**A Mathematical Formulation of The Ricardian System**

Pasinetti’s first major contribution to economics was probably "the mathematical formulation of the Ricardian system", published in 1960 in a paper now considered classical. In such work Pasinetti presented a very concise and elegant (and pedagogically effective) analysis of the basic aspects of classical economics.

At that time, Piero Sraffa had just published *The Works and Correspondence of David Ricardo*, one of the most masterful editorial works ever to be published in economics; and scholars were wondering how Sraffa’s remarkable work could clarify and enrich the interpretation of *Classical economics*. Pasinetti’s mathematical formulation provided a rigorous and clear answer to that question, in particular with *reference* to two major Classical problems: the *theory of value* and the *theory of income distribution*.

On this subject, a major stimulus came from a famous paper written by Nicholas Kaldor, in 1956, where Kaldor presented a review of the history of several theories of distribution, covering the period from Ricardo to Keynes. Although Ricardo's theory (in Kaldor's paper) was without equations, it was the starting point after which economists began to explicitly see the Ricardian model as a coherent whole, susceptible to mathematical formalization.

Another influence came directly from Sraffa and concerned the relative prices of goods produced in the economic system made to depend only on the amount of labour embodied in them – the well known *labour theory of value*. In fact, an early draft of Pasinetti’s paper was read by Sraffa, who gave his approval to almost the entire paper:

“ | “I myself remember that, when I returned to my college after submitting to his attention an early draft of my mathematical formulation of the Ricardian system, a friend immediately asked me: ‘Have you now thrown it into the paper basket?’ At my answer, ‘I have, but only the first section; the major part of the work seems to stand’, the surprised reply was, ‘Well, if it has gone through Sraffa’s scrutiny, it will hold for good”. | ” |

—L. Pasinetti |

Pasinetti explains that*“the more constructive approach is taken of stating explicitly the assumptions needed in order to eliminate the ambiguities”* in the Ricardian model, hence, the reason for the mathematical formulation.

In this view, the most straightforward mathematical Ricardian model that can be formulated, with *minimal economic* complications, is the one in which only one commodity is produced (‘corn’ for instance), and where there are three social classes: capitalists who earn profits, workers who earn wages and landowners whose income comes from the rent of land. This is the way to express Kaldor’s model mentioned above. Although the Pasinetti model is more general and comprises two sectors (agriculture and manufacturing), it is illuminating to start with the simplest version –a one-commodity model expressed by the following equations:

*(1.1)*

*(1.1a)*

*(1.1a)*

*(1.1a)*

Equation *(1.1)* shows that output, *Y*, depends only on the number of workers, *N*, engaged to work on the land. Three conditions *(1.1a)* are necessary and help to restrict the meaning of *(1.1)*. Specifically, the first displays that land, when workers are not employed, can produce something or nothing at all. The second condition shows that the marginal product to kick start the cultivation of land must be greater than *μ*, the subsistence wage; otherwise the system will never start working. The third condition shows the diminishing returns of labour. The following equations show the determination of the quantities of different categories of income:

*(1.2)*

*(1.3)*

*(1.4)*

*(1.5)*

where *W* are the total wages, *x* is the wage per worker, *K* is the capital of the economy, *R* is the total rents perceived by landowners and *B* are the total profits that go into the hands of capitalists. The latter are represented as a residual income once the rents and wages have been paid. In these models all the capital is working capital; it is assumed to be composed entirely of advances to workers as wages. Notice, moreover, that expression *(1.4)*, supplemented by technical conditions *(1.1a)*, expresses what still today is called the Ricardian theory of rent.

Hitherto we need two more equations to close the model. They are:

*(1.6)*

*(1.7)*

Equation *(1.6)* shows that long-term wages tend to subsistence level. Equation *(1.7)* shows the stock of capital at the beginning of the year. Throughout the construction of his model, Pasinetti insists that he is interested in "natural solutions" of the Ricardian model, i.e., those to which the system tends to in the long term.

Thus, equation *(1.6)* does not preclude short-termwage deviations with respect to its natural level. Eventually, note that *μ* is the only magnitude which is determined from outside the system: it is an amount set by customs and habits of society. This means that the number of workers employed (proportional to population), is determined by the system itself, unlike what happens in modern theories of economic growth, where the rate of population growth is taken as exogenously given.

