Can a market economy solve problems of scarcity more effectively than a command/market economy?

A market economy is one where there is no interference by the government. In a market economy, market prices and output are determined by the market forces of the supply and demand, whereas in the command economy, there is interference by the government.

No, the market economy cannot solve the problem of scarcity. Instead, the market economy is one of the reasons for the scarcity of resources. Market economy gives a free choice to private players to invest in the project, where resources used are scarce. A command economy is one where there are regularizations by the government on using such scarce resources for the production of goods and services.

Thus, the market economy is not effective in solving the problem of scarcity.