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Università Cattolica del Sacro Cuore
ISTITUTO DI TEORIA ECONOMICA
E METODI QUANTITATIVI
Pasinetti on Ricardo
Enrico Bellino
January 2012
ISBN 978-88-343-2180-5
VITA E PENSIEROUniversità Cattolica del Sacro Cuore
ISTITUTO DI TEORIA ECONOMICA
E METODI QUANTITATIVI
Pasinetti on Ricardo
Enrico Bellino
January 2012
VITA E PENSIEROEnrico Bellino, Istituto di Teoria Economica e Metodi Quantitativi,
Università Cattolica del Sacro Cuore, Milano
enrico.bellino@unicatt.it
ist.temq@unicatt.it
www.vitaepensiero.it
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accordance with the provisions of the law n. 633 of 22 april 1941 (art. 68, par. 4 and
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© 2012 Enrico Bellino
ISBN 978-88-343-2180-5Acknowledgments. I am grateful to Luigi Pasinetti for the conversa-
tions about his interest for Ricardo since its genesis. I also thank
Luciano Boggio, Pierluigi Porta and Rodolfo Signorino for the
comments on a previous version of this note. As usual, the responsi-
bility of the whole content is my own.
Entry written for the Companion to David Ricardo, ed. by Heinz
Kurz and Neri Salvadori, Edward Elgar, forthcoming.
34
1. Introduction
The mathematical formulation of the Ricardian system proposed by
Pasinetti (1960) has become the primary reference for all scholars
interested in Ricardian economics ever since its publication. It was
written in the Academic year 1957-58 when Pasinetti was at Harvard
University attending a series of seminars for graduate students orga-
nized by Franco Modigliani, who was at Harvard that year on leave
from North Western University. Modigliani asked his students to
chose their favourite economist and express his theory in mathemati-
cal terms. Pasinetti chose Ricardo. His interest for Ricardo had arisen
from his previous Academic year spent at Cambridge, U.K.,1 where
he had the opportunity to read the Kaldor (1956) paper, attend his
lectures and read the Introduction to Ricardo’s collected works writ-
ten by Sraffa (1951). Pasinetti recalls that in its first formulation, his
paper dealt separately with first the case of just one commodity
(‘corn’), naturally following Kaldor’s diagrams which were put into
equations, and then with the case of two commodities (‘corn’ and
‘gold’). When the work was later submitted to Sraffa for discussion,
the latter expressed strong disagreement with such a distinction and
convinced Pasinetti to immediately start with the case of two com-
modities.2 A basic overview of Pasinetti’s article is provided in Sec-
tion 2. The article formalizes the distribution and value theory con-
tained in Ricardo’s Principles but in Section 3 we shall see how Pa-
sinetti’s model actually formalizes the distribution theory contained
in Ricardo’s early writings, which after Sraffa’s rehabilitative read-
ings provided in (1951) and (1960) – see Section 4 – can be regarded
as the ‘core’ of the Ricardian distribution and value theory.
1
Pasinetti spent the Academic year 1956-57 at Cambridge, U.K., as a PhD
student; in 1957-58 he was at Harvard University , where he was also regis-
tered as PhD student. At the end of that year, he was asked to chose only
one option, and he decided to return to Cambridge, U.K. I got these details
in private conversations with Luigi Pasinetti, who is gratefully acknowl-
edged. I apologize for any possible error or inaccuracy.
2
Later on, in his Lectures (Pasinetti, 1977, chap. 1) he re-proposed this dis-
tinction, essentially as a didactic device.
52. The mathematical formulation
Consider an economic system composed of three classes: land-
owners, capitalists and workers. Capitalists organize the production
process by employing workers and the lands rented by land-owners.
Two categories of goods are produced, which in the mathematical
formulation are reduced to one good for each category: ‘necessary’
goods (say ‘corn’) used as wage-goods for worker, and ‘luxury’
goods (say ‘gold’). Both productions cycles take exactly one year
and capital is constituted just by the wage-good advanced to workers.
