Question regarding transferring and pay

[ariastar]

What is the name of your state (only U.S. law)? California/Colorado

We are moving to Colorado from California in April (hence the two states listed). Now the cost of living in less in Colorado than in Silicon Valley, and my fiancé was prepared to take a pay deduction (staying with the same company, same position, just different location). However my mother's boyfriend, who is a higher up in a large department store chain said that when transferring and staying with the same company, that the pay at the new location must at least equal the salary at the previous location, caveat being that the employee doesn't get yearly raises until the salary of co-workers in the same position and with similar experience have caught up. He's pretty sure that this is law and not just something his company does. But I have no clue where to begin looking into this.

Is this correct? If my fiancé makes X+1 here and the salary for someone of the same position with the same qualifications in the new, lower-priced area is X, do they have to pay him at least X+1 or can they decrease his pay?

[cbg]

Your mother's boyfriend is wrong. There is no such law.