Structural Change, Demographic Transition and Fertility Di erence

Transcription

1 Structural Change, Demographic Transition and Fertility Di erence T. Terry Cheung Washington University in St. Louis Preliminary and Incomplete February 4, 207 Abstract This paper asks if the force that drives the demographic transition in the developed countries and the force that makes less developed countries having higher fertility rate are the same one. The hypothesis is that the sectorial fertility goes a long way in explaining the fertility transition and the cross-country fertility di erence. In less developed stage, as agricultural productivity is low and as physical labors play a much important role in the farming sector, the economy is usually populated with agricultural workers who have higher birth rate. When the economy develops, the share of agricultural employment and sectorial fertility both decline, hence having an aggregate implication in the overall fertility rate. By using detailed U.S. census data, this paper shows that such pattern is consistent with historical description in U.S. from the mid-9th century to the eve of the baby boom. The general equilibrium model in this paper, when calibrated by using the U.S. data, is able to replicate the features of the sectorial fertility rates and sectorial employment share in the U.S. for 90 years. JEL Classi cations: O, O4, J, J3, J24

2 Introduction An average family in the U.S. has 2 children in 205 while. an average family in 880 had more than 5 children; and 2. in Niger, an average family had more than than 7 children in These observations are well documented. We reiterate two of them here to facilitate the discussion of the objective of the paper (and we will postpone the detailed discussion in Section 2.). The rst observation is that developed countries usually go through demographic transition, meaning that the fertility rate declines as time passes by. The U.S., for example, started such fertility decline from the late 9th century. Another well documented fact is that the less developed countries tend to have higher fertility. This paper asks one question: Is the force that drives demographic transition in the U.S. and the force that makes the poorer countries having higher fertility the same one? Our hypothesis is that the size of the agricultural sector (and hence the agricultural productivity) goes a long way to explain the demographic transition and the cross-country fertility di erences. For example, well above 50% of the U.S. workers were working in the agricultural sector in 880 while it has been around 2% for recent years. The agricultural employment share was as high as 60% in Niger in If an average agricultural family gives birth to more children when compared to the non-agricultural ones, this will have aggregate implication to the observed birth rate. We will show in detail in Section 2.2 that the U.S. data clearly indicates that there exists fertility rate gap between families from agricultural and non-agricultural sectors. The gap can be as high as 2 children. This number has signi cant implication when the composition of the economy changes (i.e. the relative importance of the agricultural sector declined). Take the same U.S. data as an example, the average number of children a family gave birth to in 900 was 3.9. There were around 40% of the labor forces working in the agricultural sector which, on average, gave birth to 5.5 children per family. On the other hand, mothers in non-agricultural sector only had 2.8 children on average in the same year. In the year 2000, the agricultural sector employment only accounted for 2.5% of the labor forces. A simple calculation will show that, even without accounting for the decline in the sectorial fertility rate, merely plugging the year 2000 labor market structure into year 900 s sectorial birth rate will result in a decrease in fertility rate that closed around 53% 2

3 of the birth rate di erence between 900 and Even without accounting for the decline in sectorial fertility rate, this simple exercise accounts for more than half ( child, or 53%) of the total decline in the fertility (.9 children per woman). So by and large, the e ect of sectorial change in understanding fertility decline is crucial. Some evidence point to the fact that the problem of child labor may be the cause of high fertility in the agricultural sector while in subsection 2.3, we explore these evidence and see how the problem of child labor is developed in the poorer countries (which with lower agricultural productivity). Then we will discuss how the problem of child labor leads to the problem of high fertility that we have observed in less developed countries. We will also present in the same subsection that child labor had been more pronounced in agricultural sector in the U.S. before the child labor has been banned. We have made two contributions to the broad literature. One is empirical while the other is theoretical. As far as we know, this is the rst paper that documents the fertility rates are di erent for agents working in the agricultural and non-agricultural sectors 2. This is important because of two reasons. First, it shows that within a broader phenomenon of fertility transition, there can be heterogeneity in di erent sectors. So this points to the direction that using one-sector model to understand the overall fertility decline might omit some important features. For example, before 900, the agricultural fertility rate did not fall despite the gradual fall of non-agricultural fertility rate. Second, it has implication to understand the cross-countries fertility di erence. While some authors argue that role of mortality rate decline (Kalemli-Ozcan, 2002 and Soares, 2005), the impact of introduction of labor-saving technology (Doepke, 2004) and the implication of longevity on quality-quantity trade-o (Manuelli and Seshadri, 2009) are important to understand cross-country fertility di erence, this article shows that sectorial composition of the economy may also be useful in understand fertility di erence across the nations. That is, the countries with larger agricultural sector, other things being constant, will imply higher Assuming the sectoral birth rate of 900 and applying the labor market structure in Then we will have: 5:5 2:5% + 2:8 97:5% = 2:9. When compares this gure to the actual gure being 3.85 and 2 in 900 and 2000 resepectively, our back-of-envelope calculation shows 3:9 2:9 that such exercise closed = 53% of the gap. 2 3:9 2 Two similar constructions are Greenwood and Seshadri (2002) and Jones and Tertilt (2008). Greenwood and Seshadri (2002) separate white women in years of age into rural and urban and counts the number of children under 5 that they have. Jones and Tertilt (2008) use the same de nition of fertility as we do (children ever born). However, they devide the women into rural versus urban, instead of agricultural versus non-agricultural. 3

