Question: is the marginal cost curve the supply curve

Answer: No, The portion of the marginal cost curve above its intersection with the average variable cost curve is the supply curve for a firm operating in a perfectly competitive market. (the portion of the MC curve below its intersection with the AVC curve is not part of the supply curve because a firm would not operate at price below the shutdown point) This is not true for firms operating in other market structures. For example, while a monopoly ``has'' an MC curve it does not have a supply curve. In a perfectly competitive market, a supply curve shows the quantity a seller's willing and able to supply at each price -- for each price, there is a unique quantity that would be supplied. The one-to-one relationship simply is absent in the case of a monopoly. With a monopoly, there could be an infinite number of prices associated with a given quantity. It all depends on the shape and position of the demand curve and its accompanying marginal revenue curve.