"appropriate" is a loaded and subjective word. Further, some government activities and programs could be widely viewed as appropriate up to a point.

That said, we generally expect our federal government to:
establish a monetary system
maintain economic stability with respect to employment and inflation rates
enforce public and private contracts
protect markets from corruption and malfeasance.
allow markets to operate openly and efficiently
provide safety nets for the misfortunate.
etc.

What is the appropriate role for government in the economic system?

That depends upon which economic system you embrace. Please post your answer and we'll be glad to critique it.

The appropriate role for government in the economic system is a complex and debated topic. Different economic systems have different views on the extent of government involvement. Here are some general perspectives on the roles and functions of government in the economy:

1. Market-based economies: In capitalist or free-market systems, the general belief is that the government's role should be limited. The government's main responsibilities are to ensure the functioning of markets, protect property rights, and enforce contracts. The idea is that free competition and the pursuit of self-interest will lead to efficiency and overall societal welfare.

2. Mixed economies: In mixed economies, which most countries have to some extent, the government takes a more active role. It regulates markets, enforces laws against fraud and corruption, and provides infrastructure and public goods such as transportation, education, and healthcare. The government also often intervenes to promote economic stability through fiscal and monetary policies, and may provide social safety nets for vulnerable populations.

3. Command economies: In command or centrally planned economies, the government plays a dominant role. It decides what goods and services should be produced, sets production targets, and controls prices and distribution channels. The government's goal is to achieve economic equality and stability, but it can result in a lack of individual freedom and lower efficiency compared to market-based systems.

In summary, the appropriate role for government in the economic system depends on the economic ideology and the goals of a particular society. There is no one-size-fits-all answer, as different countries and cultures have different values and priorities. Ultimately, it requires a careful assessment and balancing of various factors to find the most appropriate role for government intervention in an economic system.