Lands used in corn production may differ in fertility. Capitalists be-
have rationally and organize the production of corn on the various
plots of land in order of decreasing fertility: lower quality land is
used as the production of corn is increased. This technology can be
represented by the function,
Qc = f(Nc), (1)
where Qc is the output of corn and Nc are the workers employed in
corn production. Let us suppose that: i) f(0) ≥ 0, ii) f ′(Nc) > 0, iii)
f ′′(Nc) < 0 and iv) f ′(1) > 0.3 Assumption i) is trivial; ii) and iii) to-
gether reflect the fact that as corn production is extended the addi-
tional product decreases as additional workers are employed on less
and less fertile plots of land. We thus have decreasing returns to
scale due to an extensive use of lands of varying quality.4 Assump-
3
Assumption ii) was not explicitly introduced by Pasinetti. Moreover, in
(1960), assumption iii) takes the form f ′(0) > 0: “at least when the econom-
ic system begins to operate and workers are employed on the most fertile
piece of land, they must produce more that what is strictly necessary for
their support”: Pasinetti, 1960, p. 82; the formulation of iii) used here is bor-
rowed from Pasinetti (1977, equation (I.3.8-b)).
4
From the formal point of view, conditions ii) and iii) can also be inter-
preted in intensive terms, according to the usual assumption of decreasing
marginal returns of factors (see Pasinetti, 1977, p. 10, fn. 8). This interpre-
tation of conditions ii) and iii) is not however very relevant for Ricardian
analysis. Partially connected to this point, Morishima (1989, pp. 50-1) ob-
jected that a unique corn production function, summarizing the input-output
6tion iv) is a ‘viability’ condition: when the first worker is employed
on the most fertile plot of land he must be able to produce more that
his subsistence. Given the different qualities of lands, the owners of
cultivated lands will be able to claim payment from capitalists, the
rent. On each plot rent will at most be equal to the difference be-
tween the corn produced on that plot and the corn produced on the
least productive cultivated plot – called ‘marginal land’ – given by
f '(Nc). Capitalists would have in fact the alternative of cultivating on
the marginal land or worse lands, where cultivation is free. Total
rents (R) are thus given by the difference between the total quantity
of corn produced, f(Nc), and the corn that would be produced if all
lands had the same fertility as that of the marginal land, Nc f '(Nc):
R = f(Nc) – Nc f '(Nc). (2)
Gold is produced by labour under constant returns to scale; its tech-
nology is described by
Qg = αNg, (3)
where Qg is the output of gold, Ng are the workers employed in gold
production and α is the quantity of gold produced by one worker.
Also, the unit wage x is fixed at the subsistence level x on the basis
of the Malthusian principle:
x=x ; (4)
x is not what is physiologically considered as the necessary mini-
mum needed to support workers. It is the level that in a given country
and in a specific stage of society keeps the population constant. Total
wages (W) are:
W = (Nc + Ng)x. (5)
Capital (K) consists just in wages advanced to workers:
K = W; (6)
relation of all lands, could be used “to explain the rent of a land as the sur-
plus which it yields”; Kurz and Salvadori (1992, § 3) argued in favour of
Pasinetti’s approach.
7the total amount of corn that can be advanced to workers is given:
K= K . (7)
Profits are determined as a surplus, that is, as the difference between
the gross product (net of rents) and the ‘necessary consumption’
needed to repeat the production process year after year at least at an
unchanged scale. In both industries, necessary consumption consists
just of wages. Profits in the corn industry (Pc) are determined in
physical terms, as the difference between homogeneous quantities of
the same commodity (corn):
Pc = (Qc − R) − Ncx. (8)
On the contrary, profits in the gold industry (Pg) must be calculated
in value, being the difference between amounts of heterogeneous
commodities:
Pg = pg Qg − pcNgx, (9)
where pc and pg are the prices of corn and of gold. To explain what
determines pc and pg, we must elaborate a theory of value. Coher-
ently with the analysis developed in the Principles, Pasinetti resorts
to a pure labour theory of value, according to which the value of
each commodity (net of rent, if any) is equal to the quantity of labour
required to produce it:5
5
Quite similar steps are proposed in (Pasinetti, 1977, p. 14): instead of in-
troducing the labour theory of value through equations (10) and (11), Pasi-
netti states: “Ricardo argues that what fundamentally determines the ‘value’
or ‘natural price’ of produced commodities is their cost of production”, that
is, “wages plus profits at the ruling rate of profit” (see Pasinetti, 1977, p.
