on human capital exceeds that on physical capital - thereby the capital output ratio rises and the fertility rate falls. Notice that social security leads to a decrease in fertility rates only be- cause the presence of human capital makes the supply of labor elastic. Absent human capital –or if the social security program was ﬁnanced using lump- sum taxes– there is only one eﬀect at play: a rise in social security receipts would lead to a fall in the stock of capital, which would lead to a fall in the capital output ratio and hence raise the fertility rate. Indeed, this is the argument in Boldrin et. al. (2005). Thus the addition of human capital, and the major role played by taxation into the Barro-Becker model, implies that more generous social security regimes ﬁnanced by higher taxes on labor income have a negative net eﬀect on fertility. 6 Conclusions This paper integrates a life-cycle model of human and physical capital accu- mulation with the Barro-Becker framework. This permits an analysis of the eﬀects of changes in life expectancies and productivity on the fertility rate. The model is able to capture the wide variation in fertility rates seen across the income distribution. Further, the model suggests that a substantial part of the lower fertility rates in Europe are due to the higher labor income tax rates. 35 References [1] Barro, Robert J. and Lee, Jong-Wha, 1996,“International Measures of Schooling Years and Schooling Quality,” American Economic Review, May (Papers and Proceedings), 86(2), pp. 218-23. [2] Becker, Gary S. Human Capital : A Theoretical and Empirical Analysis, with Special Reference to Education. New York: National Bureau of Economic Research, 1964. [3] Becker G.S. (1960), “An Economic Analysis of Fertility,” in: A Report of the National Bureau of Economic Research, Demographic and Economic Change in Developed Countries: A Conference of the Universities- National Bureau Committee for Economic Research (Princeton Univer- sity Press, Princeton, NJ) 209-231. [4] Becker, G.S. and R.J. Barro (1988), “A Reformulation of the Economic Theory of Fertility,” The Quarterly Journal of Economics, 103(1):1-25. [5] Becker, Gary S., Murphy, Kevin M., and Tamura, Robert, “Human Capital, Fertility, and Economic Growth,” Journal of Political Economy, 98, 5 (1990) pp. S12-S37. [6] Ben Porath, Y., 1967, “The Production of Human Capital and the Life Cycle of Earnings,” Journal of Political Economy 75, pt. 1, 352-365. [7] Boldrin, Michele, Mariacristina De Nardi and Larry Jones, “Fertility and Social Security," NBER Working Paper No. 11146, February 2005. 36 [8] Davis, Stephen J. and Henrekson, Magnus. “Tax Eﬀects on Work Ac- tivity, Industry Mix and Shadow Economy Size: Evidence From Rich- Country Comparisons.” Mimeo, University of Chicago Graduate School of Business, 2004. [9] Doepke, M. (2001), “Accounting for Fertility Decline During the Tran- sition to Growth,” Journal of Economic Growth, forthcoming. [10] Fernandez-Villaverde, Jesus, “Was Malthus Right? Economic Growth and Population Dynamics.” Mimeo, Department of Economics, Univer- sity of Pennsylvania, 2001. [11] Gale, William G. and Scholz, John