4 fertility. The theoretical contribution is a reply to the (constructive) criticism of Guinnane s (Guinnane, 20) that macroeconomic approach setting aside microstructure of the economy (e.g. marriage status and heterogeneities in other dimensions). To this end, our model has two sectors: agricultural sector and non-agricultural sector. Di erent from Greenwood and Seshadri (2002), our model exhibits di erent trends of fertility rate movements for di erent sectors. In sum, our model has the feature that the non-agricultural sectorial fertility rate declines gradually in the beginning and follows by a quick decline of agricultural sectorial fertility. The trick is to add educational cost into the model so that the relative cost of educating children will be di erentiated for di erent groups of agents. Hence, their pace and timing of quantity-quality tradeo will also be di erent. In particular, this model implies that agents in the nonagricultural sector will be carrying tradeo before the agents in the agricultural sector. The mechanism of our model mainly depends on three e ects, all of them come from the interaction between the subsistence level of agricultural consumption and productivity. These three forces together give the overall decrease in the total fertility rate and the rst increase then decrease pattern in the agricultural sector. Labor Market E ect The rst force is that while the agricultural productivity increases, the relative demand for agricultural goods (relative to the total consumption) decreases. This will then reduce the demand for the agricultural labor and will drive down the overall agricultural employment share in the economy. As the non-agricultural sector has lower fertility rate, the increase in the importance of this sector will lower the average fertility rate. This is related to the overall decline in aggregate fertility that we observed. Income E ect The second one is the income e ect a ecting non-linearly on the spending on child-rearing. When the income is very close to the subsistence level (yet above it), a small increase in the income will decrease the agricultural consumption (relative to total consumption) by a large extent. These residual income will then be allocated to non-agricultural consumption and child-rearing spending, making them growing out of proportion (with respect to income) in the beginning. As income grows higher and higher, more and more resource is allocated into non-agricultural consumption (as increase of children is highly discounted) and it results decrease in the child-rearing spending (with respect to income). This is related to the initial increase then decrease in the agricultural 4

5 sectorial fertility. Quality-Quantity Tradeo E ect The third force is the quality-quantity tradeo. In our model, the quality-quantity tradeo will be done similarly as the one in Doepke s model (Doepke, 2004). Only the high income agent will start to restrict the family size and educate their kids in the beginning as they have relatively lower education cost. However, when income increases relative to the xed subsistence level, the middle class and even the farmers found it better to educate their kids. This is related to the fact that non-agricultural fertility rate declines before that of the agricultural one. When we calibrate the model parameter using the US data, the model explains reasonably well the pattern that we have observed in the US demographic transition. In particular, the steady-state comparative statics analysis of the model can replicate the pattern of both the agricultural and non-agricultural fertility rate as well as agricultural employment share reasonably well. The subsequent work will be organized as following: in Section 2, we will organize the empirical data and show the reasons that sectorial fertility is important to understand total fertility trend. Section 3 is for literature review and we will discuss how our model add value to the existing literature. Then in section 4 we will lay out our model with some analytical as well as some computational results. Finally, in Section 5, we will conclude. 2 Motivating Facts 2. Larger Agricultural Sector, Higher Fertility The size of agricultural sector and fertility are positively correlated. In the countries with higher agricultural sector, the fertility rates are usually higher. This is also true across time. The aim of this subsection is to provide evidence for that. From Figure, we can see that the average fertility of the USA has been steady from 840 to 880. Disregarding the baby boom in the 940s, the overall trend of the number of children per family has been decreasing. From the gure, we will see that the total fertility of the USA decreased from average about 5 children per family in 880 to around 2 children per family in 970, i.e. a reduction of 3 children per family in 90 years time. At the same time, the agricultural employment share had been steady from 850 to 880 and had declined since 880 from around 52% of the total work force to around 2.5% in the late 20th century. 5

6 [Figure Here] This phenomenon is not an isolated event that just happened in the USA. On the contrary, it is a rather universal pattern that happened in most of the developed countries and happening in some developing countries. Table below shows the panel data evidence that total fertility rate is positively correlated with agricultural labor share. One of the objections of this regression exercise is the problem of endogeneity that both GDP per capita as well as human capital is correlated with both agricultural employment share and fertility rate. However, even if we control for log real GDP per capita as well as human capital, we can see that the total fertility rate is still positively and signi cantly correlated with agricultural employment share. One of the irregularity is that when we only control for human capital, the e ect of agricultural employment share seems to be insigni cant. However, when we introduce interaction term (of agricultural employment share interacts with human capital), the correlation between agricultural employment share and total fertility rate becomes positive and robust again. This suggests that sectorial size as a determinant of total fertility rate depends on schooling 3. In a place with higher human capital, the e ect of agricultural employment share seems to be of less important. This consistent with the US experience in Figure as when the economy grows, the agricultural fertility converges to that of the non-agricultural one. This might be due to, among others, that education leads to the convergence of rural and urban areas in for example Caselli and Coleman (200), which we will discuss in more detail in Section 2.3. [Table Here] The above two pieces of evidence show the size of agricultural sector is one of the factors that determines the fertility rate. However, one should bear in mind that agricultural sectorial size should also be a factor to be explained as most of the literature in the eld of structural change (or structural transformation) suggested. From the evidence that we presented, It is clear that agricultural sector size is positively correlated with fertility rate. It does not, however, say why the size of the agricultural sector is important and how important is the sectorial size. We will dedicate the next subsection to discuss the reasons for the size of the agricultural sector to be an important determinant and how important is the 3 A similar result is obtained when the interaction term is log per capita real GDP and agricultural employment share (not shown). One reason might be that log per capita real GDP and human capital is very much correlated. 6

