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Market Economy - It can be generally defined as an economy which markets (unless significant distortion or market failure occurs and prevails) determine the equilibrium price. Government plays minimal role in the economic transactions and activities and it is primarily responsible for enforcing laws and regulations to ensure the proper and efficient operation of the market system.
Mixed Econmy - It is an economy that is in the middle of the spectrum. It is combined with market economy and planned economy. Market does play a role in determining the equilibrium. However, in the system, government may have some (political or economic or otherwise) objectives to achieve and thus government will use various policy tools at its disposal to achieve these objectives. As a result, government interventioin may be able to set the equilibrium price below/above that of the market economy. The best example will be China or the East Asian Tigers in the 1960s-1980s.
Planned Economy - It is an economic system of which market does not exist or does not play any role in setting equilibrium price (i.e., market does not have the power to allocate resources to their most efficient use!) Government is the major (if not only) player in an economy. Quite often the case, government assumes itself to have all the information required to make the allocation decision. Government also has the ability to set the equilibrium price and the formal market is either nonexistent or suppressed and thus there exist the so-called 'black market'. The best example would be former USSR.