Mónica -

Hi.  Well, I do have a reference list which I collected for the information they have on ratio estimation, and I expect that many, if not most of them, will have what you want as well. 

Here it is:

https://www.researchgate.net/publication/317914104_Handout_Bibliography_for_Comparison_of_Model-Based_to_Design-Based_Ratio_Estimators_Poster

Hope some of it is useful for you. 

Cheers - Jim

PS -

By the way, I don't think that what Cochran has would be considered outdated in the area you described. That book is a classic. It is well written and still very useful.

It's also good to know that he was also apparently a very nice person.

:-)

Mónica - 

Looking at this a bit more, I see that there is remarkably little in the textbooks I have checked which addresses your issue: "multistage stratified sampling and sample size calculation."  Actually, in my quick check, I saw nothing specifically addressing this.  Sample size estimation would definitely be complex because you could make tradeoffs between the two stages.  Also, sample size is not the only consideration.  Cluster sampling can be done as a cost saving, or data collection logistics easing method, which actually increases sample size, but makes that sampling more practical.

In Sampling, 3rd ed, 2012, Steven K. Thompson, Wiley, there is a chapter on Multistage Designs, Chapter 13, but the section which may be closest to being of interest to you might be Section 13.5. "Derivations for Multistage Designs," pages 179 to 182, especially 179 and 180, I think. 

Otherwise, most textbooks seem to leave it to you to go from the information they give on multistage sampling and on stratified sampling and inference, to put those things together yourself.  Perhaps you should look for a journal or conference paper which specifically addresses this. 

If you have auxiliary data for a model-assisted approach, you might look at "Optimal Regression Estimator for Stratified Two-Stage Sampling," by N. Nangsue and Y.G. Berger in Contributions to Sampling Statistics, Mecatti, Conti, and Ranalli, eds., Springer, 2014, pages 167-177.  The abstract starts with this: "Regression estimators are often used in survey sampling for point estimation.  We propose a new regression estimator based upon the optimal estimator proposed by Berger, et.al.(2003).  The proposed estimator can be used for stratified two-stage sampling designs when the sampling fraction is negligible and the primary sampling units (PSU) are selected with unequal probabilities.  For example .... We assume ...."  You get the picture.  This is one of many different scenarios one might find. 

In Model Assisted Survey Sampling, by Särndal, Swensson and Wretman, Springer, 1992, softcover printing 2003, page 319, they have Section 8.8, "Stratified Clusters and Poststratified Clusters," pages 319-322, where on page 320, they discuss "Stratified Cluster Sampling," saying "In some applications, it is a good idea to let each group form a stratum, and to select a stratified sample of clusters, possibly followed by subsampling within each of the selected clusters."  But I think you mean to randomly select clusters and then do stratified sampling within clusters, not stratify and then pick clusters, which is what I think the quote from Särndal, Swensson and Wretman meant. 

So there are a lot of possibilities.  To find exactly what you want may mean finding the right paper, or deriving it yourself from information such as that in Cochran or Thompson. 

Sorry I did not see anything more promising. 

Best wishes - Jim