7 agricultural sector when compare to the other explanatory factors. 2.2 Agricultural Sector has higher Fertility Rate This subsection is to discuss why the fertility rate is high in countries and times when the size of the agricultural sector is relatively large. We will also provide evidence on the reasons to focus on the sectorial fertility. To this end, we would need more disaggregated data. So, in this subsection, we will use the US census data from IPUMS-USA (Flood, King, Ruggles & Warren 205) to establish the result. The method that we used the US census is similar to that employed by Jones and Tertilt (2008). As IPUMS-USA data contains detailed census data from that asked the (female) respondents the number of children they had ever given birth to, thus a synthetic cohort birth rate can be constructed. Also, we follow Jones and Tertilt (2008) and add 27 years to the cohort birth rates to make them comparable to the fertility rate of a particular year. We separate the population into two di erent groups: agricultural versus non-agricultural by the de nition of Herrendorf, Rogerson and Valentinyi (204) 4. By their classi cation, the term agricultural sector used here might be misleading as the sector actually means subsistence sector lumping up agriculture, hunting, forestry and shing. The result of the decomposition has been shown below in Figure 2. [Figure 2 Here] Figure 2 shows two di erent features. First, the number of children ever born by female in agricultural sector is consistently higher than that in nonagricultural sector. For example, in the late 9th century, there were on average 5 children for agricultural family and only 3 children for non-agricultural family a di erence of 2 children between agricultural and non-agricultural families. The gap persisted throughout the rst half of the 20th century despite the fact that the gap is less pronounced. One of the main reason for the little convergence is that starting from the late 20th century, the agricultural sectorial fertility rate started to decline. The di erence has further converged since around 950. Second, the trends for di erent sectorial fertility rates are di erent. Even though the average children ever born remained roughly constant from 850 to 880 as in Figure, the number of children ever born by a non-agricultural 4 One of the protest about the measure of fertility in the US is concerning the switch of industries betweem the time the children was born and the time the measure (census) is made. However, if one is willing to accept that the agricultural sector has higher fertility than the non-agricultrual sector, then such switch only makes the fertility rate gap larger (not smaller). 7

8 sector woman was falling gradually in the same period. However, the fertility rate of the agricultural sector only started to fall in the eve of the 20th century. Jones and Tertilt (2008) pointed out a similar observation 5 that the fertility transition had begun early and had been gradual among the upper income families (or urban households) and began later and was much quicker for the lower income families (or rural households). Their analysis has a crucial di erence with ours in the way that they do not separate women according to the sectors they were working in. Instead, they separate them by their income and their area of residence (Farm Status). One can see from Figure 3 below that the agricultural and non-agricultural fertility were initially tracking very closely with the farm and non-farm fertility respectively. However, the di erence begins to grow larger in the later half of the 20th century. This might be mainly due to the fact that the originally farm and non-farm status are good proxies to one s sectorial employment. However, when time goes by, the structural change makes the rural and urban residencies less predictive on one s sectorial employment. For example, one may live in rural area while not engaged in jobs related to agriculture. Or as transportation developed quickly after the Second World War and made the gap between rural and urban less pronounced. Our analysis is also better when we think in modeling terms since the production functions of agriculture and non-agriculture are better de ned when compared to that for rural and urban. [Figure 3 Here] As can be seen from both Figure 2 and 3, the decline of fertility between the two period can be decomposed into two di erent forces: rst, the within sectorial decline of fertility (the total fertility rate declined in both sectors; within di erence) and second, the change of the sectorial components (the decline of the importance of the agricultural sector; between di erence). The following decomposition summarizes the decomposition of fertility di erence between two time points t 0 and t into the change in sectorial component and the change 5 However, they consider di erent groups of agents when compare to our categorization (which is agricultural vs. non-agricultural). They use high income vs. low income and rural vs. urban. 8

9 in fertility rate F. fertility t0!t = X X it F it it0 F it0 i i = X X it F it it0 F it + X i i i it0 F it = X ( it it0 ) F it + X it0 (F it F it0 ) i i {z } {z } between within X it0 F it0 Figure 4 below shows an example of such decomposition exercise between 980 and speci c years from 900 to 970. The gure is a combination of bar chart and line chart. Each bar corresponds to the di erence of children ever born between a speci c year and the year 980 while components of the bar show the absolute importance of the between and within changes 6. [Figure 4 Here] There are two things that can be read from Figure 4 above. First, sectorial composition of economy is important to understand the decline in fertility rate as it accounts for a large portion of the decline in fertility. Second, the importance of the agricultural share depends on its absolute size of change, i.e. the term ( it it0 ). This is very intuitive, if not even tautological, that when the change of the employment share in agricultural sector is large, it will lead to a larger share of between di erence. However, reading it clearly, one will notice that it usually implies long run changes. For example, in Figure 4 we can see that the absolute (and relative, if looking at trend) importance of the sectorial contribution declined when one moves closer and closer to the year 980 (the reference year). This is mainly due to the fact that the relative labor forces moving out from the agricultural sector was small. For example, the employment share in the years 970 and 980 were both around 3%. So the contribution of the between e ect was insigni cant. The next question to be asked will then be the reasons that the average agricultural family having more children than that of the average non-agricultural family. We think that the reasons are in two-fold: rst, the need of physical labor and second, the educational cost. i These two factors together give the 6 The negative of the contribution of sectorial decline of fertility in the year 920 and 930 is because the fact that (at least one of) the sectorial fertiltiy is lower in those years when compared to the year

10 problem of child labor Child Labor prevalent in Agriculture The main aim of this subsection is to establish the fact that child labor is more prevalent in agricultural sector and give a possible explanation to that. The fact that the child labor is more prevalent in the agricultural sector is relative easy to establish as it is quite established. Edmonds (2005) concludes that "[m]ost working children participate in agriculture" (p.82) after studying the data from Vietnam. In the year 993, for example, among the 69.4 percent of 5-year-old who worked, more than three quarters of them were in agriculture. Lebergott (964, p.53) documents some similar evidence in the USA. He observes that from 870 to 930 (before the child labor law was established), the child labor was more prevalent in agricultural sector. For example, in the year 990, two in every ten children aged 0-5 worked. And among those who worked, one in every two children were in the agricultural sector. For world-wide evidence, the report of International Labor Organization also documents that around 60% of the child labor was in the agricultural sector. The reasons for why the child labor is more prevalent in agricultural sector is more di cult to establish as the reasons can be manifold. In this paper, we consider a pair of demand and supply channels that lead to such observation: the need for physical labor (demand factor) and the high education cost (supply factor) Physical Labor Demand First, we will present the fact that the agricultural sector is more labor-intensive in poor economy, which induces high demand for physical labor (including child labor). As shown in Table 2 (Panel A) that the labor input share is higher in the countries with lower real GDP per capita and/or lower agricultural productivity. This indicates that the less developed countries (or countries at a time with lower agricultural productivity) are usually correlated with higher input in physical labor. Panel B supports this nding by using panel data xed e ect regression. This is consistent with the ndings by Restuccia, Yang and Zhu (2008). They point out that there have been barriers for the developing countries from using 7 Mortality might not be the reason as city s mortality rate is usually higher in history. For example, given the poorer health condition in the UK cities, the mortality rate in rural areas are usually better. 0

