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There are several reasons why countries trade with one another. Trade among nations is taken as a sign of good intent and a means of maintaining non-hostile diplomatic relations. Trade is used to empower allied nations by providing them with valued resources such as oil, grain, or bullets, as well as crippling and weakening rivals by imposing economic sanctions on goods & services such as: military armaments, food, or medicine. Bans such as these are used to punish nations or motivate a change in their political and economic behavior. A course of action that the United States of America has pursued several times when suspect of nations endoursing terrorism  . Moreover, trade unifies neighbouring countries with shared economic ideals by creating a common currency and trade laws that bolster each party's economic power. The establishment of the euro for example in 2002 united 12 countries and is now used in 22 countries, currently overpowering the US dollar. 
Essentially countries trade in order to purchase goods and/or services that would not have been available within their borders either due to insufficient resources or underdeveloped technology  . Therefore through trade, countries are able to obtain any desired good or service that would have otherwise been unattainable or would have placed a burden on economic activity. International trade may be described as a interdependent web of sustainability among countries. International trade therefore mirrors specialisation  , this being a key concept underlined in the law of comparative advantage.
The law of comparative advantage involves the opportunity costs of two or more parties (a firm or company, in this case a country) in their production of a good or service and highlights their ability in producing it at the highest possible efficiency in relation to all the other possible goods or services that could have been produced in its place  . In this sense, there is merit in trading with other countries when international differences are present in the opportunity cost of given goods  . An isolated economy with limited resources is able to produce tractors and hats for instance. The more resources allocated in the production of tractors, the less are available for the production of hats and vice versa. The opportunity cost of tractors is the quantity of hats sacrificed seeing that resource allocation was focused on the production of tractors and not hats.  This situation can be illustrated by the diagram overleaf:
Diagram (a) shows the maximum combination of tractors and hats this economy can produce. In the event that all resources were used to produce maximum tractor output while sacrificing the production of hats all-together, then the outcome would be shown by point A. Similarly, point D represents the event that the economy sacrificed the production of tractors to achieve maximum hat output.  Points B and C correspond to relative trade-offs. Point E represents inefficient use of resources, while point F requires more resources than the economy has at hand and can only be achieved by development of the given economy. The curve A-D is known as the ââ‚¬Å“production possibility curveââ‚¬  .  11
Using the principle of comparative advantage, countries derive whether it would be beneficial to start trading and if so, if it should export or import.
Take for instance the market for wheat. The wheat industry is large seeing that it is produced in many countries making it a good example in terms of analysing the gains and losses a country may experience as a result of trade.
For example, Country A's market for wheat is isolated from the world market. There are no transactions be it exports or imports and the market for it is comprised uniquely by its domestic buyers and sellers  . The diagram overleaf depicts the market equilibrium without international trade:
In an economy like Country A's, domestic supply and demand are balanced by adjusting the price. In the absence of international trade consumer and producer surplus are in equilibrium. 
To determine whether or not Country A should trade with other countries the domestic price of wheat should be compared to that of other countries, commonly known as the world price  . If the domestic price of wheat is lower than the world price then Country A becomes an exporter of wheat seeing that domestic wheat producers take advantage of the increased foreign prices and begin selling to other countries. By contrast, if Country A's domestic price of wheat were higher than the world price then it becomes an importer of wheat since consumers are eager to buy cheaper wheat from abroad. 
The principle of comparative advantage is a key element as far as trade is concerned. By considering the domestic price in relation to the world price of wheat Country A derives whether or not it has a comparative advantage in producing it.
The opportunity cost of wheat is derived from the domestic value. In other words, how much of another good Country A has to sacrifice in order to produce one unit of wheat. A low production price of wheat states that Country A has a comparative advantage to the rest of the world. Conversely, if Country A has a high production value, other countries have a comparative advantage. 
Diagram (b) shows the domestic equilibrium price and quantity for wheat during pre-trade conditions. Once Country A starts trading, the domestic price increases to reach the world price level. This is to say that domestic producers will now sell at this new increased price which in turn forces consumers to pay more. This is shown by the diagram below:
Quantity demanded and quantity supplied differ when in trade. The new excess quantity is used as exports to other countries. Before trading, the price level adjusted itself so that domestic supply and demand could balance. Consumer surplus being areas A + B and producer surplus area C. Total surplus summing up to areas A + B + C. Now that a new price has been set, consumer surplus is A while areas B + C + D are the producer surplus. The new total surplus is A + B + C + D.
Producers surplus increases by areas B and D making them better-off. While consumer surplus is reduced by area B. Due to producer gain trumping consumer loss, total surplus in Country A increases. This example shows how trade bolsters the economic state of a country and reinforces the pro-trade argument. 
Following these points, one concludes by saying that trade among nations is beneficial seeing that it allows each party to allocate its resources accordingly in order to specialise in what it does best, while obtaining other desired goods at a lower rate.
When countries decide what to specialise in upon entering trade with one another, their opportunity costs are taken into consideration seeing as relative production costs differ from country to country.  The following model puts into practice the example with hats and tractors. Take for instance two countries producing hats and tractors, Greece and Britain.
British workers earn 6 pounds an hour whereas Greek workers earn 3 euro and have an absolute advantage in terms of both goods. Table 1 shows that less British unit labour hours are needed for both goods. Britain is relatively more productive in terms of tractors seeing that it takes 1.5 times longer for Greece to produce one. However, it takes Greece only 5/4 times longer to make a hat.  Britain holds the comparative advantage for tractor production and Greece in hats. By sacrificing 10 hats, Britain acquires 40 extra labour hours to make a Tractor.  The opportunity cost of a tractor in Britain is 10 hats and 12 hats in Greece. The opportunity cost of Greece however ( 1/12 of a tractor) is less than the opportunity cost in Britain ( 1/10 of a tractor.).  Once again, this proves that Britain has a comparative advantage in tractors and Greece in hats. Specialisation and trade allows these countries to produce and trade each good more efficiently. Greece focuses on hat production and Britain on tractor.
Trade has proven to be a very beneficial course of action for countries to take part in. Benefits from trade range from maintaining good-willed relations between nations, to empowering allies with precious resources, to weakening foes by stripping them away and finally to allow each country to obtain goods and services in demand that would have otherwise been impossible to attain. Through the principle of comparative advantage countries determine a number of factors. Initially, with the use of a production possibility frontier diagram, they derive opportunity costs for different combinations of producing goods. Efficient resource allocation paves the way to specialisation of goods. Secondly, they see if they have a comparative advantage in entering international trade and exporting (buying). Trade, exports to be more precise, increase countries' economic power as it increases Total Surplus. Finally, when faced with the option of multiple good production, countries compare their comparative advantage in relation to each good and settle for the most efficient outcome. As a result of specialisation all parties benefit from reduced costs.