Karl. “Inter-generational Transfers and the Accumulationof Wealth.” Journal of Economic Perspectives, Autumn 1994, 8(4), pp. 145—60 [12] Galor, Oded, 2005, “From Stagnation to Growth: Uniﬁed Growth The- ory,” forthcoming in P. Aghion and S. Durlauf (eds) Handbook of Eco- nomic Growth, North Holland. [13] Heckman, J., L. Lochner, and C. Taber, 1998, "Explaining rising wage inequality: Explorations with a dynamic general equilibrium model of labor earnings with heterogeneous agents," Review of Economic Dynam- ics 1, 1—58.997, pp. 73-102. [14] Hsieh, Chang-Tai and P. Klenow, 2003, "Relative Prices and Relative Prosperity," NBER Working Paper 9701. [15] Kalemli-Ozcan, Sebnem, “Does Mortality Decline Promote Economic Growth?”, Journal of Economic Growth, (2002), 7. 37 [16] Lucas, Robert E, 2002. “The Industrial Revolution: Past and Future.” In Robert E. Lucas, Lectures on Economic Growth. Cambridge: Harvard University Press. [17] Manuelli, R. and A. Seshadri, 2005, “Human Capital and the Wealth of Nations,” mimeo, University of Wisconsin-Madison. [18] Mincer, J., 1974, Schooling, Experience and Earnings, Columbia Uni- versity Press, New York, 1974. [19] OECD, 1996, Education at a Glance: OECD Indicators. [20] Prescott, Edward, “Why Do Americans Work So Much More Than Eu- ropeans?,” Federal Reserve Bank of Minneapolis Staﬀ Report No. 321, November 2003. [21] Soares, Rodrigo, “Mortality Reductions, Educational Attainment, and Fertility Choice”, American Economic Review, 95(3), June 2005. [22] Solow, Robert M., 1956, “A contribution to the theory of economic growth”, Quarterly Journal of Economics, Vol.70 No.1, pp: 65 - 94. 38 7 Appendix Proof of Proposition 1. : We show that the ﬁrst order conditions corresponding to both problems coincide. Since the problems are convex, this suﬃces to establish the result. Consider ﬁrst the ﬁrst order conditions of the income maximization problem given the stock of human capital at age 6, h(6) = hE . Let q(a) be the costate variable. A solution satisﬁes whn ≤ qγ 1 zh (nh)γ 1 xγ 2 , with equality if n < 1, (30a) x = qγ 2 zh (nh)γ 1 xγ 2 , (30b) q = rq − [qγ 1 zh (nh)γ 1 xγ 2 h−1 − δ h ] − w(1 − n), ˙ (30c) ˙ h = zh (nh)γ 1 xγ 2 − δ h h, (30d) where a ∈ [6, R]. The transversality condition is q(R) = 0. Let Φ be the Lagrange multiplier associated with the budget constraint (1). Then, the relevant (for the decision to accumulate human capital) prob- lem solved by a parent is Z R max Φ{ e−r(a−I) [wh(a)(1 − n(a)) − x(a)]da I Z B+I +ef e−r(a−I) [whk (a)(1 − nk (a)) − xk (a)]da B f −rB −e e bk − ef e−r(B+6) xE } + e−α0 +α1 f e−ρB V k (hk (B + I), bk ), where, in this notation, a stands for the parent’s age. It follows that the ﬁrst order conditions corresponding to the choice of [h(a), n(a), x(a), qp (a)] are identical to those corresponding to the income maximization problem (30), including the transversality condition qp (R) = 0 for a ∈ [I, R]. It follows that qp (a) = q(a).Simple algebra shows that the ﬁrst order conditions 39 corresponding to the optimal choices of [hk (a), nk (a), xk (a), qk (a)] also satisfy (30) for a ∈ [6, I). However, the appropriate transversality