But the shift isn't just healthy - it's also a golden opportunity for economists to do what social scientists love best, which is to go on a giant raid and conquer the other social sciences! The new empiricism is the amphibious assault ship that will carry hordes of Econquerors (heh) to the vulnerable shores of sociology.
The first big econ raid on sociology came from Gary Becker and other theorists in the 70s and 80s, who applied rational choice theory (partial equilibrium optimization, game theory, etc.) to issues like crime and marriage that had traditionally been the domain of sociology. By the turn of the century, economists were brashly trumpeting their dominion over their sister field. This earned the undying animosity of sociologists, of course, since social scientists tend to guard their intellectual "territory" quite jealously.
But the attack on soc-land was ultimately repulsed. Sociologists were basically free to ignore the incursion by A) not learning anything about econ models, and B) not publishing econ models in their journals. That was quite an effective tactic. Since university administrators and other dark overlords of academia (not to mention the general public) also don't understand optimization models, economic imperialists found themselves mostly talking to each other.
The conquest was also a lot more difficult than economists predicted, since the defenders got an assist from Extant Reality. A lot of the imperialists' models were just flat-out wrong. A spectacular (and spectacularly tragic) example was Gary Becker's prediction that harsh sentencing could substitute for consistent law enforcement. Oops.
And in the end, economists had to pull back their occupying troops to fight battles closer to home. The financial crisis and Great Recession showed that economists didn't really understand the economy, which made a lot of people wonder why they were going off and trying to explain the division of household chores. (Of course, there's a good reason for this; it's a lot easier to test theories about household chores than it is to understand the business cycle...but hey!) By the 2010s, economic imperialism was kind of a joke.
But now the empirical revolution, especially the Credibility Revolution, is giving econ a second chance to conquer the neighbors. The new techniques - regression discontinuity, difference-in-difference, synthetic controls - are mostly pretty easy and quick for any smart person to grasp. The results of these sorts of studies are also usually very easy to interpret, unlike the output of many optimization models. And most importantly, the techniques can be applied to any social science subject.
That means that if empirical economists feel like writing a paper, it's really easy to go pick a soc topic. Want to study education policy? Just pull out your empirical econ toolkit! The relationship between crime and poverty? Go for it!
The new econ imperialists will succeed where the first wave mostly failed, because they're armed with better weapons. Back then, it was basically a war of theory against theory, and the fights looked like:
Economist: [optimization model no one except economists understands]
Sociologist: [slew of jargon no one except (possibly) sociologists understands]
Public: Hmm, let me go with my political priors on thi - Hey, look, CNN is talking about Monica Lewinski! Sorry guys, gotta go.
Now, when empirical economists come with actual evidence, sociologists will be in a bind. See, the public doesn't get theory, but it gets empirical results. "X causes Y" is easy to understand, even for the most distracted of university administrators or Quartz columnists. That means empirical economists will soon be treated as experts on sociological topics.
In order to rebut economists, both in the public sphere and in the court of their own intellectual consciences, they will either need A) empirics of their own, or B) a good understanding of empirics AND the ability to clearly explain their own theories in order to use theory to question economists' interpretations of their results.
Sociologists will have at least two barriers to doing this. First, many sociologists aren't nearly as proficient at stats and math as economists. That's not a huge problem, because Credibility Revolution techniques are actually not nearly as hard to learn as stuff like structural econometrics or Bayesian time-series methods. So sociology will have to beef up its grad students' quant skills, but it's hardly an insurmountable task.
Second, sociology is much more closed-off than econ, since few sociologists publish working papers. Soc is going to have to become a lot more open if it's going to hold off against the coming econ onslaught, both in the court of public opinion and in the general intellectual world.
Even if sociology does this, it's going to have another problem - many sociology theories will be found to be "not even wrong". As empirical evidence becomes the gold standard for social science arguments, jargon-heavy non-mathematical theories will just be less and less useful. That's probably going to make some soc theorists even angrier than it made econ theorists - at least econ theories, being quantitative, can often be tested with data.
(Political scientists, meanwhile, will just keep doing what they always do, which is to cheerfully copy techniques from anyone and everyone.)
Anyway, in the end, I expect the new age of econ imperialism to be a good thing, for the world and for social science in general. It's time for the siloes to come down, for the borders between the arbitrary domains to be erased. Econ hordes: Conquer, pillage, burn!!