14). In formal terms:
pc(Qc − R) = pcxNc(1 + r) (10′)
pgQg = pcxNg(1 + r). (11′)
Given these equations, Pasinetti just writes
pgQg /Ng = pc(Qc − R)/Nc ,
8pc(Qc − R) = Nc, (10)
pgQg = Ng. (11)
The rate of profits of the system (r) is given by:
pc Pc + p g Pg
r= . (12)
pc K
We have thus 12 equations in 13 unknowns: Qc, Qg, Nc, Ng, R, x, W,
K, Pc, Pg, pc, pg, r. The remaining degree of freedom is closed by
formulating a theory of expenditure. Like Ricardo, Pasinetti supposes
that rents are entirely spent on luxuries (with the exception of a neg-
ligible part, not considered here). Hence
pgQg = pcR. (13)
This equation implicitly entails that the output of corn equals the in-
comes of the other two classes taken together, i.e. profits plus wag-
es.6
The configuration described by equations (1)-(13) is called
by Pasinetti the ‘natural’ equilibrium of the Ricardian system. This
configuration is actually a ‘moving’ equilibrium. In fact, besides a
theory of distribution, a theory of value and a theory of demand, it is
quite easy to enucleate a theory of growth from the economic system
here considered. As capitalists save (the main part of) their profits
and accumulate them into the stock of corn K available at the begin-
ning of each period, a higher number of workers can be employed in
which is obtained by dividing (10′) and (11′) by Ng and Nc respectively and
by equalizing the left-hand side members. This equation (which equalizes
the value of the product per worker – net of rents – of the two industries)
together with pgα = 1 (which coincides with equation (11) after substituting
(3)), replace equations (10) and (11).
6
Substituting equations (8) and (9) into (12) we obtain:
rpcK = pc(Qc − R − xNc) + pgQg − pcxNg.
Thanks to (5) and (13) one gets
rpcK + pcW = pcQc.
9both industries as time goes by. This entails the cultivation of more and more plots of land of increasingly lower quality. Rents thus augment, squeezing profits from a certain point on, till they are ze- roed out (or below that level which induces capital accumulation), with total wages increasing proportionally. The capital accumulation process thus stops and the system reaches the (Ricardian) stationary state. A simple way to study this process is to study the sign of the derivatives of the natural equilibrium value of the endogenous va- riables of the model with respect to K.7 Pasinetti obtains thus that all these magnitudes increase as a consequence of capital accumulation with just two exceptions: the rate of profits which decreases, and to- tal profits which increase at the beginning of the process of capital 7 More in general, Pasinetti points out that there are at least four dynamic processes at work in Ricardo’s system: i) capital mobility, which tends to equalize the rates of profits of the industries by channeling capital towards the most profitable industry; ii) a demographic dynamics, determined by the Malthusian mechanism, which pushes the wage rate to its natural level; iii) capital accumulation, as just described in the text, and iv) technical progress, which delays – without subverting – the convergence to the statio- nary state through shifts of the corn production function. In Pasinetti we do not find an explicit analysis of dynamics i) and iii): beyond temporary oscil- lations, a uniform rate of profits is supposed as permanently achieved on average, while technical progress is supposed by Ricardo to not alter the main conclusions of his analysis. More emphasis is devoted to processes ii) and iii) which require the setting of a truly dynamic system which is pre- sented by Pasinetti in the Appendix of his article. The convergence to the stationary state with a natural wage rate is formally proved there, even if from the economic point of view the description of the transient process, i.e. the sequence of ‘natural’ equilibria, is more relevant than the resting point of the process (the Ricardian steady state): “Ricardo, however, inves- tigates the properties of his system at a very particular stage of the whole movement, which he considers the relevant one. Most of the analysis is car- ried on as if the demographic mechanism has already fully worked through, while the capital accumulation process has not yet been completed” (Pasi- netti, 1960, p. 87). For this reason, the rough analysis based on the sign of partial derivatives proves to be more informative than the rigorous analysis in the Appendix. 10
accumulation and decrease when the system approaches the statio-
nary state.