condition for this problem is 1 ∂V k (hk (B + I), bk ) qk (B + I) = e−[α0 +(1−α1 )f ] e−(ρ−r)B . Φ ∂hk (B + I) However, given (5), and the envelope condition ∂V k (hk (B + I), bk ) = Φk qp (I), ∂hk (B + I) evaluated at the steady state Φ = Φk , it follows that qk (B + I) = qp (I). Thus, the program solved by the parent (for a ∈ [I, R]) is just the contin- uation of the problem he solves for his children for a ∈ [6, I). It is clear that if (5) does not hold, then there is a ‘wedge’ between how the child values his human capital after he becomes independent, qp (I), and the valuation that his parent puts on the same unit if human capital, qk (B + I). For simplicity, we prove a series of lemmas that simplify the proof of Proposition 2. It is convenient to deﬁne several functions that we will use repeatedly. Let ∙ ¸ 1−γ 1 γ γ γ 2 γ 1 zh wγ 2 Ch (zh , w, r) = 2 1 , (r + δ h )γ and m(a) = 1 − e−(r+δh )(R−a) . The following lemma provides a characterization of the solution in the post schooling period. 40 Lemma 3 Assume that the solution to the income maximization problem stated in Proposition 1 is such that n(a) = 1 for a ≤ 6 + s for some s. Then, given h(6 + s) the solution satisﬁes, for a ∈ [6 + s, R), µ ¶ γ2w £ ¤ 1 x(a) = Ch (zh , w.r) 1 − e−(r+δh )(R−a) 1−γ , a ∈ [6 + s, R), r + δh (31) Ch (zh , w.r) −δh (6+s−R) h(a) = e−δh (a−6−s) {h(6 + s) + e (32) δh Z eδh (a−R) r+δ h γ (1 − x δh ) 1−γ dx}, a ∈ [6 + s, R), eδh (6+s−R) and w q(a) = [1 − e−(r+δh )(R−a) ], a ∈ [6 + s, R). (33) r + δh Proof of Lemma 3. : Given that the equations (30) hold (with the ﬁrst equation at equality), standard algebra (see Ben-Porath, 1967 and Haley, 1976) shows that (33) holds. Using this result in (30b) it follows that ∙ γ2 γ1 ¸ 1 µ ¶ γ 2 γ 1 zh wγ 2 1−γ γ2w £ ¤ 1 x(a) = 1 − e−(r+δh )(R−a) 1−γ , (r + δ h )γ r + δh which is (31). Next substituting (31) and (33) into (30d) one obtains a non- linear non-homogeneous ﬁrst order ordinary diﬀerential equation. Straight- forward, but tedious, algebra shows that (32) is a solution to this equation. The next lemma describes the solution during the schooling period. Lemma 4 Assume that the solution to the income maximization problem stated in Proposition 1 is such that n(a) = 1 for a ≤ 6 + s for some s. Then, given h(6) = hE and q(6) = qE , the solution satisﬁes, for a ∈ [6, 6 + s), 1 r+δ h (1−γ 1 ) γ (a−6) x(a) = (hE1 qE γ 2 zh ) 1−γ2 e (1−γ 2 ) , a ∈ [6, 6 + s) (34) 41 and ³ ´ 1−γ (1 − γ )(1 − γ ) 1 −δh (a−6) −(1−γ) γ 2 γ 2 2 1 2 h(a) = hE e [1 + hE qE γ 2 zh (35) γ 2 r + δ h (1 − γ 1 ) γ 2 r+δ h (1−γ 1 ) 1 (a−6) (e (1−γ 2 ) − 1)] 1−γ 1 , a ∈ [6, 6 + s) Proof of Lemma 4. : From (30b) we obtain that 1 1 x(a) = (q(a)h(a)γ 1 ) 1−γ 2 (γ 2 zh ) 1−γ2 . (36) Since we are in the region in which the solution is assumed to be at a corner, (30a) implies ³ γ ´ 1−γ2 1 1 1 1−γ γ h(a) ≤ (γ 2 2 zh ) 1−γ q(a) 1−γ (37) w In order to better characterize the solution we now show that the shadow value of the total product of human capital in the production of human capital grows at a constant rate. More precisely, we show that For a ∈ [6, 6 + γ s), q(a)hγ 1 (a) = qE hE1 e[r+δh (1−γ 1 )](a−6) . To see this, let