A firm in a perfectly competitive market invents a new method of production that lowers its marginal costs. What happens to the price it charges?
If the firm invents such a method, it will have two choices. It can lower its prices (and therefore its marginal revenue) to the point where marginal revenue equals marginal costs or it can increase its production to the the point where its marginal costs go back up and marginal revenue and marginal cost are the same. In the long run, what it does will not matter because it will not make any economic profit.
Part of the definition of perfect competition is that there is perfect information within the market. No firm can produce its product any better than any other firm in the long run. Therefore, it will soon be the case that all other firms in the market will have the same procedures and will have the same marginal costs.
In the short run, this firm may make economic profit either by lowering prices or raising production. However, it will not matter in the long term because the firm will get no economic profit.