When someone wants to open an account in a credit union he has to be a member of a specific group or organization.
Why not allow anyone to join? Who restricts them from opening their business to anyone and why?
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Credit unions are mutually-owned (i.e. customer owned) financial institutions that provide banking services. They take deposits from their members (customers) and loan them to other members. Members vote on a board of directors who manage operations.
They are considered not-for-profit, but they pay interest on deposits. They get some preferential tax treatment and regulation and their deposits are insured by a separate organization if federally accredited. State-chartered credit unions don't have to maintain deposit insurance at all.
Their charters specify who can join. They can be regionally based, employer based, or based on some other group with common interests. Regulators restrict them so that they don't interfere too much with banks. Otherwise their preferential tax and regulatory treatment would leave banks uncompetitive.
Other organizations with similar limits have gone on to be competitive when the limits were released. For example, there used to be an insurance company just for government employees, the Government Employees Insurance Company. You may know it better as GEICO (yes, the one with the gecko advertisements). Now they offer life and auto insurance all over.
Credit unions would like looser limitations (or no limitations at all), but not enough to give up their preferential tax treatment. Banks oppose looser limitations and have as much political clout as credit unions.
It's required by law. 12 USC 1759 (b) requires that membership in a credit union be limited to one or more groups with a "common bond", or to people within a particular geographic area.
For lots more gory details on how this is interpreted and enforced, you can read the manual given to credit unions by the National Credit Union Administration, which is their regulatory agency.