Which of the 3 support items can earn you the most gold?

Solution 1:

This answer was given on 05-02-2014. The current patch is 4.1. The answer will probably get less accurate and/or relevant in future patches.

I'm gonna have to make some assumptions to create a proper example here.

1. I'm not taking in account any gold increases from masteries.
2. I'm taking the first ten minutes of the game as 'laning phase', The values may change if the laning phase is longer, but you really can't know that up front so early decision making should not change.
3. I'm not taking into account any jungle farming
4. I'm assuming the 'level 1' version of the items, buying the upgrade at some point will ofcourse change the data

Relic shield:

Minions start spawning from 1:30. So at ten minutes the 18th wave will spawn. This means that for the first ten minutes you have 17 waves to farm (the 18th will be underway at ten minutes). A cannon minion spawns in the wave every 3 waves and the Relic shield has a 60 second internal cooldown. So you can last hit the following:
wave 1 (melee minion) +23 +5
wave 3 (cannon minion) +40 +5
wave 5 (melee minion) +23 +5
wave 7 (cannon minion) +40 +5
wave 9 (melee minion) +23 +5
wave 11 (cannon minion) +40 +5
wave 13 (melee minion) +23 +5
wave 15 (cannon minion) +40 +5
wave 17 (melee minion) +23 +5

total 320 gold

Spellthief's edge:

The internal cooldown is 10 seconds per enemy. In a standard game you will be able to start hitting your opponents from about 2:00 when both teams go to their lane after helping their jungler. That means you can hit your opponents 48 times (both of them, as the cooldown is not shared between enemies hit) so 96 procs on the item. It gives you 4 gold per proc. Spellthief's edge also gives a passive 2 gold per 10 seconds. This is 120 gold for ten minutes.

total 504 gold

Ancient coin:

This one is easy, there are 17 waves (as said in the Relic shield part) to be farmed. Four of them have an extra minion. totaling 106 minions. The coin gives 2 gold. It also has the 2 gold per 10 seconds passive.

total 332 gold

Conclusion:

Spellthief's edge is the best item for pure gold gain, but it only grants gold to you. The Relic shield also gives gold to your adc (85). Ancient coin seems like the worst choice gold wise, but it's a lot more passive (and easy, 'safe pick' etc). Depending on when you buy the upgrades this coud change a lot. I think it's too complex to go into that in an answer here, if you really want to know about the upgrades as well, I suggest looking up a support guide that covers these items.

sources:
Relic shield wiki entry
Spellthief's edge wiki entry
Ancient coin wiki entry
Minions wiki entry

Solution 2:

There are 3 build paths (I neglected the final item, since it has no effect on the gold income):

Path: Frost Queen's Claim

Spellthiefs Edge: Autoattack on a champion gives 4 Gold, up to once every 10 seconds. +2 Gold per 10 seconds.

Frostfang: Autoattack on a champion gives 8 Gold, up to once every 10 seconds. +4 Gold per 10 seconds.

Path: Talisman of Ascension

Ancient coin: Being near a minion's death gives 2 Gold

Nomad's Medallion: Being near a minion's death gives 4 Gold. +2 Gold per 10 seconds.

Path: Face of the Mountain

Relic Shield: Killing a minion + 5 Gold - Max. 2 Charges which recharge every 60 seconds.

Targon's Brace: Killing a minion + 10 Gold - Max 3 Charges which recharge every 30 seconds.

Assumptions

Let's say, first minion wave arrives at 2.00 minutes. Minion spawn time is 30 seconds.

• Caster minions: 15 Gold +0,5/3min
• Melee minions: 20 Gold +0,5/3min
• Siege minions: 40 Gold +1/3min (spawns every 3 minion waves)
• Considering every 3rd wave with siege minion makes + 1/3 minion per wave
• 2 waves per min = 2*6,3 = 12,6 minions / minute

Some other stuff we assume:

• First back at minute 6
• At this base we bux the upgraded version
• We neglect other items
• We neglect the time you lose while porting back and walkign down to lane

Some Math:

Path: Frost Queen's Claim

Let's assume first hit on an enemy is also at minute 2. We can trigger the passive 6 times per minute. This means:

• 6 minutes of gp/10 (12 Gold/Minute) = 6 min * 12 Gold = 72 Gold
• 4 minutes of passive (6 times/Minute) = 24 times * 4 Gold = 96 Gold

Bonus Gold at 6 minutes: 168

Now we assume to have Frostfang. Since this item doesn't depend on minion gold value we can easily calculate gold per minute.

