Technically, this statement is not true, as the short-run supply curve is the firm’s entire marginal cost curve only when the prices of the goods and services are above the minimum point of the average variable cost.
Variable costs are defined as costs that vary in relation with the changes in the firm’s output production, such as the costs for the raw materials that go into each product, or the amount of compensation given to the workers for each completed unit.
In the short run, a firm must cover its variable costs. If it fails to do so, then it will jeopardize the production of goods and services. If the selling price falls below the average variable cost, then the firm’s output will be equal to zero. Thus, it is better for the firm to stop the production of goods and services, altogether. In economics, this is known as the shutdown price; in this case, the short-run supply curve would be the upward sloping portion of the marginal cost curve from the shutdown price.