11 the modern intermediate inputs (like chemical fertilizers) in agriculture. So that workers can only rely on their physical labor. [Table 2 Here] In the thesis of Restuccia et al. (2008), the barriers to modern intermediate inputs in agriculture and the barriers in the labor market lead to low aggregate agricultural productivity. Together with the subsistence requirement, a high share of employment (with low labor productivity) is found in agriculture. Lagakos and Waugh (203) provide similar story. They established the fact that women are less productive in agricultural work and the fact that they are more prevalent in agricultural sector in the developing countries, contributed to the low measured aggregate productivity in developing countries. However, when viewing the evidence slightly di erently, this also points to the fact that with subsistence level, even less productive labors need to be appropriated into the subsistence sector (agricultural sector).this result is supported by the following Table 3. [Table 3 Here] From Table 3, we can see that the labor employment share is negatively correlated with both agricultural productivity, income as well as human capital. The reason that these has to do with the child labor is best summarized by comment from Rosenzweig and Evenson (977):...[T]he basic conditions motivating... families to bear relatively large numbers of children... was the high return to use of raw labor power of children compared to investments in skills obtained in schools. This is aligned with the evidence that the fertility and average percentage of child labor are closely related as we can see in Figure 5 below. Agree with what Rosenzweig and Evenson (977) argued, it is clearly that the total fertility rate is positively and signi cantly correlated to the average percentage of children aged 5 to 4 working. [Figure 5 Here] Cost of Education Supply In what follows, we show that the countries with higher prevalent of child labor are usually those with lower human capital. As showed in Figure 6 below, a

12 signi cantly negative relation can be found between human capital and child labor prevalence. [Figure 6 Here] This maybe due to a lot of reasons. One of the main reasons that was provided by many development economists is that labor working displaces time allocated to formal schooling. However, from the USA, the education attainment data only available since 940. This was after the child labor had been abolished. So, consistent and systemic data on how child labor a ects human capital accumulation and school attainment cannot be studied using the U.S. census data in hand. However, historians and economic historians have studied the problem and come up with some interesting ndings. For example, Baker (205) uses certain type of agricultural pest infection in early twenty century Georgia as an instrumental variable and concluded that 0 percent reduction of cotton yield reduced the demand of child labor and led to a 2 percent increase in black enrollment rate. Another way to attack this problem is to look at the developing countries. Fafchamps and Wahba (2006), for example, use detailed survey data from Nepal to study the relationship between child labor and urban proximity. They nd that children living near towns and cities are more likely to attend school as school quality and return from education are better in urban areas. Moreover, proximity to urban area also tends to reduce the transportation cost to and from school. As farming is usually away from the cities and towns, the children who lived in rural sector tend to receive little education. As a result, they provide a large pool of physical labor. Together with the demand side factor that agricultural sector in lower-income countries needs more labor. This creates the result that more child labors (as a form of "cheap" labor) being observed in the agricultural sector. 2.4 Remark This section establishes some facts that we would like to be explained by the model. To facilitate the subsequent discussion, I will recap the pieces of fact that we have established. The size of the agricultural sector and the fertility rate are positively correlated across time and space; 2

13 2. The agricultural sector tends to have higher fertility rate because children are a form of valuable labor source; and 3. The trend and movement of the fertility rates of agricultural and nonagricultural sectors are di erent. The hypothesis to explain the facts is that in the early stages, economy grew slowly, only a fraction of those who were well-o and were in the cities had the chance to receive education. Given a certain income, they would have fewer children as some resources were devoted into education. As the economy grew slowly, the speed of such substitution between quality and quantity was also slow. For the farmers who required more physical labor, it was of less interest for them to have educated children. So, we observe high fertility rate in this sector. Nonetheless, in the 20th century, the economy grew quicker and more agents (even in rural areas) found it pro table to receive education and a larger portion of agents quickly substitute quantity for quality. Then we observe the overall and sectorial fertility rates declined quickly. 3 Related Literature This article should be placed between two main areas of macroeconomics: demography and structural change. Virtually all economic analysis of fertility nowadays starts from Becker s model of children demand (e.g. Becker, 960; Becker and Lewis, 973; Becker 98). In the earlier models, the demand of children is derived from consumer theory. As fertility tends to be negatively correlated with cross-country income since the beginning of the fertility transition 8, Becker and Lewis model (Becker & Lewis, 973) featured a standard substitution e ect. As the children are normal goods, the increase in income will increase the demand of children (income e ect). However, the rise in child quality will, in turn, increase the shadow price of quantity that reduces the demand for number of child (substitution e ect). So, if the substitution e ect dominates, then the demand for children will decrease. Up to that point, the analysis of fertility choice theory is con ned. Then the joint work of Barro and Becker (Becker & Barro, 988 and Barro & Becker, 989) 8 Guinnane (20) points out that prior to the transition, the income elasticity of fertility with respective to income is positive; while after the transition, the elasticity tends to be zero or even negative. 3

