If you definitely plan on getting the MEI, it is worth considering doing the ME portion first.
What Flyhayes hinted at is that the first instructor check ride has to be done in a complex aircraft and the MEI certification will complete that requirement.
When you add the CFI (ASEL) it will be an add-on so there will be some things you won't need to do and you can do it in a non-complex (i.e. cheaper) aircraft.
I am not certain that it will ultimately save money as the same applies to doing ASEL first; then AMEL is an add-on with less to accomplish.
But, I think the real benefit lies in the maneuvers to be completed for each rating.
There are a lot of factors to consider. If you plan on spacing out the ratings (for monetary or other reasons), it may be best to knock out ASEL first as you can start employing that one right away, like rickair mentioned, while you build time and money in the bank.
Download the PTS (
https://www.faa.gov/training_testing...-S-8081-6D.pdf) and you can see the differences by referencing the add-on table on page 21 and page 96. This also includes an update, change 4, updated Jan 2015 clarifying the use of complex aircraft.
Cheers,
Heed