Final answer:
In a purely competitive market, a firm maximizing profit at an output level where long-run ATC is minimized is producing at its minimum efficient scale, ensuring productive and allocative efficiencies.
Step-by-step explanation:
If a purely competitive firm is maximizing profit at some output at which long-run average total cost (ATC) is at a minimum, we can conclude that the firm is producing at its efficient scale. This implies that the firm is operating at its minimum efficient scale (MES), which is the lowest level of output at which it can achieve the full economies of scale, thus minimizing its long-run average total costs. This also indicates that the firm is producing in a way that is productively efficient, meaning it is producing without waste and at the lowest cost possible on the production possibility frontier in a perfectly competitive market.
Moreover, because of the process of entry and exit in the market, the price is equal to the minimum of the long-run ATC curve, suggesting that the firm is making neither economic profits nor losses in the long run, but is instead breaking even. This also contributes to allocative efficiency, as resources are being distributed in such a way that consumers are getting the products they want at the price they are willing to pay, which is the cost of production.