Is it grammatically correct to use "after" or "before" at the beginning of a simple sentence? I suppose it is technically an adverb, but it somehow seems like an incomplete preposition phrase to me. Example 1: We sat in the sun. After, we went swimming. Example 2: It was easier to find a job in the past. Before, there was less competition.
Before is commonly used as an adjunct, usually in final position in its clause:
I've never been to this restaurant before, so you may have to help me with the menu.
Your example sentence could be recast accordingly:
There was less competition before.
Your objection then is whether before as adjunct may come at the beginning of a sentence rather than at the end. Since the first sentence ends with in the past, it makes stylistic sense to begin with before to signal the reader that the rest of the clause is also in the past.
This would also hold if you prefer to parse before in both sentences above as an adverb rather than an adjunct.
After, however, has very few adverbial uses, mostly restricted to the day/morning after, and no dictionary I consulted lists a usage as an adjunct.
Instead, afterward[s] is the adverb of choice:
Afterward[s] we went swimming. We went swimming afterward[s].
In everyday speech, after is sometimes used in the sense of afterward:
I would meet people for dinner, the cinema or a gig and if we went to a bar after I would just have a couple of minerals or non-alcoholic beers and then go home.
The upshot is that standard usage would support initial before far more readily than after in the same position since there already is a related word that does the job perfectly.