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This statement is not entirely true. While it is true that part of the marginal cost curve is the same as the supply curve not ALL of the marginal cost curve is part of the supply curve.
Instead, ONLY the part of the marginal cost curve that lies above its minimum point is part of the supply curve.
For firms in perfect competition, the price of their good is always going to be the same as their marginal cost. That is why the marginal cost curve is the same as the supply curve above that minimum point.
Here's a link to more of an explanation.
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