There are many different definitions of a market economy, some of which allow for government intervention. In a laissez-faire free-market economy, the government plays no role in economic decision-making.
Government Intervention in a Market Economy
Many would consider the United States to be a market economy, despite its heavy levels of government control and regulation.
In a certain sense, a government can intervene in a market economy up to the point that it is no longer considered a market economy. Elements of capitalism still exist as long as private individuals are allowed to own property and profit from its use.
Three Types of Economic Systems
Economic systems are divided into three broad categories: free market, mixed and command. The determining factor comes down to who owns and controls property and the factors of production.
In a free-market economy, private individuals or groups are in control. The government is in control of a command economy. Mixed economies have elements of both. Most economies in the world today are mixed, though some are command.
An example of a command economy would be communist North Korea. The North Korean government owns and controls all property, production decisions and allocation of resources. The old Soviet Union was also a command economy. These are not considered market economies.
The purest free-market economy would conceivably lack a monopolistic government and coercive taxation. Historical evidence struggles to come up with concrete examples of a government-less free-market system. The closest well-documented examples in modern history would be Hong Kong in the 1950s and the U.S. during the 19th century (excluding the Civil War period).
Clearly, even the most free-market economies by historical standards have some level of government influence. Some libertarian and free-market proponents, known as minarchists, suggest that a true market economy would only have three government functions: courts, police, and military.