Ricardo did not explain how equilibrium prices are determined.
Labor is the only factor of production, the usage of this factor determine the cost of production.
Because their is only two goods the relative usage of the factor determine their exchange rate. And because we have constant return to scale and their is only one factor of production the PPF is a strait line -> exchange rate are constant and independent of the quantity demanded.
Also, because you have only two good, the idea of price doesn't make that much sens. You can only exchange good A for good B and vice-versa.
In other word, in Ricardian model, the price is the opportunity cost.
To know the quantity consumed, you need to know the utility function. But it will not change the price in autocracy.
I need to check :). But I believe the demand will ony tell use where we are on the world PFF. Then the slop of the world PPF at this point will give the "price".