M(a) = q(a)hγ 1 (a). Then, q(a) ˙ ˙ h(a) ˙ M(a) = M(a)[ + γ1 ]. q(a) h(a) However, it follows from (30c) and (30d) after substituting (36) that ˙ h(a) = zh h(a)γ 1−1 x(a)γ 2 − δ h , a ∈ [6, 6 + s) h(a) ˙ q(a) = r + δ h − γ 2 zh h(a)γ 1−1 x(a)γ 2 , a ∈ [6, 6 + s). q(a) Thus, q(a) ˙ ˙ h(a) + γ1 = r + δ h (1 − γ 1 ). q(a) h(a) The function M(a) satisﬁes the ﬁrst order ordinary diﬀerential equation ˙ M(a) = M(a)[r + δh (1 − γ 1 )] 42 whose solution is M(a) = M(6)e[r+δh (1−γ 1 )](a−6) which establishes the desired result. Using this result the level of expenditures during the schooling period is given by 1 r+δh (1−γ 1 ) γ (a−6) x(a) = (hE1 qE γ 2 zh ) 1−γ2 e (1−γ 2 ) , a ∈ [6, 6 + s). Substituting this expression in the law of motion for h(a) (equation (30d), the equilibrium level of human capital satisﬁes the following ﬁrst order non- linear, non-homogeneous, ordinary diﬀerential equation 1 γ 2 [r+δ h (1−γ 1 )] ˙ γ γ γ γ (a−6) h(a) = (hE1 2 qE2 γ 2 2 zh ) 1−γ2 e (1−γ 2 ) hγ 1 (a) − δ h h(a). It can be veriﬁed, by direct diﬀerentiation, that (35) is a solution. The next lemma describes the joint determination, given the age 6 level of human capital hE , of the length of the schooling period, s, and the age 6 shadow price of human capital, qE . Lemma 5 Given hE , the optimal shadow price of human capital at age 6, qE , and the length of the schooling period, s, are given by the solution to the following two equations " # 1−γ 1 γ (1−γ 2 ) γ 1 γ 2 γ 1 (1−γ 1 )(1−γ 2 ) γ11 γ 2 zh w −γ 1 qE = hE (38) (r + δ h )(1−γ 2 ) 1−γ 2 e−(r+δh (1−γ 1 ))s m(s + 6) 1−γ , 43 and γ2 γ1 γ2 µ ¶ 1−γ 2 1−γ 2 −δh (1−γ 1 )s (1 − γ 1 )(1 − γ 2 ) γ 1 qE hE e (γ 2 2 zh ) 1−γ 2 (39) γ 2 r + δh (1 − γ 1 ) γ 2 r+δ h (1−γ 1 ) s 1−γ 1 −δh (1−γ 1 )s [e (1−γ 2 ) − 1] + hE e Ã ! 1−γ1 (1−γ ) γ 1−γ γ1 2 γ22 1−γ 1 1−γ 1 = (zh wγ 2 ) 1−γ [m(s + 6)] 1−γ . (r + δ h ) Proof of Lemma 5. To prove this result, it is convenient to summarize some of the properties of the optimal path of human capital. For given values of (qE , hE , s) the optimal level of human capital satisﬁes ³ ´ 1−γ (1 − γ )(1 − γ ) 1 −δ h (a−6) −(1−γ) γ 2 γ 2 2 1 2 h(a) = hE e [1 + hE qE γ 2 zh (40) γ 2 r + δ h (1 − γ 1 γ 2 r+δ h (1−γ 1 ) 1 (a−6) (e (1−γ 2 ) − 1)] 1−γ 1 , a ∈ [6, 6 + s) Ch (zh , w, r) −δh (6+s−R) h(a) = e−δh (a−s−6) {h(6 + s) + e (41) δh Z eδh (a−R) r+δ h γ (1 − x δh ) 1−γ dx}, a ∈ [6 + s, R). eδh (6+s−R) Moreover, during at age 6 + s, (37) must hold at equality. Thus, ³ γ ´ 1−γ 2 1 1 1 1−γ γ h(6 + s) = (γ 2 2 zh ) 1−γ q(6 + s) 1−γ . w Using the result in Lemma 4 in the previous equation, it follows that 1−γ 1−γ γ h (r+δ (1−γ 1 ))(6+s) (h 1 qE ) 1−γ 2 e 1−γ 2 q(6 + s) = E ¡ γ ¢γ γ γ1 . (42) 1 w 1 (γ 2 2 zh ) 1−γ2 Since w q(6 + s) = [1 − e−(r+δh )(R−s−6) ], r + δh 44 it follows that " # 1−γ 1 γ (1−γ 2 ) γ 1 γ 2 γ 1 (1−γ 1 )(1−γ 2 ) γ11 γ 2 zh w −γ 1 qE = hE (r + δ h )(1−γ 2 ) 1−γ 2 e−(r+δh (1−γ 1 ))s m(s + 6) 1−γ , which is (38). Next, using (40) evaluated at a = 6 + s, and (37) at equality (and substituting out q(6 + s)) using either one of the previous expressions we obtain (39). We now discuss the optimal choice of hE . Since qE is the shadow price of an additional unit