Marxist Socialism
Marxist Socialism refer to a Marxian school of economics which emerged soon after Marx's death, led by his companions and co-writers, Friedrick Engels and Karl Kautsky.
The questioning of the status quo of the economic system was not a new idea; Jean-Jacques Rousseu sowed its first seeds and was followed by a plethora of thinkers whose ideas are better described in the article Economic Heterodox Tradition. Each one of those philosopher-economists, during the XVIII-th and early XIX-th centuries, pointed out some of the flaws they identified in the prevailing economic system. Their rethoric was strong and gathered some large audiences; from time to time they would succeed in promoting big riots and even some short lived revolutions. But none of those thinkers developed a systematic and encompassing political economy theory system which had the strenght to possibly change the world.
Some of them were too inconsistent, or too utopic; some were too romantic, or too idealistic, or even "half-mad". Most of the experiments which tried to implement their theories in the real world did not last long. Even those which succeeded, like Robert Owen's (1771-1858) example of the viability of co-operative factory communities, the New Lanark Mills in Scotland, represented only punctual local experiments, unable to influence the economic activities beyond the limits of their own nearby village.
On October 1st, 1867 the publication of the first volume of a single book would change that situation. The book was Karl Marx's Das Kapital - Kritik der Politschen Ökonomie [1]. A very long, dense, complex, logical, sometimes tedious, but a very systematic work Das Kapital has been read by very few but, since its publication in 1867, has been discussed by all. Marx died before he could complete the work; his friend Friedrick Engels edited Das Kapital's second and third volumes based on Marx's notes.
The marxian economists have been able to inovate and modify most of Economics but in one aspect: the "natural propensity" of economists to disagree among themselves. Immediately after its birth, marxian economics splat in two main currents: the orthodox, led by Friedrich Engels (1820-95), Karl Kautsky, Rosa Luxemburg, Georgy Plekhanov and others and the revisionists, led by Bernstein (1850-1932), Jean Jaurès (1859-1914), G.D.H. Cole (1890-1959), Sidney Webb and the Fabian Socialists, Mikhail Ivanovich Tugan-Baranovsky (1865-1919), Werner Sombart and the (Youngest) German Historical School, among others.
Karl Marx work suggested that the number of conditions required for steady-state growth were too numerous for capitalism to avoid its own breakdown. This "inevitablibilty" of the capitalism failure was challenged by Bernstein (1899) who thought that if socialism is to exist, it must be a conscious choice.
The ortodox marxian economics
Marxist, neoclassical, and Keynesian economics all have their roots in the "Classical Political Economy" of the early XIXth century. One key point to understand is that Marxist economics is an economic theory of capitalism. Another important point to have always in mind is that the "soviet-type" of marxism, that is to say, the implementaion of "marxism" by the governments of the Soviet Union and other communist contemporary coutries bear little relation to the system as devised by Marx; they only represent the "soviet" interpretation of marxism.