What are the ethics and economics of cashing a check for 10 cents?

I overpaid a bill – I owed $74.90 on an account I rarely use. I paid $75.00, so the next month I had a credit balance of ten cents.

I didn’t use that account for 30 days, and the bank mailed me a check for a dime. Nice of them.

Here’s the background of my question:

1) the bank spend 55 cents to mail me the check plus the cost of the envelope and the person who stuffed it.

2) If I deposit the check through cell-phone deposit, it will cost them bank some amount of money to process and verify the amount, then proof it and store the image and so on. Plus whatever processing the Federal Reserve does.

3) If I go into the bank to deposit the check, then I’m taking some minutes of the teller’s time, plus the same back-office processing.

From the standpoint of resources and effort, my bank and the merchant’s bank are spending some overhead in order to process my 10 cent check.

Ethically, should I be placing them in a position of expending their resources for a measly 10-cent credit to me?