Socialism vs social democracy as income-equalizing institutions

Socialism is defined as a normative property of an allocation: that the allocation of labor and output be Pareto efficient, and that output received by individuals be proportional to the value of the labor they expended in production. Social democracy is an institution: the redistribution of income through taxation, with a system of private ownership of capital. We present a stylized parameterization of the US economy and compute its (unique) socialist allocation, and the Gini coefficient of the income distribution in that allocation. We compute the Gini coefficient of after-tax income in the present US "social democracy" and show that it is lower than in the socialist allocation. Hence, socialists must choose between two mutually exclusive alternatives: eliminating exploitation in the Marxian sense (achieving socialism, as defined above), or equalizing income. We propose that egalitarians must go beyond socialism, as it has been classically conceived.