Some economists justify the use of math in economics as a means of keeping their model straight. What if it has another purpose? What if it’s a signal?
Ideally, the signal should have a cost. Restated, it should be costly for someone to attain that signal. In our example, the optimal cost is one that is costly enough to dissuade candidates who don’t fit the employers’ criteria and not too costly that it dissuades those who do fit the criteria.
Math could be interpreted as a signal, at least as far as its use in economics in concerned. Suppose we’re interested in differentiating good economic theorists who can make enlightening insights on complex topics from average economic theorists who aren’t so good at doing that. Math, specifically the kind of math you learn for economics, is not easy to learn. There’s a cost, and that’s the amount of time spent studying it (time you could have spent studying/doing something else). The cost is high enough to weed a lot of people out.
At my alma mater, math is used as a signal to differentiate those who intend to move on to grad school and those who don’t. You can get an economics degree from San Diego State University with only precalculus, trigonometry, and “business calculus.” But, if you want to go to grad school, the straightforward economics degree isn’t usually good enough. Instead, you get a “specialization is quantitative economics,” which necessitates a higher level calculus class and then a mathematical economics class, which pretty much sums up 1–3 semesters of math (derivatives, integrals, and matrix algebra, pretty much). That specialization serves as a signal to boards which regulate the entry of masters and PhD candidates.
In the academic world, perhaps math signals certain capabilities, including the ability to think about complex subjects and derive accurate results. Note, this is a generalization, and I’m sure there’s plenty of very good economists who aren’t so good at math, or at least don’t use math much in their work — in fact, I know good economists who fit this characterization. But, maybe they’re the exception to the rule?