# 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:

$$N_{t} = N_{0}(1+n)^{t}.$$

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