Why would the company pay more when they can just not pay more? The only things I can see happening is they might lower prices as competition ramps up, or in general as there is more supply for the same cost.

If there's sufficient demand, that's just what happens.

To try and explain one path: Company A doesn't raise wages but makes 5% more money. Company B pivots from Industry B into construction (because suddenly construction is having 5% fatter margins), and hires workers at more competitive wages to poach them from Company A. Company A forces to raise wages.

If there's a demand ceiling on housing it's a different story though.

If labour supply is fixed and productivity goes up then the value and demand for labour goes up, driving up wages