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A fixed exchange rate is one in which the currency is pegged, and the government intervenes to ensure that this value is maintained if it is threatened.
Governments or central banks can do this by buying or selling currency reserves or by increasing or decreasing interest rates. A government buying its own country's currency or raising interest rates will create demand for the currency and push up its value. Alternatively, selling its currency or cutting interest rates below other nations will increase the supply of the currency and push its value down. The Chinese Renminbi is operating a fixed exchange rate system against the U.S. dollar.
The main advantage of a fixed ER is the certainty that it provides surrounding trade. Companies are aware of the ER and are not discouraged from cross-border business activity due to ER fluctuations. For example, a building company in NI may not wish to carry out a project in Eire if there is currently a lot of ER volatility and a potential for the € to lose a lot of its value (as that is the currency that any rewards will be paid in). The rapid post-war expansion in trade is linked by many economists to the security of the adjustable peg system established after the war.
A fixed ER system also encourages governments to follow prudent economic policies which will limit inflation and ensure that the economy remains competitive with fellow nations. A fixed ER system does not have an automatic mechanism to correct the value of the currency in the face of high inflation, but this means that it is much more difficult for inflation to reach truly high levels. A fixed ER system is also a lot less vulnerable to speculation than a floating one (however, unless speculators believe there is a possibility that the government cannot support the currency, eg Black Wednesday in 1992 when the Bank of England failed to maintain the value of the £ against the Duetshemark and hence cleared its reserves)
One of the major disadvantages of a fixed ER system is that currencies frequently become over or undervalued. If a country is persistently running a current account deficit then it's central bank will continually be buying up its currency and it will need to either devalue or deflate the economy. However, it is frequently very difficult to devalue a currency as this will erode the main premise that the system is built upon, and any countries in a fixed system do not want to devalue as it is seen as an admission of economic failure. This means that to deal with a current account deficit a country will have to deflate its economy. This will mean large scale unemployment and a decrease in living standards, which is typically something a government wants to avoid at all costs. Thus this reveals the main benefit that should come with a floating system, the inclusion of automatic mechanism to deal with a current account deficit. If a country is running a deficit as it is importing too many goods, its currency is likely to be overvalued, people will be supplying too much of the currency and the value will fall.
A floating ER system should also depreciate in the face of high inflation to ensure that the economy remains competitive, however, this can mean that a government has no incentive to tackle inflation, triggering a wage-price spiral.