if an exempt employee quits on a wednesday, do we have to pay the employee for the remainder of the week

Absent a contract, employment is employment at will. Unless there was an agreement otherwise, you do not need to pay someone for other that the hours that they actually work.

If they quit on Wednesday, pay stops on Wednesday.

No, the pay stops on Wednesday. Make sure the matter is written down and placed in the former employee's file. With no contract, no union and no agreement, you are correct to proceed.

Good luck to you.

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None of the prior answers have provided a square answer to key component of the question you raised: Does the fact that the employee is exempt for FLSA purposes bar you from docking his/her pay for the remainder of the week during which s/he does not work after the termination? That answer is no.

In the normal course, if you wish to avoid the obligation to pay overtime to a salaried employee, you can only dock the employee's pay if s/he performs no work for an entire work week, where the reason for no work is not the employee's choice (e.g., no work, bad weather, etc.). If the employee is absent by his/her own choice (this includes vacation and sick time), then you may dock pay only on days when the employee provides no services. However, you may dock the employee's PTO bank for partial days (hours) missed, as long as the employee continues to receive full pay taking PTO into account.

Nonetheless, when an employee terminates employment, docking pay is permitted during the last week in order to pro-rate salary in proportion to the time actually worked.