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I understand that monopoly profit is the return on capital (=profit) of the monopolist, which is larger than the normal profit in a competitive market. As monopoly rent I understand the income in excess of the factor cost (=rent) of the monopolist.
In this sense both would refer to the same thing. Is this correct?

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Economic (not just monopoly) rent is what is otherwise known as produces surplus. And producer surplus is related to profit via

$$\text{Producer Surplus}=\text{Profit}+\text{Fixed Cost}$$

assuming that fixed cost is sunk.

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