14 extended the setup to the altruistic parents facing dynastic trade-o between current consumption and children s future well-being. This linked fertility choice to economic growth. In the seminal work of Becker, Murphy and Tamura (990), they explore the interaction between human capital accumulation and fertility and their implication to economic growth. However, the model is silent about the transition of economy from Mathusian state to the modern state. Recently, Galor and Weil (2000) uses a uni ed version of Quantity-Quality tradeo model to explain the transition from Malthusian Economy, in which the fertility is positively correlated with slow economic growth, to Modern Economy, in which the fertility is negatively correlated with fast economic growth. So, they extend the discuss to a dynamic implication of such tradeo and consider the mutual feedback between fertility choice and economic growth. As economic historian Timothy Guinnane points out, however, such discussion mainly focus on the macroeconomic variables such as the aggregate fertility rate and the economy-wide productivity growth 9 while demographers stress on the heterogeneity in the way fertility declines. This article adds value to this aspect of the Uni ed Growth Theory 0 by adding some structures into the fertility model. In our model, the fertility rate of agricultural sector is di erent from that of the non-agricultural sector. The change in the relative contribution in the aggregate economy has implication on the aggregate fertility rate observed. Since our work is related to the change of the relative importance of sectors, it also related to the structural change (or structural transformation) literature. Structural change is a large literature (see Herrendorf, Rogerson and Valentinyi, 204 for review). The topic that we are interested in is the implication of increase in agricultural productivity on agricultural labor allocation. To this end, the most related literature will be Matsuyama (992), Gollin, Parente and Rogerson (2002) and Lagakos and Waugh (203). However, all of them treat fertility as exogenous (or as a constant). fertility choice. In this work, nonetheless, we will endogenize This research is mostly related to three pieces of previous articles: Kogel and Prskawetz (200), Greenwood and Seshadri (2002) and Doepke (2004). While we share similarities with them, we will discuss below that some of our results are very di erent. Kogel and Prskawetz (200) argue that the increase in the agricultural pro- 9 Some noticeble exceptions are e.g. Kogel and Prskawetz (200) and Greenwood and Seshadri (2002). 0 For the review of the Uni ed Growth Theory, see Galor (2005a) (2005b) (20) 4

15 ductivity pushes the agricultural labors out of the agricultural sector. Those labors will then work in the non-agricultural sector. Due to the increasing return to scale of labor in the non-agricultural sector, the productivity increase in the non-agricultural sector will further attract labors from the agricultural sector. As children are normal goods in their analysis, the increase in wealth tends to increase the birth. However, if entering into the non-agricultural sector requires education that incurred cost from parents, then there will be the quality-quantity tradeo and birth rate may decline. Their framework can be used to capture some of the important features of the Industrial Revolution. However, in their work, there is only one economy-wide fertility. So it does not capture the fact that the fertility rate di erence across sectors as we observed from the data. In Greenwood and Seshadri (2002), the authors established the fact that rural area had higher fertility when compared to urban area. They then use a two-sector model with exogenous technological progress to generate fertility decline. However, as the relative cost of education is the same for both rural and urban households, the sectorial fertility rates co-move as whenever it is optimal for the skilled agent to have skilled children, it is also pro table for the unskilled agents to have skilled children and vice versa. So the model will have a hard time to be used to understand the observed di erence in the trends of sectorial fertilities as shown in Figure 2. In particular, it will be di cult to reconcile that the trends and the timing of fertility transition are di erent in di erent sectors. Doepke (2004) is an extension of Hansen and Prescott (2002) to endogenize fertility. He uses one sector model with time and goods cost to explain how the abolishment of the child labor law a ects the return of having extra child. The application of his model is useful in explaining some real world phenomenon. Doepke (2004) has only one sector and hence the model cannot be used to understand the di erences in the fertility di erences between the agricultural and non-agricultural sectors. Given the fact that the child labor is more pronounced in the agricultural sector, such di erence needs to be accounted. For example, Liao (20) uses a similar model to explain the fertility decline and its impact on economic growth in Taiwan. 5

16 4 Model 4. Production (Labor Demand) The economy consists of two sectors (i): agricultural and non-agricultural (or manufacturing) sectors (i = a; m). These two sectors are di erent in two dimension. First, their production functions are di erent in terms of input intensity. Second, agricultural sector is the subsistence sector which will be captured in the consumption side. The economy is populated with N d working (adult) individuals and, among them, N s are skilled workers. We call them agents and adults interchangeably. We de ne the (endogenous) share of unskilled labor being allocated to the agricultural sector be a. So that we will have N a = a (N d N s ) N m = ( a ) (N d N s ) where N a and N m are the endogenous share of agents in agricultural (farmer) and unskilled occupation (unskilled worker) of the manufacturing sectors respectively, or if we de ne as labor share: s = N s N d a = a ( s ) m = ( a ) ( s ) There are two types of agents (j) in the economy: skilled (s) and unskilled (u). The agricultural sector needs only unskilled labor (L a;u ) as input while the non-agricultural sector needs both unskilled labor (L m;u ) and skilled labor (L m;s ). So, the production functions for agricultural (Y a ) and non-agricultural products (Y m ) are given as: Y a = A a L a;u Y m = A m L m;ul m;s where A a and A m denote productivity in agricultural and non-agricultural sectors respectively. Also, as the production functions are constant return to scale, they can be 6

17 expressed in intensive forms: y a = Y a L a;u = A a = A a l a;u N d N d y m = Y m L m;ul m;s = A m = A m l N d N m;ul m;s d The CRS production functions can also be represented by a stand-in representative rm whose problem is de ned as: max py a w a;u L a;u () max Y m w m;u L m;u w m;s L m;s (2) That rm s maximization problem gives the market prices of factors: w a;u = pa a w m;u = A m lm;s l m;u w m;s = ( )A m lm;u l m;s where p is the relative price for agricultural goods in terms of non-agricultural goods. The free mobility of factors gives w a;u = w m;u = w u and for simplicity, w m;s = w s. So the wage premium is given by w s w u = lm;u which is high when the relative supply of skilled labor is lower, a downward sloping demand function. l m;s 4.2 Household (Labor Supply) The two types of agents live for two periods: childhood and adulthood. Each adult can give birth to both types of children (k): skilled (s) and unskilled (u). Children, if they are not to be endowed skills, only consume their parents time endowment. If the agent chooses to endow their children with skills, they have to pay goods cost in terms of non-agricultural goods. Adults make all the 7