3. Principles versus Essay and the “early writings”
System (1)-(13) is a stylized version of a Ricardian economy
suitably simplified in such a way as to present the core of Ricardo’s
Principles while avoiding all those complications that prevent him to
provide univocal and rigorous results. The crucial assumption that
makes this experiment possible is the supposition that just one com-
modity (‘corn’) is both consumed and used as capital good. Not sur-
prisingly, but quite interestingly, this simplification coincides with
that adopted by Ricardo in the Essay (Ricardo, 1815) and in some
other early writings which Sraffa (1951 and 1960) has allowed us to
re-appraise for their theoretical content and insight. The crucial de-
vice of these works is the substantial homogeneity between outputs
and inputs (wage goods or, simply, ‘corn’) due to the primacy of
agriculture. This homogeneity makes it possible to determine the rate
of profits of agriculture in physical terms with the consequence that
competition among capitalists will induce other industries to align
their rates of profits to that obtained in agriculture. A simple re-
styling of Pasinetti (1960) provides us with an analytical formulation
of this distribution and value theory contained in Ricardo’s early
writings which seems even more effective than the one provided by
Pasinetti of Ricardo’s Principles. Let us replace the five equations
(8)-(12) with the following four equations:
Qc − R − xN c
rc := , (8E)
xN c
p g Qg − pc xN g
rg = , (9E)
pc xN g
rg = rc, (10E)
pgα = 1 (11E)
11(the letter (E) following the equation numbers stands for Essay and
Early writings.) Equations (8E) and (9E) define the rates of profits of
the two industries. The physical nature of the rate of profits of the
corn industry emerges immediately: it is a ratio between quantities of
corn, Qg, R, and xNc; after substituting (1) and (2) rc can be re-
f ' ( Nc ) − x
expressed as rc = . The rate of profits of the corn indus-
x
try can thus be known before the determination of prices. On the
contrary, the rate of profits of the gold industry depends on prices.
Capital mobility will tend to align the rate of profits of gold to the
rate of profits of corn, as stated by equation (10E). Thus, the rate of
profits of the entire system is
f ' ( N c ) − x 1 − x / f '( N c )
r= ≡ , (14)
x x / f '( N c )
where x/f ′(Nc) is the quantity of corn paid as wage to the amount of
labour required to produce 1 unit of corn on the marginal land. This
result echoes the famous ‘basic principle’ that “it is the profits of the
farmer that regulate the profits of all other trades” contained both in
Ricardo (1815) and in his correspondence in 1814 and early 1815
with other economists (see Sraffa, 1951, p. xxxi). This equalization
takes place through suitable changes of the relative price of gold in
terms of corn: after substituting (8E) and (9E) into (10E) and using
(1), (2) and (3) one obtains:
pg 1/ α
= . (15)
pc 1/ f ' ( Nc )
Lastly, equation (11E) fixes the quantity of gold produced by one
worker (α) as the unit of account. Observe that 1/α and 1/f ′(Nc) ex-
press the quantities of labour required to produce one unit of gold
and one unit of corn on the marginal land, respectively. The labour
theory of value does not enter here as an assumption, like in equa-
tions (10) and (11), nor does it play a particular role in the theory
here considered. It is just a consequence of the assumption that the
12capital of both industries is constituted by a single commodity; capi-
tal intensity is thus uniform between industries, xNc/Nc = xNg/Ng = x,
and the labour theory of value holds.8 For further reference let us de-
note by (E) the model constituted by equations (1)-(7), (8E)-(11E)
and (13). It contains 12 equations in 12 unknowns: Qc, Qg, Nc, Ng, R,
x, W, K, pc, pg, rc, rg.
4. The direct connection with Sraffa (1960)
Model (E) constitutes the crucial link in the chain from Ri-
cardo’s early writings to Sraffa (1951 and 1960). Sraffa’s works ap-
pear thus as the generalization of Ricardo’s distribution and value
theory contained in his early writings.9 The unacceptable restriction
that corn was the only commodity required for its own production as
well as for the production of all other commodities is now totally re-
moved. By introducing some additional Assumptions in the Sraffa
system we can see the direct connection with system (E). Ass. 1:
wages paid ex-ante; the price system become thus
pT = (1 + r)(pTA + wlT), (16)
where p is the price vector, A is the input coefficient matrix, l is the
direct labour coefficient vector and w is the money wage rate. Ass. 2:
wages are constituted by a composite commodity represented by vec-
tor x; then
8
Curiously enough, the assumption that capital consists of corn anticipated
to workers only, leads us also to the opposite extreme of the labour theory
of value i.e. the pure capital theory of value: by multiplying both the nu-
merator and the denominator of the right-hand side of (15) by x we see that
the relative price of gold in terms of corn is regulated also by the ratio of the
quantities of capital-wage required to produce one unit of the two goods.
