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My understanding is that wage differentials should not exist in a perfect labour market for occupations, but the following exam question seems to imply that this can be the case. A brief marking scheme also implies this.

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and its marking scheme explanation:

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I have always thought that if occupations belonged to a perfect labour market, it would mean that due to the assumptions of perfect mobility and homogeneity of labour, wages will always adjust so that wages are always equal across industries. The marking scheme seems to imply that this does not have to be the case.

One possible caveat I suspect would make the question's situation possible would be if the firms do not operate in perfectly competitive product markets, though I am not sure about this, and neither did the marking scheme mention this.

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The question (implicitly) assumes that there are relevant skill differences in the population. As you say, it is true that in a perfect labor market there are no wage differences if workers can move freely across sectors.

If your course defined perfect labor markets as having homogeneity of labor and perfect mobility, then the given answer is wrong. However, typically these two factors are not necessarily a feature of perfect labor markets. That usually just means there is no market power on either side.

As the question asks about the case of different labor supplies, there can be no homogeneity of labor nor perfect mobility. Different labor supply is only possible if not everyone can do each type of work, meaning there are more available workers for certain jobs, thereby implying not all workers have the skills to do all jobs. Hence, to the extent that the question assumes supply differences are possible, it also assumes that workers cannot move freely across sectors. That is the reason wages do not equalize according to the answer.

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