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,

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is the critical level of potential income above which the individual chooses to consume more than the subsistence level and below which the individual consumes the subsistence level.
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As long as the potential income of a member of generation £, zt = wthu is below z, then the fraction of time necessary to assure subsistence consumption, c, is larger than 1 — 7 and the fraction of time devoted for child rearing is therefore below 7. As the wage per efficiency unit of labor increases, the individual can generate the subsistence consumption with smaller labor force participation and the fraction of time devoted to childrearing increases.

As will become apparent from Proposition 1, the entire increase in the fraction of time devoted to childrearing is used to raise the quantit}’ of children without a change in quality. As long as potential income of a member of generation t, zt = wtht, is higher than the subsistence consumption constraint is not binding and a constant fraction of the unit time endowment, 7, is devoted to child rearing regardless of how high wages are. Hence further increases in wages are devoted entirely to increased consumption.

Figure 3 shows the effect of an increase in potential income zt on the individual’s choice of total time spent on children and consumption. As is apparent from the diagram the income expansion path is vertical until the level of income passes the critical level that permits consumption to exceed the subsistence level. Thereafter, the income expansion path becomes horizontal at a level 7 in terms of time devoted for childrearing.

Regardless of whether potential income is above or below z, increases in wages will not change the division of child-rearing time between quality and quantity. What does affect the division between time spent on quality and time spent on quantity is the rate of technological progress, which changes the return to education. Specifically, using (12), the optimization with respect to et+i implies that independently of the subsistence consumption constraint
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