These were manufactured for various Massachusetts PDs in the 20s and 30s; for example, one for the Boston PD:
https://www.flyingtigerantiques.com/rare-ca-1920-boston-police-patrolman-badge-in-the-clamshell-or-sunburst-radiator-style.html

This is not engraved with any specific town's police dept.; it seems to be a display/yardstick version. It is made of nickel-plated brass.