Preliminaries
In this model, time is discrete and extends from $t=0, 1, \ldots, \infty.$ Individuals make decisions at points in time. We shall have initial conditions detailing the state of the economy at $t=0.$
Individuals live for two periods
The main difference with respect to the Ramsey model is that in the OLG model, individuals live for two periods. Note: this means that, at every point in time, two generations are alive and overlap.
This is relevant: the economy goes on forever but individuals only operate during some periods. Hence, there will be infinite two-period-lived generations. In particular, at $t=0$, we will have a young and an adult generation. This adult generation will die at the end of $t=1$, the young generation will become adults and have children: the new young generation of $t=2.$ Hence, we can represent the generations diagrammatically —in brackets I have denoted the year in which each generation was born.
| $t=0$ | $t=1$ | $t=2$ | $t=3$ | $t=4$ | $t=5$ |
|---|---|---|---|---|---|
| Old (t=-1) | Die | ||||
| Young(t=0) | Old (t=0) | Die | |||
| Young(t=1) | Old (t=1) | Die | |||
| Young(t=2) | Old (t=2) | Die | |||
| Young(t=3) | Old (t=3) | Die | |||
| Young(t=4) | Old (t=4) | ||||
| Young(t=5) |
To simplify the model, we assume that each adult born in $t-1$ has $n>-1$ children.
Note: we assume the number of children to be constant.
More complex set-ups include endogenous fertility.
These are the young population at time $t$
Therefore, the total population $N$ at time $t$ is composed of adults and young people.
$$N_{t} = \underbrace{N_{t-1}}_{\mathrm{Adults}} + \underbrace{N_{t-1}n}_{\mathrm{Youngs}} = N_{t-1}(1+n).$$
The total population at any time $t$ is:
$$N_{t} = N_{0}(1+n)^{t}.$$