## 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

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.

Individuals: Fertility, Human Capital, Saving, and Consumption

In each period t a generation that consists of Lt individuals joins the labor force. Each individual has a single parent. Individuals within a generation are identical in their preferences and their level of human capital. Members of generation t live for three periods.16 In the first period of life (childhood), t — 1, individuals consume a fraction of their parent’s time. The required time increases with children’s quality.

In the second period of life (parenthood), t, individuals are endowed with one unit of time, which they allocate between childrearing and labor force participation. They choose the optimal mixture of quantity and quality of children and supply their remaining efficiency units of labor in the labor market. They earn the competitive market wage per each efficiency unit of labor and save their income for future consumption. In the third period of life (old age), t + 1, individuals do not work. They consume their savings from the previous period along with accrued interest.

Preferences

The preferences of members of generation t, which are defined over consumption in old age, above a subsistence level с > 0, as well as over the potential aggregate income of their children are depicted in Figure 2.17 They are represented by the utility function

where nt is the number of children of individual t, ht+1 is the level of human capital of each child, and wt+1 is the wage per efficiency unit of labor at time t + l.19

The utility function is strictly monotonically increasing and strictly quasi-concave, satisfying the conventional boundary conditions that assure that, for sufficiently high income, there exists an interior solution for the utility maximization problem. However, as depicted in Figure 3, for a sufficiently low level of income the subsistence consumption constraint is binding and there is a corner solution with respect to the consumption level.