18 economic decisions. In particular, the adults make the following decisions:. Consumption (both agricultural and non-agricultural) 2. Number of children to have and whether to educate them So the utility function for the agent will be: (c i;j a c) + ( ) ci;j m + n i;js Vs 0 + n i;ju Vu 0 (n i;js + n i;ju ) " where c i;j a is the agricultural consumption for type j agent working in sector i. c i;j m is de ned similarly. c is the subsistence level only de ned in agricultural sector; is the altruism coe cient and " is the curvature on number of children, we follow the literature and assume that 0 < " < 2. Finally, Vs 0 and Vu 0 are the utility that skilled children and unskilled children will enjoy as adults, respectively. The income I i;j of the agents are given as: I i;j = The e ective cost to raise children (e i;jk ) is di erent for di erent k: ( ( w u w s e i;js = s w j + s i=a + s ( i=a) e i;ju = u w j i=a is an indicator function to determine if the agent is working in agricultural sector 3. s and u are the time costs for educated and non-educated children respectively 4. We assume that s > s being the goods cost for agricultural and non-agricultural sectors respectively. In here, we take a simplifying assumption that child labors works in the same sector as their parents. This is consistent with Edmonds (2005) that relative few children work outside their family. 2 See Jones and Schoonbroodt (200) for implication 3 This is consistent with our argument in section as it is more costly for the children in the rural area to go to school. In the case of America, Goldin (998) has documented that the industrialized regions in the North and in Midwest had higher enrollment rates in early 20th century. 4 One can think of u = b u where b u is the true time cost of child rearing while is the potential gain from child labor. This is consistent with Doepke (2004). 8

19 The budget constraint is pc i;j a + c i;j m = w j X k=u;s e i;jk n k (3) More speci cally, pc a;u a pc m;u a pc m;s a + c a;u m = w u u w u n u ( s w u + s ) n s + c m;u m = w u u w u n u s w u + s n s + c m;s m = w s u w s n u s w s + s n s So the Bellman Equation of the agent is given by: V (w j ; i) = c i;j a subject to (3) max ;c i;j m ;n s ;n u 8 < : [(c i;j a c)] + Lemma The problem can be rewritten as: V (w j ; i) = max E i;j ;f i;j 8 < : s:t: w j = C i;j + E i;j where C i;j X pc i;j a ( )(ci;j m ) + (n i;js +n i;ju ) " n i;js V (w 0s ; i 0 ) + n i;ju V (w 0u ; i 0 ) + E i;j " + c i;j m, (p; ) = ( fi;j b s + fi;j b u (p; ) (wj E i;j pc) 9 = ; (4) h f i;j )" b V s (w0s ; i 0 ) + f i;j b V u (w0u ; i 0 ) p + ( ), E i;j = k=u;s e i;jk n k and f i;j being the share of child-rearing expenditure (E i;j ) spent on skill endowed children. Proof. Proof in Appendix Notice that i 9 = ; pc i;j a = = C i;j + p C i;j + p pc + pc 2 + pc 4 + p 3 5 9

20 while c i;j m = p 2 4 C i;j + p We can see that such expressions exhibit structural change for positive price. That is, the expenditure share in agricultural consumption decreases while that of non-agricultural goods increases as the total expenditure increases. This Lemma not only says that the value can be reduced to a indirect-utilitylike problem but also says that the problem can be simpli ed to only look at the child-rearing expenditure E i;j pc and its share of it spent on skill endowed children f i;j. Without the term, the above expression is clearly ( f i;j b s + fi;j b u )" linear in f i;j. When the term presents, the value function is convex in f i;j and 3 5 the agent will only have one type of children. This observation is summarized by the Proposition below: Proposition 2 In optimality, one agent will give birth to only one type of child, but not both. That is, either k = u or k = s but not both for an agent. Proof. Proof in Appendix The Proposition simpli es the analysis as we only need to compare the value of two types of children and do not need to consider any mix within one household 5. Given the child-rearing cost structure and wage income of the agents (i.e. w s > w u and s > s ), the skilled agents has the lowest relative education cost while the farmer has the highest relative education cost. That is: s w s + s u w s < s w u + s u w u < s w u + s u w u This cost structure is consistent with the historical picture and cross-country experience. The fertility decline rst happened in agents with higher income then it happened in agents with lower income. While development economists showed evidence that the urban proximity increase school attendance while children live far away from the urban area tend to work more (see for example Fafchamps & Wahba, 2006). The above cost structure provides a partial explanation for the observations as the one with the highest income (the skilled agents) has the lowest relative cost of raising educated children as the goods cost becomes smaller a share when income increases. The distance between urban and rural area tend to increase 5 This result is not new: see Doepke (2004) for discussion and Liao (20) for application 20

21 the cost of education so that s > s and we have the result that the agents live far away from the urban area tend to have the highest education cost and reduce their children s propensity to go to school. So, this model predicts that rich agents starts educating their children (and restrict fertility) rst. This is followed by the working class in the cities then by farmers. As from the above Proposition 2, the household problem can be further simpli ed as the follows: 8 V (w j ; i) = >< max E i;j ;k >: (p; ) Ci;j pc s:t: C i;j = w j E i;j So the agents only choose child-rearing expenditure E i;j children they want to raise k. 9 + E i;j " V (w 0k ; i 0 ) >= e i;jk " >; and the type of Although the value function depends on the price, but similar to Doepke, we will have the agents to be indi erent to wither types of the children if and only if V (w 0s ; i 0 ) e i;js " = V (w0u ; i0 ) e i;ju " This completes the optimality condition for the agents. In order for us to proceed from here, it is necessary for us to de ne the equilibrium to restrict our focus. 4.3 Equilibrium De nitionn 3 The steady state equilibrium is a vector of price fp; w u ; w s g, an allocation c i;j a ; c i;j m ; n i;jk ; i;jko, an aggregate allocation fl i;j ; a ; y a ; y m g and value function V (w u ; a), V (w u ; m) V (w s ; m) and migration cost B such that given the productivity A a, A m and skilled endowment ratio s s. Given the price vector, the rm maximizes its pro t according to () and (2) 2. Given the price vector, the households maximize their value according to (4) subject to (3) 2

