The amount that A and B finally exchange at is determined buy their individual sense of values at the moment of the exchange. 

Producer A decides that he values whatever amount of money is being offered, more than he values the additional widget in his inventory, and consumer B decides that he values the widget more than the amount he is paying to acquire it.

Either party may also decide at any time that they value the staus quo more than the exchange.  In other words, producer A may decide that he still values his additional widget more than what consumer B is willing to pay, and consumer B may decide that he values his money more than what producer A is asking for his widget.

The costs to producer A may be a factor in his decision not to accept a lower price for his widget, but his costs do not set the price (value) of the widget.  The price (value) is set at the price point of the voluntary exchange. 

At that point of agreed exchange, each party still subjectively views himself to have come out ahead in the deal - each having acquired something he (at the moment at least) valued more, in exchange for something he valued less.

But why would producer A agree to sell something at a loss - or in other words, for less money than it cost him to produce the item?

A fair question, and there are many reasons why it might be desirable or even advantageous for producer A to sell his inventory of widgets at a loss.  We shall examine this question further in the future.

Suffice it to say however, that producer A cannot indefinitely afford to continue making widgets if he will be losing money each time he sells one.  Either he will need to come up with a less expensive method of making widgets, or he will need to produce something else that consumers such as B will value (be willing to pay) more than the cost of production.

Through this process, resources are constantly re-directed by the market, from lower-valued uses, to uses that are valued more.