  Autarky Equilibrium in the Immobile Factor Model by Steven Suranovic ©1997-2004 Suppose two countries, the US and France, have the exactly the same number of winemakers and cheesemakers. This means and . Suppose also that the US has an absolute advantage in the production of cheese while France has the absolute advantage in the production of wine. This means and . Also assume that the preferences for the two goods in both countries are identical.

For simplicity let aggregate preferences be represented by a homothetic utility function. These functions have the property that for any price ratio, the ratio of the two goods consumed is equal to a constant. One function with this property is where is the aggregate quantity of cheese demanded and is the aggregate quantity of wine demanded. This function says that the ratio of the quantity of wine demanded to the quantity of cheese demanded must equal the price ratio.

As an example suppose that consumers face a price ratio gallons of wine per pound of cheese. In this case consumers will demand wine to cheese in the same ratio, 2 gallons per pound. Suppose the price ratio rises to say, . This means that cheese becomes more expensive compared to wine. At the higher price ratio consumers will now demand 3 gallons of wine per pound of cheese. Thus as the relative price of cheese rises the relative demand for wine rises as consumers substitute less expensive wine for more expensive cheese. Similarly, as the price of wine falls the relative demand for wine rises. The PPFs for the two countries in this case are plotted in the Figure. The US produces more cheese than France while France produces more wine than the US. Because the factors are immobile the ratio of wine to cheese production in the US must be .

In autarky the quantity demanded of each good must equal the quantity supplied. This implies that the ratios of quantities must also be equalized such that, .

Substituting from above yields the autarky price ratio in the US, Similarly, France's autarky price ratio is given by, Since by assumption the two countries have identical labor endowments and the US has a comparative advantage in cheese production, it follows that,      