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Malthusian regime: The Dynamical System


Hence, the rate of technological progress between time t and t 4- 1 is a positive, increasing, strictly concave function of the size and level of education of the working generation at time t. Furthermore, the rate of technological progress is positive even if labor quality is zero read more.

As will become apparent, the dynamical system of the described economy is rather complex. Population size does not play a qualitative role in the evolution of the economy, except for its significant role in the takeoff from the Malthusian Regime. Hence, in order to simplify the exposition without affecting the qualitative results, the dynamical system is analyzed initially under the assumption that population size has no effect 011 technological progress. In particular, let
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Malthusian regime: The Basic Structure of the Model 5

As is apparent from (17). e”(gt+1) depends upon the third derivatives of the production function of human capital. A concave reaction of the level of education to the rate of technological progress appears plausible economically, hence it is assumed tha
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Malthusian regime: The Basic Structure of the Model 4

Optimization

Members of generation t choose the number and quality of their children, and therefore their own savings and old-age consumption, so as to maximize their intertemporal utility function. Substituting (8)-(10) into (7), the optimization problem of a member
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The optimization with respect to nt implies that, as long as potential income at time t is sufficiently high so as to assure that c*+i > c, the time spent by individual t raising children is a fixed fraction 7, whereas the remaining fraction 1 — 7 is devoted for labor force participation. However, for low levels of potential income, the inequality constraint binds. The individual consumes the subsistence level, c, and uses the rest of the time endowment for childrearing. That is further,

Malthusian regime: The Basic Structure of the Model 3

Budget Constraint: Quantity-Quality of Children Vs. Consumption

Following the standard model of household fertility behavior (Becker, 1960) it is assumed that the household chooses the number of children and their quality in the face of a constraint on the total amount of time that can be devoted to child-raising and labor market activities. We further assume that the only input required to produce both child quantity and child quality is time. Since all members of a generation are identical in their endowments, the budget constraint is not affected if child quality is produced by professional educators rather than by parents.

Let rq + ree/+i be the time cost for a member of generation t of raising a child with an education level e^+1. That is. rq is the fraction of the individual’s unit time endowment that is required in order to raise a child, regardless of quality, and re is the fraction of the individual’s (or of an equally educated teacher’s) unit time endowment that is required per each unit of education of each child.

Malthusian regime: The Basic Structure of the Model 2


The multiplicative form in which technology, At, and land, Xt, appear in the production function implies that the relevant factor for the output produced is the product of the two, which we define as “effective resources.” review

Output per worker produced at time t, yu is therefore
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The total return to land (including appreciation) at time pt, and the rate of return to capital at time r*, are equal to one another, since individuals may save either by purchasing capital or land. Hence, given the constancy of world interest rate at the level r, it follows that pt—rt = r.

Malthusian regime: The Basic Structure of the Model

In the Modern Growth regime, technology and output per capita increase rapidly, while population growth is moderate.

The rest of this paper is organized as follows. In Section 2, we formalize the assumptions about the determinants of fertility and relative wages presented above, and incorporate them into an overlapping generations model. Section 3 derives the dynamical system implied by the model, and analyzes the evolution of the economy along transitions to the steady state. Section 4 concludes by discussing possible extensions of the model.

The Basic Structure of the Model

Consider a small, open, overlapping-generations economy that operates in a perfectly competitive world where international capital movements are unrestricted and economic activity extends over infinite discrete time. In every period the economy produces a single homogeneous good that can be used for either consumption or investment. The good is produced using physical capital, efficiency units of labor, and land.

Malthusian regime: Introduction 5


New technology will create a demand for the ability to analyze and evaluate new production possibilities, which will raise the return to education. Schultz (1975) cites a wide range of evidence in support of this theory. Similarly, Foster and Rosenzweig (1996) find that technological change during the green revolution in India raised the return to schooling, and that school enrollment rates responded positively to this higher return. Such an effect would be a natural explanation for the dramatic rise in schooling in Europe over the course of the 19th century.

The effect of technology on the return to human capital in which we are most interested is the short run impact of a new technology. In the long run, technologies may either be “skill biased” or “skill saving.” But we would argue that the introduction of new technologies is mostly skill biased.10 For example, Williamson (1985; Table 3.7) concludes that early industrialization raised the return to skills. The ratio of average wages of skilled workers to unskilled workers in Britain rose from 2.45 in 1815 to 3.77 in 1851, whereas the 60 years after 1851 saw a significant reduction in wage inequality. If technological changes are skill-biased in the long run, then the effect on which we focus will be enhanced, while if technology is skill-saving then our effect will be diluted.

Malthusian regime: Introduction 4

The emergence from the Malthusian trap raises intriguing questions. How is it that the link between income per capita and population growth, which had for so long been a constant of human existence, was so dramatically severed? And how does one account for the sudden spurt in growth rates?

The existing literature on the relation between population growth and output has tended to focus on only one of the regimes described above. The majority of the literature has been oriented toward the modern regime, trying to explain the negative relation between income and population growth either cross-sectionally or within a single country over time. Among the mechanisms highlighted in this literature are that higher returns to child quality in developed economies induce a substitution of quality for quantity (Becker, Murphy, and Tamura, 1990); that developed economies pay higher relative wages of women, thus raising the opportunity cost of children (Galor and Weil, 1996); and that the net flow of transfers from parents to children grows (and possibly switches from negative to positive) as countries develop (Caldwell, 1976). The negative effect of high income on fertility is often examined in conjunction with a model in which high fertility has a negative effect on income due to capital dilution add comment.

Malthusian regime: Introduction 3

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Figure 1 shows the growth rate of total output in Western Europe between the years 500 and 1990, as well as the breakdown between growth of output per capita and growth of population. The figure demonstrates that the process of emergence from the Malthusian trap was a slow one. The growth rate of total output in Europe was 0.3 percent per year between 1500 and 1700, and 0.6 percent per year between 1700 and 1820. In both periods, two-thirds of the increase in total output was matched by increased population growth, so that the growth of income per capita was only 0.1 percent per year in the earlier period and 0.2 percent per year in the later one. In the United Kingdom, where growth was the fastest, the same rough division between total output growth and population growth can be observed: Total output grew at an annual rate of 1.1 percent in the 120 years after 1700, while population grew at an annual rate of 0.7 percent.

Malthusian regime: Introduction 2


The Malthusian model implies that, in the absence of changes in the technology or in the availability of land, the population will be stable around a constant level. Further, improvements in technology will, in the long run, be offset by increases in the size of the population. Countries with superior technology will have denser populations, but the standard of living will not be related to the level of technology, either over time or across countries.

The Malthusian model’s predictions are consistent with the evolution of technology, population, and output per capita for most of human histoty. First, the standard of living was roughly constant. Maddison (1982) estimates that the growth rate of GDP per capita in Europe between 500 and 1500 was zero. Lee (1980) reports that the real wage in England was roughly the same in 1800 as it had been in 1300. Clark (1957) concludes that income per capita in Greece in 400 BC was roughly equivalent to that in Britain in 1850 or Germany and France in 1870 add comment.

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