The previous model allows us to see the basic features of Ricardo’s distribution theory. However, no reference can be made to anything about the theory of value, since all the magnitudes are measured in terms of ‘corn’. As Pasinetti says: *“where any question of evaluation has not yet arisen, corn being the single commodity produced”*. However, by enlarging the model to a two-sector model, more remarkable features of Ricardian economics emerge. The two sectors include: the basic goods sector (wage goods called ‘corn’) and a luxury goods sector (called ‘gold’). The whole new model appears as:

*(2.1)*

*(2.1a)*

*(2.1a)*

*(2.1a)*

*(2.2)*

*(2.3)*

*(2.4)*

*(2.5)*

*(2.6)*

*(2.7)*

*(2.8)*

*(2.9)*

Equations *(2.1)* to *(2.7)* are identical to those of the single sector model, but now they bear added subscripts to differentiate the production of ‘corn’ from that of ‘gold’. Equation *(2.8)* presents the gold production function which, unlike the ‘corn’ production function, exhibits constant returns to scale. The parameter is the physical amount of ‘gold’ produced by a worker in a year. The next two equations show in monetary (nominal terms) the wage rate (‘corn’) per worker and the rate of profit:

*(2.10)*

*(2.11)*

where *p _{1}* and

*p*represent the price of ‘corn’ and the price of ‘gold’ respectively. Pasinetti supposes that the wage rate and the rate of profit are identical in both sectors thanks to free market competition. Note also from

_{2}*(2.10)*that only

*p*(the price of corn) enters wage determination, since it is assumed that workers in both sectors only receive only 'corn' as wages; in a terminology later developed by Sraffa, ‘corn’ is the only basic commodity produced in the system. The same consideration can be made, from the opposite view, looking at

_{1}*(2.11)*, since the only capital involved is represented by advances in the form of ‘corn’. Hence

*K*appears as multiplied by

*p*. The following two equations are Ricardo's implicit assumptions -that the "natural price" of any good is given by its cost of production:

_{1} *(2.12)*

*(2.13)*

Equation *(2.12)* allows the determination of the total monetary profits of the gold industry. Equation *(2.13)* is much more important. It shows that on balance, the value of output per worker is the same in both sectors. It should be noted that the product of sector 1 (agriculture) is considered free of rents. To achieve this result, both the wage rate and the rate of profit have been considered homogeneous in both sectors, and the capital/labour ratio must also be considered as identical.

Pasinetti closes the model with two further equations:

*(2.14)*

*(2.15)*

Relationship *(2.14)* simply adopts the amount of ‘gold’ as numeràire; hence, it is equal to unity. Relation *(2.15)* displays the income distribution between the different social classes. Ricardo supposes that workers spend their entire wages to buy 'corn', capitalists reinvest their profits in capital accumulation, and landowners spend all their rents on luxury goods. The simplicity of this argument about consumption functions for each social class enables Pasinetti to close the whole circuit of income distribution with a single equation. Specifically, *(2.15)* shows that landowners spend all their income received as rents, *p _{1}R*, in the buying of luxury goods,

*p*. There is no need for more equations, since

_{2}X_{2}*“the determination of the demand for one of the two commodities (gold in our example) also determines, implicitly, the demand for the other commodity (corn), since total output is already functionally determined”*.

The system presented above (15 equations with 15 unknowns) displays the natural solutions of the Ricardian economic system, i.e. those solutions attained in the long term and corrected for market distortions and imbalances in the short term. Ricardo did not deny the existence of such deviations, but for his analysis, they were not the relevant issues. These solutions, furthermore, as demonstrated mathematically by Pasinetti exist, they are unique, and, moreover, their steady state solutions are stable. On the other hand, it can be demonstrated that if we take the partial derivatives of all variables with respect to *K*, (because the process of capital accumulation is what matters for the dynamics of the model) the evolution of the variables is consistent with the conclusions which Ricardo reached; especially with the tendency of the whole economic system in the long run towards a stationary state.

The above model has several outstanding aspects. The foremost of which is a theory of income distribution entirely independent of the theory of value. The inclusion of a new sector -and consequently the whole structure of relative prices- have not even slightly changed the way income is distributed among landowners, workers and capitalists. At the same time, prices though of course not equal to, are exactly proportional to the quantity of labour embodied in each commodity. This is a perfectly clear *labour theory of value*.

The attentive reader may notice that the two major results (on income distribution and on value respectively) depend on two assumptions that are implicit in the formulation of the model, i.e. of the above set of equations, namely: 1) that capitalists appropriate the whole surplus of the economic system, after paying rents to landowners, and conventional wages to workers; and 2) that the proportion of labour to capital is exactly the same in all sectors. These two assumptions have given rise to endless discussions on value and distribution ever since.

Further work undertaken by Pasinetti’s has been concerned with reversing the causal chain of the first assumption and in rendering the second assumption superfluous.

Read more about this topic: Luigi Pasinetti, Theoretical Contributions

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