Summary
In 20th-century discussions of Karl Marx's economics, the transformation problem is the problem of finding a general rule by which to transform the "values" of commodities (based on their socially necessary labour content, according to his labour theory of value) into the "competitive prices" of the marketplace. This problem was first introduced by Marx in chapter 9 of the draft of volume 3 of Capital, where he also sketched a solution. The essential difficulty was this: given that Marx derived profit, in the form of surplus value, from direct labour inputs, and that the ratio of direct labour input to capital input varied widely between commodities, how could he reconcile this with a tendency toward an average rate of profit on all capital invested among industries, if such a tendency (as predicted by Marx and Ricardo) exists? Marx defines value as the number of hours of labor socially necessary to produce a commodity. This includes two elements: First, it includes the hours that a worker of normal skill and dedication would take to produce a commodity under average conditions and with the usual equipment (Marx terms this "living labor"). Second, it includes the labor embodied in raw materials, tools, and machinery used up or worn away during its production (which Marx terms "dead labor"). In capitalism, workers spend a portion of their working day reproducing the value of their means of subsistence, represented as wages (necessary labor), and a portion of their day producing value above and beyond that, referred to as surplus value, which goes to the capitalist (surplus labor). Since, according to Marx, the source of capitalist profit is this surplus labor of the workers, and since in this theory only new, living labor produces value, it would appear logical that enterprises with a low organic composition (a higher proportion of capital spent on living labor) would have a higher rate of profit than would enterprises with a high organic composition (a higher proportion of capital spent on raw materials and means of production).
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Law of value
The law of the value of commodities (German: Wertgesetz der Waren), known simply as the law of value, is a central concept in Karl Marx's critique of political economy first expounded in his polemic The Poverty of Philosophy (1847) against Pierre-Joseph Proudhon with reference to David Ricardo's economics. Most generally, it refers to a regulative principle of the economic exchange of the products of human work, namely that the relative exchange-values of those products in trade, usually expressed by money-prices, are proportional to the average amounts of human labor-time which are currently socially necessary to produce them within the capitalist mode of production.
Prices of production
Prices of production (or "production prices"; in German Produktionspreise) is a concept in Karl Marx's critique of political economy, defined as "cost-price + average profit". A production price can be thought of as a type of supply price for products; it refers to the price levels at which newly produced goods and services would have to be sold by the producers, in order to reach a normal, average profit rate on the capital invested to produce the products (not the same as the profit on the turnover).
Exchange value
In political economy and especially Marxian economics, exchange value (German: Tauschwert) refers to one of the four major attributes of a commodity, i.e., an item or service produced for, and sold on the market, the other three attributes being use value, economic value, and price. Thus, a commodity has the following: a value, represented by the socially necessary labour time to produce it (Note: the first link is to a non-Marxian definition of value); a use value (or utility); an exchange value, which is the proportion at which a commodity can be exchanged for other entities; a price (an actual selling price, or an imputed ideal price).
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