I know that we can allocate observations from the total sample to each strata either using proportional allocation ($n_{h}/N_{h} = n/N$) or optimal allocation:
$$n_{h} = \left [ \frac{\frac{N_{h}S_{h}}{sqrt (c_{h})}}{\sum_{l=1}^{H}\frac{N_{l}S_{l}}{sqrt (c_{l})}{}} \right ]n \tag{1}$$
Where $N_{h}$ and $n_{h}$ are the population size and sample size in each stratum respectfully. $c_{h}$ is the cost of sampling one unit from stratum h, $S_{h}$ is the population standard deviation of stratum h and $n$ is the total sample size. What I am confused about is how do we actually choose $n$. The book I'm using (Lohr's SDA) says that
$$n = \frac{n_{0}}{1 + \frac{n_{0}}{N}\sum_{h = 1}^{H}\frac{N_{h}S_{h}^{2}}{Nv}} \tag{2}$$
where
$$n_{0} = \frac{z_{\alpha/2}^{2}v}{e^{2}} \tag{3}$$
($e$ is the margin of error) is the sample size if all strata fpcs could be ignored and
$$v = \sum_{h =1}^{H}\frac{n}{n_{h}} \left ( \frac{N_{h}S_{h}}{N} \right )^{2} \tag{4}$$
What confuses me is that $n$ is calculated using $v$ and $n_{0}$ which themselves are calculated using $n$. Could someone provide some clarification?
