If the Fed believes the natural rate of unemployment is 4% instead of 5%, how may the Fed come to realize that its belief about the natural rate was mistaken?

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The most likely way that the Fed will find this out is if inflation rates change.  If the Fed has been wrong and the natural rate really is 5%, inflation will rise and the Fed will know that it was wrong.

The reason for this is that a tight labor market leads to inflation.  If there are "too few" unemployed people, there will be more competition to hire workers.  If this happens, wages will rise.  As wages will rise, prices will have to rise as well in order to allow producers to continue to make a profit as its costs rise. 

Therefore, if the Fed thinks the natural rate of unemployment is lower than it is, inflation might increase.

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