In my opinion, to answer the question OP original stated, the most EFFICIENT way to run an economy is through almost pure capitalism. If you understand supply and demand it is almost irrefutable to try to say anything but pure free markets create the most EFFICIENT means of an economy. I think the reason this scares peoples is because pure capitalism is quite literally survival of the fittest, and slavery for those who can't think of a way to rise up. HOWEVER, that doesn't mean that pure capitalism is practical in all cases. There are regulations in certain industries such as child labor, indentured servitude, and other markets for a variety of factors, mostly good ones.
Child labor creates issues where the child is unaware of the potential risks, as are parents. They are typically also desperate which makes their decision making process unreasonable, and unjust. Indentured servitude is much the same way along with others. Even with pure capitalism, it should theoretically sort these issues out over time, such as giving consumers choices about where to employ their children, and the companies that are providing the most unsuitable conditions for children (mining) would eventually be forced to abandon such a tactic as consumers boycott them, and other options for child labor arrive that provide more suitable conditions. There is a flip side to child labor in many poorer countries that isn't being discussed, and that is some of these citizens are quite literally so poor they NEED their children to make some income to help the family.
On the flip side, in favor of capitalism, look at what it has brought us. Even the poorest of poor families today, the homeless, and others, have a standard of living that is leaps and bounds what it was even 50 years ago. The internet has prospered, given us quite literally access to the same learning opportunities as anyone who goes to college. There is something to be said with how this marvelous invention, the internet is being used in a society such as the US today. People use it for Facebook, Twitter, etc. These uses provide almost no benefit to society, YET the citizens that have access to the internet DO have the opportunity to learn through Khan academy, numerous articles, journals, and many many other ways. They choose to remain ignorant, and that is mostly their own fault.
To take a look at socialism, and socialism like policies, those are in my mind the LEAST EFFICIENT way to run an economy. The destruction of incentives through free money is a horrible idea. It is worse slavery than even capitalism creates in many scenarios. People who are subjected to this type of indirect servitude become submissive. They don't want to move on to potentially better ideas as they are afraid of losing out on the free goods and services that were provided to them before. That is the free side of socialism, to look at the other side, the government mandated side, it is also highly inefficient. Much more so than any rational free market could be. Where are the incentives for drug companies to produce new medicines, if they know that the government will pay them less than cost to research, develop, and market this new drug? It drives innovation way down as the movers and shakers that actually make things happen in an economy are incentivized to do the exact opposite. Why would I start a business in Socialist Russia when I know for a fact I can make more money doing the same business in America? No rational person is going to opt to do that. It also affects supply and demand in a horrible way, the government mandating creation of houses in an area happens BEFORE the actual market transactions happen. The affect? Either the government produces to little, or too much and creates inefficiency in the supply and demand curve.
I'm not advocating pure capitalism, I am in favor of some regulations, but I hope a few examples I provided might sway opinions on socialism at least. I truly do not understand the liberal fascination with this idea that has become so prevalent as of late. It is horribly ineffective at driving an economy, especially a partly capitalist economy. Do you really think that by artificially letting every student in America go to college for free that there isn't going to be a rubber band effect? It will happen in the form of jobs paying less for students with degrees as everyone has one. It will also mean the less driven, less motivated become the future generation of teachers who in turn create less and less efficient teachings for the students that are supposed to be getting a QUALITY education. There is a cause and effect for everything.