In a perfectly competitive market, a firm that invents a new method of production that lowers its marginal costs will either have lighter losses if it is unprofitable or higher profits if it is profitable.

What happens to the price it charges for its widgets is entirely up to the firm's owners or management. Under perfect competition, their temporary marginal cost advantage will dissipate quickly, as this condition assumes no lasting information asymmetries.

Lower prices, all things being equal, should allow them to sell more units and maximize short term profits so they may lower their price to try to increase market share. Perhaps they are already competitive or dominant in the market in which case they may leave their price the same and enjoy higher profits for a time.

No matter what they do with the price under perfect competition the innovation in productivity will soon spread to competitors, however, and the marginal cost advantage it provided will vanish. Remember under perfect competition all profits go to zero, and all firms eventually must exit. That is one way we know imperfect competition and imperfect information are ubiquitous marketplace phenomena.

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.

A firm in a perfectly competitive market invents a new method of production that lowers its marginal costs. What happens to its output?

In such a case, the firm's output should increase.  The reason for this is that the firm should always produce the quantity at which the marginal revenue and marginal cost are equal.  When the firm invents the new method of production, its marginal costs fall.  This means that marginal costs are now lower than marginal revenue.  By definition, this means the firm is not making as much profit as it could be.  In order to maximize its profit, it will need to produce more goods so that it can return to the level where marginal cost equals marginal revenue.