Measure of New Firms Born in Each Period

Question

In an economy with stochastic overlapping generations of firms, how do we easily understand the measure of new firms born?

The set-up states:

"In each period, measure $\rho\in(0,1)$ of new firms are born and are endowed with net worth $w_0>0$. Firms survive to the next period with probability $(1-\rho)$ and hence the measure of firms alive in every period is 1."

$\textbf{My Question:}$

1. What is the measure of firms?
2. How does the measure of firms relate to the firm's survival probability onto the next period?

I think I am not understanding clearly when the author uses these related but distinct terms. Can someone explain easily?

Answer 1

I don't know your model. Just some general input:

It seems that you have a model with a continuum of firms. That is, there are many firms and each one is strategically small. In mathematics, a measure of a set is a way to express the size of this set. Hence, the measure of firms alive represents "how many firms are active". The authors normalize this measure to 1. If it helps, think of 1 million individual small firms and think of "measure 1" as a normalization of this million.

In your model, each period a only a fraction $(1- \rho)$ of this measure 1 survives and additional measure $\rho$ of firms are "born". Thus, the total measure of firms remains 1. Again, it may help to think of it as a normalization of a large finite number. For example, if $\rho=0.5$, only 500 000 out of 1 000 000 firms from last period survive and 500 000 additional firms are created such that there are still 1 000 000 firms.