22 3. The agricultural labor market clears l a;u = a [ a;us s n a;us ( a;us ) u n a;uu ] 4. The non-agricultural unskilled and skilled labor markets clear l m;u = m [ m;us s n m;us ( m;us ) u n m;uu ] l m;s = s [ m;ss s n m;ss ( m;ss ) u n m;su ] 5. The agricultural goods market clear that A a l a;u = a c a;u a + m c m;u a + s c m;s a 6. The manufacturing goods market clear that A m lm;ul m;s = a [c a;u m + a;us n a;us s ] + m c m;u m + s c m;s m + m;us n m;us s + m;ss n m;ss s 7. The skilled ratio is time invariant s = a a;us n a;us + m m;us n m;us + s m;ss n m;ss s a ( a;us ) n a;uu + m ( m;us ) n m;uu + s ( m;ss ) n m;su 8. The value for both unskilled agents are the same V (w u ; a) = V (w u ; m) B = V (w u ) 9. Expectations are realized V (w 0u ) = V (w u ) V (w 0s ) = V (w s ) In what follows, we will focus on the labor markets and the agricultural goods market as, by Walras Law, the non-agricultural goods sector will be cleared when all the four other markets cleared. The following proposition summarizes some of the main result of the model: 22

23 Proposition 4 Along the steady state equilibrium, the following are true. There can only be one type of agents indi erent between the type of child; 2. At least a fraction of skilled workers would give birth to skilled child while at least a fraction of agricultural workers would give birth to unskilled child. Proof. Proof in Appendix Then we will have the nal technical assumption before we go to the computation section. From here on, we will assume that > " 6. With this assumption, we will have the following (technical) proposition. Proposition 5 If > " then the value of (E ms ) and (E au ) are uniquely de ned. Proof. Proof in Appendix So in this economy, there can be ve regime: V (w s ) V (w u ) = s w s + s " u w s s w s + s u w s " < V (w s ) V (w u ) < s w u + s u w u " V (w s ) V (w u ) = s w u + s " u w u s w u + s u w u " < V (w s ) V (w u ) < s w u + s u w u " s w u + s " = V (w s ) u w u V (w u ) Notice that both V (ws ) V (w u ) < s w s + s " u w and V (w s " ) s V (w u ) > s w u + s u w are u not admissible as there will not be either skilled and unskilled workers in the next period and cannot contribute to the equilibrium For historical reasons that the skilled workers are minority and that the unskilled workers and farmers give birth to larger amount of children. So, the rst two regime that only skilled workers give birth to skilled children cannot s sustain the result that skilled ratio being constant. As skilled children will s be out-weighted by the unskilled children so that s will decline. s 6 Greenwood and Seshasdri s (2002) parameter values t this assumption. 23

24 Moreover, there will be two points to be made for the model. First, consider the situation when time cost of child-rearing is zero (i.e. u = s = 0), then the wage premium is: w s = lm;u w u l m;s = ( s ) ( a ) s Notice that a is decreasing as agricultural productivityincreases keeping lm;u other parameters constant. As is given, there exist unique l m;s ratio such lm;u l m;s we will have w s that ws w u = and for all lm;u l m;s > w u >. > is very important for the economy as it ensure the return on education is positive. If w u > w s, there will not be incentive for the parents to educate their children. However, the wage premium depends on the relative supply of unskilled and skilled labors. When the economy is with very low agricultural productivity, there will be a large number of workers allocated into the agricultural sector (i.e. a high a ). So only a small s can be sustained in order to have skilled wage premium. As agricultural productivity increases, a decreased and the economy can sustain a higher s without driving the skilled wage too low. Throughout the subsequent analysis, we will only focus on the separating market such that w s w u >. Second, this model does not only predict upward mobility but also possible to predict downward or no mobility. w s w u In this aspect, the model is similar to Mookherjee, Prina and Ray (202). In our model, the no mobility case is knifeedge case in the steady state equilibrium de ned. In historical content, it is more common for the children of unskilled agents to receive some education when the country started to develop (i.e. upward mobility). 4.4 Calibration Our model is a general model that could be, in principle, applied to every economy. However, due to the completeness of the US data, we would calibrate the model using the IPUMS-USA data to match the moments. We pick 870 as the rst steady state as the agricultural share seemed to be quite stable. Table 4 summarizes the parameter in the rst steady state. First we x the productivity of the two sectors to be in the rst steady state. That is A a = A m =. And we x the other four parameters from the literature. We follow Doepke (2004) 24

