I didn't look at my booking closely when I purchased our tickets for our last trip, so I don't know if PJIA charges both an improvement fee and the 'departure' fee. If it is just the departure fee it is not unlike many airports in North America that charge an airport improvement fee. The airport keeps that fee and uses it, in theory, to improve the airport. The fee, like a departure tax is usually included in your ticket cost so you may not notice it.
My point is that the name may be different, but the fee probably isn't that unusual. And the fee and a departure tax wouldn't be extraordinary in the Caribbean.
(Please don't mistake the above for an endorsement of these fees and taxes.)
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