Under the Producer Theory, if a technological innovation has reduced the cost to the firm of one or more key inputs, what would happen if an individual firm decided to hold its price steady? Which assumption(s) of producer theory would the firm effectively believe it can violate to maintain its original price? Are there any other reasons the firm might hold at its original price relative to the new market equilibrium?
The producer theory assumes that the firms are price takers.
In the given scenario, if an individual firm decides to hold the price steady even if one or more key input's cost has fallen, then the firm would be violating the "price-taker" assumption.
In such a scenario, assuming a competitive market, the drop in the price of the input will benefit all the other competing firms and thus maintaining the original price will mean that the firm in question will be out-competed.
The reason the firm might be able to hold its original price relative to this new market equilibrium would be a market monopoly or higher brand value or higher customer loyalty. For example, Apple has always been able to sell its products at a premium compared to other firms due to higher brand value compared to say, Samsung.