Question

Why should countries specialize in producing goods with which they have a comparative advantage rather than an absolute advantage?

Answer 1

Because of the simple principle that,
It is possible for one country to have absolute advantage in all products and services available in market(this is generally not seen in market but it is possible)because it takes into account the actual real cost of inputs involved in production.

However it is mathematically impossible for any country to have comparative advantage in all products and services available in market(because a mathematical reciprocal relationship is involved).Comparative advantage takes into account opportunity cost.Simply stating if you give time to make meat ,you cant make potatoes in that time period.You missed the opprtunity of making potatoes.That is opportunity cost.

And this works in real world.If you see international trade countries make goods in which thay have comparative advantage not absolute advantade.

Here is Mr Davi Ricardo's famous example.He is credited with coming with idea of comparative advantage.
Ricardo's example[edit]
In a famous example, Ricardo considers a world economy consisting of two countries, Portugal and England, which produce two goods of identical quality. In Portugal, the a priori more efficient country, it is possible to produce wine and cloth with less labor than it would take to produce the same quantities in England. However, the relative costs of producing those two goods differ between the countries.

Hours of work necessary to produce one unit
Country Cloth Wine
England 100 120
Portugal 90 80
In order to produce an additional unit of cloth, England must commit 100 labor hours, which could have instead produced 5/6 units of wine. Although Portugal can produce a unit of cloth with fewer hours of work (90 hours) than England, it must forego producing a greater amount of wine 9/8 units) to do so. Accordingly, England is said to possess a comparative advantage in cloth even though Portugal possesses an absolute advantage in cloth. A similar argument would show that Portugal has both an absolute and comparative advantage in wine.

In the absence of trade, England requires 220 hours of work to both produce and consume one unit each of cloth and wine while Portugal requires 170 hours of work to produce and consume the same quantities. If each country specializes in the good for which it has a comparative advantage, then the global production of both goods increases, for England can spend 220 labor hours to produce 2.2 units of cloth while Portugal can spend 170 hours to produce 2.125 units of wine. Moreover, if both countries specialize in the above manner and England trades a unit of its cloth for 5/6 to 9/8 units of Portugal's wine, then both countries can consume at least a unit each of cloth and wine, with 0 to 0.2 units of cloth and 0 to 0.125 units of wine remaining in each respective country to be consumed or exported. Consequently, both England and Portugal can consume more wine and cloth under free trade than in autarky.