Do competitive wages always have to be defined by marginal productivity? Can we have competitive wages which are not based on productivity, when the information is not perfect?
To put it in context, Suppose the firms doesn't want to pay the competitive wage because may be marginal productivity is too high (as few number of workers are able to enter the market due to high cost of education) or may be marginal productivity is too low (due to more hiring for some reason and firms believe that paying too low a wage will inhibit workers from contributing their optimum levels). We also assume that only worker know about their true productivity, firms induce efforts from the employees by paying them wages but we are still operating under competitive environment. Under such scenario, can industry agree upon a wage that's not based on marginal productivity. Are there any papers which look at competitive wages which are not based on marginal productivity and still no one is incentivized to deviate.
I apologise for this wide question. Any economics reference, where competitive wages not based on productivity, will be helpful.