of human capital at age 6, the household chooses xE to solve max qE hB xυ − xE . E The solution is υ 1 υ 1−υ 1−υ hE = υ 1−υ hB qE . (43) Proof of Proposition 2. Uniqueness of a solution to the income maxi- mization problem follows from the fact that the objective function is linear and, given γ < 1, the constraint set is strictly convex. Even though existence can be established more generally, in what follows we construct the solution. To this end, we ﬁrst describe the determination of years of schooling. Com- bining (38) and (39) it follows that Ã ! 1−γ 1 γ (1−γ ) 1 1 γ22 γ1 1 hE = eδh s m(s + 6) 1−γ (zh wγ 2 ) 1−γ (44) r + δh ⎡ γ 2 r+δ h (1−γ 1 ) ⎤ 1−γ 1 − s 1 ⎣1 − r + δ h (1 − γ 1 )(1 − γ 2 ) 1 − e (1−γ 2 ) ⎦ . γ 1 γ 2 r + δ h (1 − γ 1 ) m(s + 6) 45 Next, using (38) in (43), hE must satisfy Ã ! (1−γ)(1−υ(1−γ υ γ (1−γ ) γ γ 1 )) 1 1−υ(1−γ 1 ) υ γ11 2 γ21 2 hE = hB υ 1−υ(1−γ 1 ) (45) (r + δ h )1−γ 2 ¡ γ 1 (1−γ )(1−γ ) ¢ (1−γ)(1−υ(1−γ )) − υ(r+δh (1−γ1 )) s υ υ(1−γ 2 ) zh w 1 2 1 e 1−υ(1−γ1 ) m(s + 6) (1−γ)(1−υ(1−γ1 )) . Finally, (44) and (45) imply that the number of years of schooling, s, satisﬁes m(s + 6)1−υ(2−γ) e(1−γ)(δh +rυ)s (46) ⎡ γ r+δh (1−γ ) ⎤ (1−γ)(1−υ(1−γ1 )) 1−γ 1 − 2 (1−γ ) 1 s ⎣1 − r + δh (1 − γ 1 )(1 − γ 2 ) 1 − e 2 ⎦ γ 1 γ 2 r + δh (1 − γ 1 ) m(s + 6) µ ¶(1−γ)υ Ã γ 2 (1−γ 2 ) !−(1−υ) h1−γ υ γ2 γ1 = 1−υ γB−υ(1−γ ) . zh w 2 1 r + δh r + δh As in the statement of the proposition, let the left hand side of (46) be labeled F (s). Then, an interior solution requires that F (0) > 0, or, µ ¶(1−γ)υ Ã γ (1−γ ) !−(1−υ) h1−γ υ γ22 γ1 2 m(6)1−υ(2−γ) > 1−υ γB−υ(1−γ ) . zh w 2 1 r + δh r + δh Inspection of the function F (s) shows that there exists a unique value of s, say s, such that F (s) > 0, for s < s, and F (s) ≤ 0, for s ≥ s. It is clear ¯ ¯ ¯ that s < R − 6. Hence, the function F (s) must intersect the right hand side ¯ of (46) from above. The point of intersection is the unique value of s that solves the problem. It is convenient to collect a full description of the solution as a function of aggregate variables and the level of schooling, s. 46 Solution to the Income Maximization Problem It follows from (30a), and the equilibrium values of the other endogenous variables, the time allocated to human capital formation is 1 for a ∈ [6, 6 + s), and 1 m(a) 1−γ n(a) = 1 R eδh (a−R) r+δ h γ , (r+δh )e−δh (a−R) e−δh (a−s−6) m(6 + s) 1−γ + γ 1 δh eδh (6+s−R) (1 − x δh ) 1−γ dx (47) for a ∈ [6 + s, R]. The amount of market goods allocated to the production of human capital is given by µ ¶ γ2w 1 r+δ h (1−γ 1 ) (a−s−6) x(a) = Ch (zh , w, r)m(6 + s) 1−γ e (1−γ2 ) , a ∈ [6, 6 + s), r + δh µ ¶ (48) γ 2w 1 x(a) = Ch (zh , w, r)m(a) 1−γ , a ∈ [6 + s, R). (49) r + δh The level of human capital of an individual of age a in the post-schooling period (i.e. a ≥ 6 + s) is given by γ1 1 e−δh (a−R) h(a) = Ch (zh , w, r){e−δh (a−s−6) m(6 + s) 1−γ + (50) r + δh δh Z eδh (a−R) r+δ h γ (1 − x δh ) 1−γ dx}, a ∈ [6 + s, R). eδh (6+s−R) The stock of human capital at age 6, hE , is " # 1−γ υ γ (1−γ 2 ) γ 1 γ 2 γ 1 (1−γ 1 )(1−γ 2 ) γ11 γ 2 zh w hE = υ υ hB (51) (r + δ h )(1−γ 2 ) υ(1−γ 2 ) e−υ(r+δh (1−γ 1 ))s m(6 + s) 1−γ 47