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Why is MC=MR at the profit maximizing level of output?

MC stands for marginal (extra) cost incurred by a firm when its production raises by one unit. MR stands for marginal (extra) revenue a firm receives from producing one extra unit of output. As a firm is trying to maximise its profits, it needs to consider what happens when it changes its production by one unit. The firm will of course incur an extra cost from producing an extra unit, but will also receive revenue from that unit. If the marginal cost is bigger than the marginal revenue obtained, then the firm should realise that producing an extra unit of output was not profitable. The firm should thus cut down some of its production. If the marginal cost is smaller than the marginal revenue, then it is profitable for the firm to produce an extra unit of output. The firm should continue to raise produce extra units of output as long as the marginal revenue it receives from that unit exceeds the marginal cost. The firm should continue doing this until MC=MR, a point at which they should keep production constant, because producing an extra unit beyong this point creates a higher marginal cost for the firm that it creates marginal revenue. 

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