Prices of production 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; it refers to the price level 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 in producing them. The importance of those price levels is, that a lot of other prices are based on them, or derived from them: in Marx's theory, they determine the cost structure of capitalist production.
The concept of production prices is introduced and elaborated systematically in chapter 9 et seq. of the third volume of Das Kapital, although it is already referred to in earlier texts by Marx. The first significant discussion occurs in the Grundrisse (1857-1858), followed by numerous references in Theories of Surplus Value (1862-1863), letters by Marx to Engels of 2 August 1862 and 30 April 1868 outlining his theory, the Resultate manuscript (1863-1866), and Capital, Volume II (1865-1877).
In the third volume, Marx considers the operation of capitalist production as the unity of a production process and a circulation process involving commodities, money, and capital. Capitalist production cannot exist without markets, and therefore it is a type of production that fully depends on market trade. For the purpose of the analysis of the immediate process of capitalist production, which is the main subject of Capital, Volume I, successful market transactions are assumed and market fluctuations are initially mostly disregarded - but that gives only one half of the story. The products have to be sold at a profit, and purchased at a competitive price, through market trade and the circulation of capital. The argument in Capital, Volume III is that the sales of newly produced commodities in the capitalist mode of production are regulated by their production prices.
A lot of the controversy about Marx's concept of production prices is probably caused by the fact that Marx never finalized the text of the third volume of Capital for publication. The book was edited together posthumously by Friedrich Engels, who tried to make a polished story out of a mass of draft manuscripts Marx left behind. Marx sketched complicated issues in a shorthand way which is sometimes ambiguous and incomplete, and does not make all the implications explicit. According to the German Marxian scholar Michael Heinrich, "Marx was nowhere near solving all of the conceptual problems". However, Marx's concept is also frequently confused with similar concepts in other economic theories. For most economists, the concept of production prices corresponds roughly to Adam Smith's concept of "natural prices" and the modern neoclassical concept of long-term competitive equilibrium prices under constant returns to scale. However, the function of prices of production within Marx's theory differs from both classical political economy and neoclassical economics.