• 4 Gold / 10 sec = 24 Gold / Minute
• 8 Gold / 10 sec (Autoattack passive) = 48 Gold / Minute

Bonus Gold income from 6 minutes on: 62 Gold / Minute

Path: Talisman of Ascension

• 4 minutes with 12,6 minions / minute = 50,4 minions
• 50,4 minions * 2 Gold = 101 Gold

Bonus Gold at 6 minutes: 101

After the upgrade we have:

• 2 Gold / 10 sec = 12 Gold / Min
• 12,6 minions / min (with 4 Gold for every minion) = 50,4 gold / min

Bonus Gold income from 6 minutes on: 63 Gold / Minute

Path: Face of the Mountain

At minute 2 we can start with killing 2 minions = 30 Gold (15 Gold each)

Then we can hit a minion every 60 seconds. Makes 4 more minions until our first back at the 6 minute mark.

• 1 Minion for 20 Gold
• 2 Minions for 20,5 Gold
• 1 Siege Minion for 40 Gold

So its 81 Gold + 30 Gold from the first two minions.

Bonus Gold at 6 minutes: 111

Now it's getting a bit tricky. We can hit a minion every 30 seconds. This means 90 seconds we can hit two melee minions and a siege minion, which gives us a total of 83 gold. (Regarding the first increase in gold bounty for the minions)

With a bit math: 83 Gold / 90 Seconds = 166 Gold / 180 Seconds = 55,3 Gold / Minute

We also have to pay attention to the rising reward which will be accountable for 6 Gold / 3 minutes.

Bonus Gold income from 6 minutes on: 55,3 Gold / Minute, which increases by 6 Gold every 3 Minutes.

Summary

Frost Queen's Claim:

• 168 Gold at minute 6
• From minute 6 it's 62 Gold / minute

Talisman of Ascension:

• 100 Gold at minute 6
• From minute 6 it's 63 Gold / minute

Face of the mountain:

• 111 Gold at minute 6
• From minute 6 it's 55,3 Gold / minute
• Increases by 6 Gold every 3 minutes

Solution 3:

Just a little addition to the previous answers, an aspect overlooked massively is the killing of minions while ADC is backed or roaming, Supports are generally pressured against taking creep score but when an ADC is vacant it's often the best course of action to prevent massive pushing, or to continue lane pressure. If we take this into account, then where an FQC Path has highest income potential (especially on champions like morgana) its passive lockout is quite detrimental, lets assume the ADC gets ganked before you back and you start farming some minions out of need while waiting for your ADCs return. a FQC would mean that you cannot regain your gold income via the support item for 12 seconds after your last minion kill, this means (in the best cases such as emptying all stacks before farming one minion) one potential 5-10 gold moment is missed each time and with multiple minions needing to be cleared the gold income is stunted every 30 seconds, the relic shield would mean you can keep getting your passive and your ADC could captalise instantly upon returning, an amulet coin could also allow full farming potential without detriment to efficiency… this means that allowing for one backing of adc while you farm, means you can potentially gain 107-134 without penalty, (if the adc takes one wave worth of time to return to lane) in addition the gold income from a relic shield (as mentioned in previous answers) gives gold to both you and your adc. so the gold gained is beneficial to both.

I like the Relic shield on many champions, the execute is handy for speedy waveclear, the heal from the passive is great for sustain, and the shield in late game can be coupled with an ardent censer for some extra attack speed for your adc the passive heal from gold income also gives an attack-speed boosting when coupled with the AC in addition to a potential life saver of a shield. Great on thresh, galio, naut, Leona, ali, or any lane with kalista.

After that I like the FQC for champions with a lot of multi-target spells in lanes where you can afford to be aggressive, morgana, lulu, sometimes janna/soraka, Galio etc. max built its active can be useful in similar ways to the Amulet

For passive ranged supports like janna, lulu (against tough lanes), soraka. Amulet coin is amazing, great for income, the max built item is beneficial for kiteing and escaping although the speed boost on a max level janna is a bit overkill… her escape and movement speed is high enough already.

To sum up, Gold income quality order is

1. FQC
2. Relic Shield
3. Amulet Coin
4. and all are better than a dorans ring or sword (in regards to team play)