Comprehensive look at the gold gen/health gen trade offs of support items.
I will calculate the bonuses that you get from the items by 10 minutes into the game. This is the theoretical maximum amount possible. (not counting item stats like health regen)
-2 gold per 10. 20 health and the kill gold for the minion(once per 40 seconds starting with two charges
14possible procs. 6 on cannon, 8 on melee minion. 6(45)+8(20)=280 hp and 430+120g
-4 gold and 5 health per minion death near you
107(4 gold + 5 health)= 428 gold and 535 hp
-2 gold per 10 and 8 g per 10 (when attacking a champion)
63 possible procs(8g)= 504g+120g
TL;DLM (Too long, don't like math)
gives 280hp and 550 gold
gives 535hp and 428 gold
gives 624 gold
In conclusion spellthiefs gives the most gold, relic gives the median amount of gold along with 280 hp (in regenerated form), and ancient coin gives 535hp and 428 gold. They all have seemingly reasonable tradeoffs. Either gold, slightly less gold and health, and a ton of health but less gold.
