Q: Last year, my family and I had first-class tickets on American Airlines to fly from Boston to Phoenix. American delayed that flight, rebooked us and then downgraded us to coach class on another flight.

A few days later, I received an email from American that it had refunded the fare difference between first class and economy class, and advised me to contact my travel agent.

When I returned from our vacation, I contacted the travel agent at AAA, who had arranged our tour through Pleasant Holidays. The AAA travel agent contacted Pleasant Holidays, which had no record of a refund from American.

Over the course of a few months, I followed up with AAA and Pleasant Holidays on the status of this credit to no avail. Finally, I reached out to one of the American executives who you list on your consumer advocacy site. The airline contacted me a day later and referred me to a website to check the status of the refund. That website indicated that a refund had been completed.

I still don’t have our refund, though. Can you help me?

— Thomas Sennett, West Hartford, Connecticut

A: If you get bumped from first class to economy — that’s called an involuntary downgrade — you should get a prompt and full refund of the fare difference. Looking at your case, I can see the problem — and a possible complication.

The complication is the fare difference. Airlines often calculate the fare difference in a way that is advantageous to them. So, if American were to give you a refund, it would be based on the difference between a first-class ticket and an expensive walk-up fare in economy class instead of a less expensive fare purchased in advance. That way, it wouldn’t have to refund you much — or anything at all.

The second issue is that you have a tour operator and an agent involved. A good travel agent can fight for your refund, but the bureaucracy of getting the money from the airline to the tour operator to the travel advisor and finally to you — well, that’s probably why it’s taking so long.

When I asked you to send me the paper trail between you, American, AAA and Pleasant Holidays, you complied. I’m so impressed that you stayed off the phone and did everything by email. You had a thorough and very helpful paper trail that made this case relatively easy to resolve.

I see you also availed yourself of the company contacts for American Airlines that I publish on my consumer advocacy site, Elliott.org. Nice work! I also have contacts for AAA in case you ever need them.

I contacted AAA on your behalf to find out the status of your refund. It reached out to Pleasant Holidays. The company found your money and returned it to you. And, best of all, American Airlines didn’t try any funny airline math. You received a check for $1,611 from the carrier.

Christopher Elliott is the founder of Elliott Advocacy (elliottadvocacy.org), a nonprofit organization that helps consumers solve their problems. Email him at chris@elliott.org or get help by contacting him at elliottadvocacy.org/help.