9
Note that Sraffa writes: ‘It should […] be stated that it was only when the
Standard system and the distinction between basics and non-basics had
emerged in the course of the present investigation that the above interpreta-
tion of Ricardo’s theory [presented here in Section 3] suggested itself as a
natural consequence’ (Sraffa, 1960, p. 93). A criticism of this Sraffian ‘in-
terpretation’ of Ricardo’s theory is expressed by Porta (1986).
13w = pTx. (17)
Thanks to (17) the price system (16) can be re-written as
pT = (1 + r)(pTA + pTxlT) = (1 + r)pTS, (18′)
where S = A + xlT is the socio-technical matrix. In this case the en-
suing rate of profit is
1 − λ*S
r= , (19)
λ*S
where λ*S is the dominant eigenvalue of S. Ass. 3: commodities are
just required as wage goods and not as capital goods, hence A = O;
Ass. 4: wages are constituted by just one commodity, say commodity
1; then x = [x1, 0, …, 0]T, matrix S is reduced to a matrix having all
zero entries except for in the first row, which has components x1lm, m
= 1, …, M, where M is the number of commodities. Then λ*S = x1l1,
and (19) collapses into
1 − x1l1
r= (19′)
x1l1
which coincides with equation (14), as x1l1 still represents the quanti-
ty of commodity 1 paid as wage to the amount of labour required to
produce 1 unit of commodity 1. In this way the correspondence be-
tween the Sraffa system and the model of Section 3 is complete. If
some of the above assumptions are relaxed and we allow – as it is
normal nowadays – that commodities are employed as capital goods
and that wages enter as generalized purchasing power and are ex-
pressed in terms of the Sraffa Standard commodity relation (19′)
finds its correspondent in
1− w
r= , (20)
w + 1/ R
where R is the uniform physical rate of surplus of basic commodities
(see, for example, Bellino, 2004). Hence, relation (20) extends to the
14general case of C (≤ M) basic commodities the idea (conveyed by
(14)) that the rate of profits of a system can be expressed in purely
physical terms as the surplus of the production process of basic
commodities only.10
5. Concluding remarks
Pasinetti’s formulation of the Ricardian system was originally con-
ceived as a mathematical presentation of the basic structure of Ricar-
do’s Principles. But the assumption that capital is constituted just by
one commodity has two consequences: on the one hand, it establish-
es a direct connection of the model with the logical structure de-
scribed by Ricardo in his Essay, rather than in its Principles; on the
other hand, it renders unnecessary the introduction of the labour
theory of value as an assumption, as we have no necessity to measure
aggregates of commodities with different compositions in order to
calculate the rate of profits. This rate emerges as a ratio of physical
quantities of corn for the entire economic system. This result is the
point of departure of Sraffa’s generalization to any number of basic
commodities.
References
Bellino E., (2004), “On Sraffa’s Standard commodity”, Cambridge
Journal of Economics, Vol. 28, No.1, pp. 121-132.
Kaldor N., (1955-56), “Alternative Theories of Distribution”, The
Review of Economic Studies, Vol. XXIII, No. 2, pp. 83-100.
10
The same asymmetry also appears in other spheres: a technical improve-
ment in the production of a basic commodity (of corn) spreads its effects to
the entire system through the wage profit relation (20). If the improvement
concerns a non-basic commodity (gold), it merely affects the relative price
of non-basic products. The same can be said of a tax on basics or non-
basics. Again, it is easy to prove that the prices of basic commodities de-
termine the prices of both basic and non-basic commodities, while the pric-
es of non-basics do not.
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16Istituto di Teoria economica e Metodi quantitativi
Working Papers (dal 2007)
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Process and Product Innovation under different Competitive Regimes,
gennaio 2009.
54. LUIGI PASINETTI, Prospettive e Limiti dell’Economia quantitativa, gen-
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55. GIANLUCA FEMMINIS E GIANMARIA MARTINI, First mover advan-
tage in a dynamic duopoly with spillover, luglio 2009.
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NI, On Labour Shares in Recent Decades: A Survey, dicembre 2009.
2012
58. ENRICO BELLINO, Pasinetti on Ricardo, Vita e Pensiero, gennaio 2012
(isbn 978-88-343-2180-5).
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ISTITUTO DI TEORIA ECONOMICA
E METODI QUANTITATIVI
Pasinetti on Ricardo
Enrico Bellino
January 2012
ISBN 978-88-343-2180-5
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