Do wages differ in two perfectly competitive labour markets? I understand that within a single labour market, the wages would be the same between two workers as firms and workers are both wage takers, but what about when there is more than one perfectly competitive labour market: if the labour market for jobs A and B are both perfectly competitive, would the workers in A earn the same as the workers in B, provided that the two jobs are not the same?
This will typically depend on if people can switch between the two markets. If switching is possible then the marginal worker has to be indifferent between working in market A and market B and this equalizes the wages in the two markets. This also assumes you mean wage rates and not wage bills. Wage bills also depend on hours worked per worker, and that decision need not be equal across industries because of sorting and other reasons. 

If people can't switch between markets it can, but doesn't have, to be that wages do not equalize. Consider two workers on two different islands with residents that don't trade or even meet. Even if labor markets on both islands were perfectly competitive, it needn't be that their wage rates were equalized.