I don't believe that Search Unemployment (MP) actually "won" over efficiency wages. The whole discussion of search literature would be too long for this post, so I'll just skim the most important parts.
- (i) As a first, Shimer (2005) discusses that the MP model actually fails to get employment (market tightness) volatility right
- (ii) In the same AER issue, Hall(2005) proposes rigid wages. He and Shimer (in his booklet) argue that there is a set of wages which satisfy Barro (1977). Moreover, an intuition for wage rigidity is that old wages are an anchor for new negotiations
- (iii) Many papers show more or less convincingly that while average wages are rigid, wages of new matches actually are quite flexible.
- (iv) Bewley (1999) presents survey evidence where managers respond that they don't want to reduce wages because it would lead to detrimental climate.
First, to the extent that the MP model actually needs exogenous wage rigidity to get the moments in the data right, it has not won over efficiency wages, as these could be understood as a complementary theory (as being the reason for wage ridigity, as an alternative to (ii)).
Second, (iii) kind of goes against both MP and efficiency wages at the same time - at least to the extent that they explain fluctuations in hirings. Note that this is the relevant margin for unemployment fluctuations: [I think it was Shimer who] showed that the most fluctuations in labor come from volatility in hiring, and not from firing.
Finally, a personal opinion. I was interested in the story myself, but I wouldn't know how you could find efficiency wages in the data we have. Also, there's not really more to the theoretical component than was written in the past century.
The trend in the past decade has gone to microfounded, self-consistent models that are found using empirical data. I don't believe that this is really possible at the moment with the efficiency wages story. We believe efficiency wages exist, but we can't detect them and they (arguably) don't explain the Shimer puzzle, which makes them less interesting these days.