25 to pick = 0:32 and " = 0:5. Then we follow Greenwood and Seshadri (2002) to pick = =3 and = 0:9 7. For s, we take the value of 0.3. This is consistent with the value of time used for average hours per day spent on caring for and helping household children (in the past few years, it was.3 hours per day or around 9 hours per week). In around 880, based on Ramey and Francis (2009), the average weekly hours per employed person is projected to be above 55 hours per week in around 880. So, the fraction of time devoted to childrearing is 3-4% of the total time spent to work and child-rearing. Table 4 summarize the assigned parameter. [Table 4 Here] Then we calibrate the remaining 4 parameters ( u, c, and s 8 ) jointly to target four data moments. They are the data values of total fertility, agricultural employment share, non-agricultural wage premium and the fertility ratio between agricultural and non-agricultural sectors in the year around Table 5 summarizes the value for the variables. Also, the education cost s will be determined by the model to equalize the value for unskilled agents to have skilled and unskilled children 20. [Table 5 Here] The implied value of the model matched quite well with the data that we observed (see Table 6). No surprise that the agricultural and non-agricultural fertility matches the data as we matched both total birth, fertility ratio between agricultural and non-agricultural sectors and employment share jointly for calibration. So, the fertility rate should be of an exact match. However, the average agricultural consumption share, which we did not target, is also quite close to data. [Table 6 Here] In the second steady state, which we taken as 930 (before the WWII), we assume that the education costs s in terms of goods cost by the two types of agents are the same. Notice that there is an increase in the calibrated value of the child-rearing cost in the second steady state. Bar and Leukhina (200) 7 In Greenwood and Seshadri (2002), the food share is /2 of the manufacturing and we have the constraint that the two share add up to be one, we pick = =3. For, the value we pick is consistent with their value as 0:9 = 0:09. 8 Originally, we xed a de nition for s and see how the change of child-rearing cost a ects the fertility. However, this strategy su ers a drawback that the de nition of skill changes over time changed. So it is better to let the model de ne s. 9 Whenever available, we use the average values from the year s is irrelavent for this steady state as no farmers will educate their children. We only need to ensure that s > s. 25

26 summarize the possible reasons for the increase in the child-rearing cost. One of the main reason that is related to our model is that the child labor was restricted along the time that we are interested. So, the cost of raising an unskilled child is increasing (as they will be stay idled even if they do not go to school). Notice that the time cost to raise high-skilled children also increased, although to a less extent. This is because the average weekly working hours decreased during the last century (Ramey and Francis, 2009). So given that agents spent 9 hours per week taking care of their kids, the decrease in time spent on work means that the fraction of time spent on child-rearing increases. So we rst x s = 0:7 which corresponds to 9 hours of child-rearing given the working time in the year 930. Then we calibrate u such that the average birth is 2:37 birth per family in 930. Moreover, we also let the value of s to change to match the argicultural employment share in 930. The value of the parameters are summarized as Table 6 below. [Table 6 Here] The model prediction is summarized in Table 7 below. While we get the nonagricultural birth rate, we underestimate the agricultural birth. This maybe due to the fact the implied education cost di erence is too small in the model so that more farmers choose to substitute quantity for quality in the model when compare to data. [Table 7 Here] 4.5 Result After we obtain the parameter values of the two end points, we do the following comparative statics exercise. First, we make s increase linearly and extend the series to as early as 850 and as late as 940. Then we vary the cost u to match the total fertility rate in the economy in respective years. And then we vary the skilled endowment ratio s to match the non-agricultural wage premium implied by Caselli and Coleman (200). The model will predict the value for sectorial fertilities (agricultural and non-agricultural sectors) and agricultural employment share. Moreover, in the exercise, we also tradeo between two regime choices (namely regime 3 and regime 5 from the previous section). We rst de ne a loss function which is a weighted average between the data and model prediction. The argument in the loss function is total fertility and agricultural employment share (both data and predicted value). So, we minimize the joint distance between 26

27 the model prediction of agricultural employment share and the total fertility in order to choose which regime to adopt. Notice that there is a switch of regime from regime 3 to regime 5 in the year 90. By doing this we can know how our model performs in explaining the trends of agricultural and non-agricultural birth rate trends. The result is in Figure 7 and Figure 8 below. [Figure 7 Here] [Figure 8 Here] It can be seen that the sectorial fertility tracks quite well with the data, the agricultural fertility rate, however, is o. This may be due to the fact that the implied cost of education in the agricultural sector is too low for the agricultural sector so that too many of them chooses to restrict fertility and let their children to educate. This drives the fertility too low. This model then is di erent from the literature that it (correctly) replicates the di erences in both trends as well as timing of fertility transitions in di erent sectors. In particular, it predicts that the agricultural fertility have to stay roughly constant while the non-agricultural fertility have to decline gradually in the late 9th century. As towards the early 20th century, there was a drop in agricultural fertility rate, similar to the data. 5 Concluding Remark and Further Research We asked one single question to begin with: Is the force that drives demographic transition in the U.S. and the force that makes the poorer countries having higher fertility the same one? Throughout this paper, we provided an answer for that: Relative importance of the agricultural sector is one of the important factors to understand the question. Unlike other sectors, agricultural sector is a subsistence sector, meaning that it has to be ful lled before labors can be allocated to other sectors. So in poorer countries or in the U.S. when the agricultural productivity is relatively low, a large amount of labors are to be allocated into the agricultural sector. As children (or any other forms of physical labor) are a valuable source of labors, a large part of the child-rearing cost can be "subsidized" whem using child labor. This makes the relative cost of having children lower. To make the things worse, as agricultural sectors are mainly concentrated in the rural areas that are away from major cities (that usually schools are located in), the schooling cost for those farm children is relatively higher. These two factors together making the agricultural sector having a 27

28 higher birth with lower average human capital. With these in mind, we built a two-sector model with two types of workers. The agents decide how many kids they will have and if they are to educate them. By adding educational cost (in terms of goods cost, or schooling fee) in to the model, we created di erent relative costs for agents with di erent incomes and sectorial employments. This then creates di erence in the pace and the timing of the fertility transition. Calibrating the model with the U.S. data, we are able to replicate some of the features that we observed in the U.S. including the di erences in the trends for fertility transition. This model can also be used to match the sectorial employment share in the U.S. The future research will be along two avenues:. The agricultural employment share is systemically lower than the data observed. We suspect that the main reason is that the model implied education cost in the agricultural sector being too low so that the it drives the fertility rate to be too low. So research is needed on this area to correct it. 2. The model can be potentially used to understand the cross-country di erences in fertility level. So further research on this avenue will be carried. 28

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32 6 Appendix : Figure Figure. Children Ever Born and Agricultural Employment Share in USA. Data source: Children Ever Born, IPUMS-USA Agricultural Employment Share, Herrendof et. al. (204) 32