For any reason to take Cull, longsword and the 100 gold you saved does it better.

Lets look at what Cull gives for 450 gold 5 AD 3 HP on hit 1 gold per CS, capping at 100 CS were it gives you 300 gold No upgrades, but sells for 180 gold, for 130 profit at 100 CS

Okay, so from the looks of it, Cull is about being a weak starting item that offers some early game sustain and pays for itself, compared to Doran's blade which is a strong starting item with sustain that scales as you get more AD and Crit, but needs to be sold for a loss later.

However, if you are looking for a gold efficient starting item, Longsword give double the AD as Cull and costs 100 gold less. Along with making your early game stronger and CSing easier thanks to 5 more AD, you can get 2 more HP pots, which heals you the same as 100 auto attacks with Cull does - not something that's going to happen till well after your first back.

But more importantly for a gold efficient start, that extra 100 gold lets you start with a refillable potion. It only heals for 100 hp twice, or 50 more hp than cull + 1 pot does, but refiles every time you back. After three backs, it's healed as much as 4 HP potions would, while 50 gold cheaper. If you're really worried about the possibility of losing some HP vs Cull + 1 pot before your first back, but want to still have the gold efficiency of the refillable potion, you can drop one or two AD quints for Lifesteal quints to have better sustain than cull gives, while still having more AD.

Rather than selling the longsword like you would for cull, it builds into a plethora of items you want; it's a 100% gold efficient buy, and will be ready to upgrade well before your cull has paid for itself. You'll be farther along in your build and stronger at every point in the game with the longsword start than a Cull start.