No shortage is needed, only multiple employers. Let's do the math:
Your productivity is worth $100k to an employer. Your current employer (A) pays you $80k.
That means another employer (B) can offer you $85k, and make a profit of $15k/year, while A loses $20k/year, and you gain $5k/year.
A bit simplified, but the idea should be clear.
That assumes that there aren't other candidates available and competing for the same positions. If there are 100 candidates qualified for one or two positions, the employers have greater leverage: if you cost too much, they can fire you and hire someone else who would much rather have a low paid job than no job at all. You'd also rather have a job and an income at all so you can secure a place to sleep and food to eat than not, so you likely decide not to cost too much.
Especially if the work is unskilled and workers are easily replaceable this can lead to a downward spiral towards a pay that's barely manageable by the workers, far removed from their productive output.
Wages are determined by supply and demand of that position in the labor market.
Productivity boosts profits, which will permit an employer to pay higher wages if there is a shortage of a particular position in the labor market. But without the shortage, there's no impetus to pay more.