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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?

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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.

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Your question can be rephrased as follows:

Are workers with different abilities paid differently in a competitive labour market? In other words, is each worker paid their marginal contribution to total production?

This is perfectly possible.

To see this, assume a continuum of workers, with index $i$, where $\eta_{i}$ denotes the ability of worker $i$. The distribution of ability in this population is given by $G(\eta)$. Since the market is competitive, we can forget about firms, and solve the problem from the social planner perspective (because of the 1st welfare theorem).

For simplicity, assume that labour is the only factor of production. All workers supply the same number of hours. Furthermore, assume that workers' production is given by the sum of their individual contributions, times a technology level parameters, $A$. So, if there are only two workers, with ability $\eta_{a}$ and $\eta_{b}$ respectively, total output is given by $A(\eta_{a}+\eta_{b})$. In the general case, total output is given by:

$$Y = A \int_{i \in L}^{}\eta_i\,\mathrm{d}i$$

(Notice that, for simplicity, this production function assumes constant returns to scale)

Now, let's postulate the hypothesis that workers are paid proportionally to their marginal product. It is clear that this is equivalent to $\eta_{i}$, as not having this worker reduces output by that amount (notice the importance of the assumption of output as the sum of workers' ability). If the wage per unit of ability is $\omega$, then the real wage of worker $i$ is

$$w(i) = \omega\eta_{i}$$

Now, under perfect competition (and without capital), all product is paid to workers. This is,

$$Y = \int_{i \in L}^{}w(i)\,\mathrm{d}i$$

Replacing the definition of wages and rearranging leads to:

$$Y = \omega\int_{i \in L}^{}\eta_{i}\,\mathrm{d}i$$

Given the production function assumed, this implies:

$$A=\omega$$

Given our assumption of constant returns to scale, this is of course reasonable. The wage per unit of ability must be equal to the technology level (as it is in the case of $Y=AL$). This completes the proof.

Notice the above result is true, irrespective of which is the distribution of ability. Notice further that equal ability ($\eta_{i}=\eta$) means $w(i)=w$, which brings us back to the Econ 101 labour market.

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