When a person is demoted on the job do they get a reduction in pay?

There are no Federal Laws, such as the Fair Labor Standards Act (FLSA), that prohibit an employer from decreasing an employee's wage rate, unless the reduction would result in the employee being paid less than minimum wage. As far as state laws are concerned, it is unlikely that they prohibit a reduction either so long as the resulting wage rate is not below the state's minimum wage. Some states do require an employer to notify an employee prior to reducing their wage rate. You would need to check with your specific state to determine its laws concerning wage reductions. An employer would be prohibited from reducing a wage rate if it would violate the terms of a employment contract or if it was motivated by a prohibited reason, such as age, race, sex, national origin, religion, disability, union activity, etc.