Generalization of Sraffa's price model

Piero Sraffa, a classical economist, aims with his seminal book « Production of Commodities by means of Commodities » (1960) to improve David Ricardo’s theory of value. The historians of economic thought consider this book as the initiation of the neo-Ricardian school, a further development of classical economics. The problematic of circularity and of production is paramount. How have the quantities and prices of commodities, the constant profit rate r and the constant wage rate w to behave, in order to guarantee a reproduction with surplus? Sraffa’s price model is an equilibrium model, where the commodities are at the same time products and means of production.

There still is a large debate on the nature of Sraffa’s book, "compressed and mathematically incomplete" said Newman (1962). Schefold showed (1976) that the theorem of Perron-Frobenius underlies Sraffa’s work. Sraffa considers in the first part of his book production economies of n sectors, each one producing only one commodity. The profit rate of the entrepreneurs is assumed to be constant and equal to r in all sectors, as the wage rate is also constant and equal to w for all the workers, performing labour in all the sectors. Sraffa writes n equations, comprising, the prices of all commodities, the quantity of labour realized in each sector, the constant rate of profits and the constant wage rate. Sraffa attains the so-called price model, by completing the n equations by an equation to calculate the value of the surplus, the national income Y. Sraffa’s price model enables calculation of the n prices of the commodities, under the condition to have normalized the total national income and the total amount of needed labor to unity.

In this paper we go on with the investigation of the mathematical background, for example pointing out that a theorem published by Ashmanov (1984) on productive Leontief economies is also on the background of Sraffa's book. Modern matrix algebra notation is used.

We do no longer add normalizations of national income and labour. Starting from Sraffa’s first examples (1960) with enlarged models, we propose a series of production problems, exemplifying them, solving them and calculating all the remaining variables. It is shown that either relative prices or absolute prices are at the issue, depending on the choice of prices or national income as exogenous variables.