Panteion University of Social and Political Sciences
Ricardo’s corn model was the first model that made the unambiguous rank, comparison and choice of techniques, with respect to their profitability, possible. The problem of choosing techniques will be mathematically formulized and a solution based on the corn model will be provided. For those, who are familiar with the problem of price determination, it is well known that it is impossible to determine prices, for a given nominal wage, without determining profit rate first. From the price determination system, a relation between nominal wage and profit rate can be obtained, known as the w-r relation. To obtain the w-r relation, and fully determine prices, the price of a single commodity, or of a basket of commodities should be exogenously determined. The w-r relation as it will be shown in this paper is sensitive in the above arbitrary price determination. This vicious cycle was known to Ricardo. He concluded that these difficulties were just mathematical and technical natured and therefore ostensible. This vicious cycle broke, as it will be shown later in this paper, by determining profit rate before and independently of prices.
Key words: Classical, Sraffian, Ricardian, Political Economy
JEL classifications: B12, B24, B51, E11, P16
The w-r relation and its dependence on price normalization
of of matrix . It also holds for
Thus, it holds:
We further assume that:
Replacing w with zero in (1) it follows:
From (2) it follows that:
We furthermore denote:
From (a) and (b) we can deduce that:
Equation (4) fully determines the price vector and up to a scalar for an exogenously given , or for an exogenously given w. It is concluded that prices cannot be fully determined for an exogenously given r, or w. To fully determine prices and obtain the w-r relation, it is mandatory to normalize prices first. The arbitrary determination of the price of a commodity or of a basket of commodities is called price normalization (Stamatis, 1991). The price normalization is accomplished by equating the price of a commodity or the price of a basket of commodities to a constant positive quantity of a homogeneous extensive thing. This equation is called normalization equation, the commodity mentioned above (basket of commodities) normalization commodity/ies and the positive quantity of a homogeneous extensive thing is called fictitious money. Let the normalization commodity be y, y ≥ 0. Let also the price of this commodity be equal to b units of fictitious money, or otherwise b units of that homogeneous extensive thing. Then the normalization equation is:
When we multiply equation (4) with (5) we can obtain the w-r relation, which is:
As R* we define the maximum profit rate of the normalization subsystem and as R¯* the profit rate which guarantees positive process.
On the other hand, if the normalization commodity consists of both basic and non-basic commodities, it follows that: A* = A . Then it holds (Stamatis G. 1998 p.12):
Therefore, it is possible to have:
for a normalization commodity consisting only of basic commodities and
for a normalization commodity consisting of both basic and non-basic commodities.
Based on the above, we can substitute relations (1), (2), (3), (4) , (5) , (6) , (7) with (1a), (2a), (3a) , (4a) , (5a) , (6a) , (7a) respectively:
We have seen that the maximum profit rate of the normalization subsystem may be different from the maximum profit rate of the given subsystem. This implies that the position of the w-r curve may change with price normalization.
When normalization commodity changes, the normalization subsystem, that produces the normalization commodity, as its net product, also changes. With the normalization subsystem changing the obtained w-r relation also changes. The w-r relation does not belong to the technique [A, I] but to the normalization subsystem [A*, I*, X*], instead. It is possible in other words to have as many w-r relations as the correspondent normalization subsystems.
It is obvious that since real wages consist only of corn, material inputs and the real wages have the same composition. Accordingly we can define matrix
Let the gross product, of the reproductive subsystem be equal to a unit of corn. Therefore, the net product is equal to
consists also of labor inputs. In that case net product is also the surplus product of this economy. Equation (9) represents the price system of the corn economy
From (9) is implied:
Concluding remarksWe have seen how price normalization changes the w-r relation in decomposable production systems. The reason is quite simple: we cannot fully determine the prices without first normalizing the prices. Price normalization can change the dimensions of the production system, converting the production technique used by the given system, into a normalization subsystem. The existence of multiple subsystems can change the maximum profit rate of the normalization subsystem. Therefore, there is no need for the profit rate of the given production technique, and the profit rate of the normalization subsystem to coincide. Price normalization also changes the slope of the w-r curve (in other words the capital intensity of the normalization subsystem in price terms). The latter also changes the shape of the w-r curve. It is well known that w-r curve serves as the basis, for applying the profit maximization criterion. This criterion has been for decades a popular tool for economists to choose among techniques although, as has been shown above, they do not actually choose between techniques, but among the normalization subsystems. The fact that prices depend on the rate of profit and the rate of profit depends on the normalization of prices makes it impossible to choose techniques in the general case.
Ricardo approximately 150 years before the so-called Cambridge Controversy, has found a way to overcome the above difficulties, although even then he was not aware of it. Having identified the vicious circle of prices, profit rate and wage determination, he had ingeniously introduced the framework of "common composition". He knew, in determining prices, an income distribution variable (he choose real wages) should first be determined exogenously. Real wages and moreover labor itself consisted as one commodity (re)produced in the Ricardian production system. Considered that, real wage is determined based technological attributes only, independently of price normalization and profit rate. As a result, the material inputs, the net product (at the same time the surplus product) and the real wages had the same composition. Given all the above assumptions and conclusions, he managed to determine profit rate independently of price normalization.
In the corn model price normalization was not mandatory for determining the profit rate. Profit rate could easily be determined for each positive vector of prices. The above determination made it possible to choose unambiguously among techniques, since in corn models all the normalization subsystems coincide with the given production technique. So 150 years before Sraffa, Ricardo introduced a well-defined method of determining the profit rate.
It holds for
In the special case that the w-r is linear then it holds
otherwise it holds
5. In the Ricardian framework there are three solid social classes: i) landowners own the land, which they rent, ii) capitalists who possess the capital, by which they buy labor power and organize the production and iii) workers who sell their labor force to the capitalists. In our model we have assumed that land can be extended indefinitely, and therefore, we have included it in the material input matrix. We further assume that all land pieces are equally productive. Consequently, all landowners receive the same rent. The above assumptions do not affect the results our analysis, but simplify it instead, since in Ricardo all the income variables are expressed in corn terms. In Pasinetti (1980) the analysis is more complex since the production of corn is referred to decreasing returns to scale and the luxurious product (gold) is subjected to constant returns to scale.
10. It is obvious that corn is a reproductive commodity and the other commodities are non-reproductive commodities. Thus Ricardo’s corn system can be represented in a canonical form with a decomposable non-singular matrix.
11. It holds for the price of the non-reproductive system: