I'm pretty sound with the concept behind comparative advantage, but still don't get how to calculate the exchange rate in terms of quantities of each products being exchanged, e.g. 1 apple for 2 cars etc.
Specifically: when two counties, each having two separate comparative advantages in producing two separate goods, wish to exchange these goods, how will the exchange quantity/rate be calculated?
If I understand correctly the question, the trick to calculate the equilibrium (relative) price of trade in the Ricardian model of comparative advantage is to specify demand once the opportunity costs have been determined. Prices of internationally traded goods, like other prices, are determined by supply and demand.
Consider two countries, two goods (computers and textiles), and given unit labor requirements such that in equilibrium the world price of computers will be between 0.5 and 2 textiles per computers. The opportunity costs define the bounds of equilibrium relative prices of trade (0.5 and 2), while the structure of demand determines the equilibrium relative price (p).
For instance, p=1 can be an equilibrium price such as one country will specialize in computers and the other in textiles. This price will be determined by the interaction of the relative supply and the relative demand of computers and textiles. This interaction is discussed in the Chapter 3 of International Economics: Theory and Policy of Krugman, Obstfeld and Melitz. An interesting video with a concrete example can be found here.
Ricardo did not explain how equilibrium prices are determined.
In autocracy
Labor is the only factor of production, the usage of this factor determines the cost of production.
Because there are only two goods, the relative usage of the factor determine their exchange rate. And because we have constant return to scale and there is only one factor of production, the PPF is a straight line -> exchange rates are constant and independent of the quantity demanded.
Also, because you have only two goods, the idea of price doesn't make that much sense. You can only exchange good A for good B and vice-versa.
In other word, in a 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.
Trade
I need to check :). But I believe the demand will only tell us where we are on the world PFF. Then the slope of the world PPF at this point will give the "price".
