Does a pokemon caught at an early level reach the same potential as a pokemon caught at a high level

This question shows that increasing your trainer level allows you to train a Pokemon's CP further.

Does this mean that if I catch a Pokemon at level 1 and keep increasing it via stardust and candy every time I level up, it will have the same potential as a Pokemon I caught at level 30?

Or should I just transfer my early level Pokemon to be turned in to candies, so they can be fed to Pokemon I catch at a later level. Or perhaps somewhere in between that they'll be good for a few more levels and then taper off?

I believe it does. What changes when you progress up the Levels is the Max CP a pokemon can have. If you're feeling sentimental towards your starter you can upgrade him over and over again, with candy and stardust and will eventually be as strong as a wild of the same starter type you find at Level 30 (current Wild max CP cap).

But if you don't hold any special feelings towards a said Pokemon always transfer the lower CP Pokemon and keep just 1 or 2 with the highest to evolve later on.

At first: DON'T power up Pokémon before Level 20. Reason being: before Level 20 the single most effective thing is Leveling up, and that is best achieved by evolving, not powering up.

Due to a huge spike at level 20 (25.000 xp for level 20, 20.000 xp for level 21, 75.000 for level 22) it starts paying off more if you actually Power Up your Pokémon.

And also, a Pokémon caught on a higher level not only has stronger starting stats but also tends to have a higher potential, at least in my experience.

You can actually calculate a Pokémon's IV (Individual Value, the higher it is the higher its potential) from a number of stats.

EDIT: since some people seem to want to know it, it is always possible to catch a Pokémon which can get the best possible attributes when maxed out, however it often doesn't pay off if you level up ones that are on a low level to begin with since they eat a lot of resources.