Robert Torrens
and the Classical Theory of Growth
Taro Hisamatsu
October 2016
Discussion Paper No.1633

GRADUATE SCHOOL OF ECONOMICS
KOBE UNIVERSITYROKKO, KOBE, JAPANTorrens and the Classical Theory of Growth1

Robert Torrens
and the Classical Theory of Growth†
Taro Hisamatsu
This paper reconstructs Torrens’s theory of economic growth.
Actual wage is determined by the capital-population ratio
between the maximum and the minimum wage. In the processof growth, the maximum wage keeps being lowered by thedecreasing marginal productivity in agriculture, while theminimum wage remains constant. Based on these notions ofwages, Torrens’s economic dynamics are described as follows.
In the early stage of growth, capital increases faster than
population so that the actual wage rises above the minimum.
Thereafter, the economy grows with a tendency for the
population to increase faster than the capital while limiting theactual wage below the decreasing maximum until it enters astationary state and the actual wage and profit rate are reduced
to their minimum. This theory has been attributed to Ricardo bysome scholars, but Torrens proposed a more ingenious theory
than Ricardo’s.

1. Introduction
In the recent issue of the Cambridge Journal of
Economics, Rosell (2016) proposed a valuable
reconstruction of the theory of distribution presented
by Robert Torrens (1780?–1864) in his text entitledOn the Means of Improving the Conditions of theLabouring Classes, which is appended to the final
edition of External Corn Trade (Torrens, 1829).1 The
main conclusions in Rosell’s reconstruction indicate
the following: Torrens’s approach to distribution
considers that the wage rate is set as a result ofcompetition among capitalists or among workers,†

A part of the paper is based on Hisamatsu (2015) written in
Japanese. The author has been supported by JSPS Grant-in-Aid
for Scientific Research (B): 25780144 and (B): 16K17095. I am
also grateful to Kazuki Kumashiro and Takuro Tanaka for thevery helpful comments and suggestions. Responsibility for thispaper’s contents, however, must remain with me.
1O’Brien (1966) points out that Torrens’s letter to
Wilmot-Horton dated 23 March 1826 was in fact the draft of
the 1829 Appendix. The Appendix, according to Robbins (1958,
p. 273) was reprinted with minor alterations in four
publications: a tract entitled Colonel Torrens on the Wages ofLabour, and on the Means of Improving the Condition of theWorking Classes. A New Edition, Corrected (1832), which is,according to De Vivo (2006, xxiii), ‘a reprint (with minor
alterations) of the Appendix to the 1829 edition of ExternalCorn Trade’, On Wages and Combination (Torrens, 1834), A
Letter to Russell (Torrens, 1837) and A Letter to Ashley(Torrens, 1844).

differing from the neoclassical economist’s way ofidentifying the real wage rate as the marginal productsof labour, and an original conception of a surplus oflabour to explain profit, based on the classical notionsof necessary labour and labour commanded, without
resorting to the Ricardian labour theory of value or the
Marxist theory of value. His approach thus does not
require fixing one of two distribution variables
exogenously in order to determine the allocation of the
net produce between the capitalist’s and worker’s
income―a way which differs from both Ricardo’s
way of requiring the wage rate given exogenously andSraffa’s way of requiring the profit rate given
exogenously.
The ‘statics model’ of distribution reformulated byRosell (2016, p. 15/18) can be clearly found in the1829 text. However, most discussions in the text weredeveloped within the dynamic view. In particular,
Torrens’s dynamic theory of economic growth basedon three notions of wage―the maximum wage,
minimum wage and actual wage―is worthy ofreconstruction because it explicitly formed a moreingenious theory than that which has been attributed toDavid Ricardo (1772–1823) by Hicks and Hollander(1977) and Casarosa (1982).
This paper reconstructs the economic growth theory
in the 1829 text and clarifies Torrens’s vision of the
dynamics of a closed economy. Torrens’s view on
economic dynamics can be explicitly explained by thegrowth model, which incorporates the wage satisfying
dynamic equilibrium between the growth rates of
population and of capital, the minimum wage fixed
exogenously and the maximum wage depending on thefertility of the lands under cultivation. Torrens
repeatedly emphasised that the maximum wage rate interms of corn continues falling from the necessity ofresorting to inferior lands in the process of capital
accumulation, but Rosell omits this point from his
considerations.
Torrens’s dynamic approach requires the classicaltheory of differential rent, the core of ‘the principles ofrent’ (Torrens, 1829, p. 466), so that diminishingreturns in agricultural production must be assumed.
Torrens did consider distribution issues in connection
with both the theory of rent and the law of diminishingreturns, but such connections are never involved inTorrens and the Classical Theory of Growth2

Rosell’s ‘static model’, in which returns to scale areassumed to be constant.
It should be also noted that in 1829, unlike before,
Torrens argued a long-run tendency in population toincrease faster than capital. The argument has animportant role in clarifying his vision of economic
dynamics. Such a long-term analysis is not involved in
Rosell’s model, in which the system is assumed to belimited to a single period. Also, as Torrens claimed to
Nassau William Senior (1790–1864), the credit ofbeing the first to prove the tendency, the change of his
position was never trivial for him.
This paper is organised as follows. Section 2
introduces Torrens’s concept of wages, based on
which Torrens’s model of economic growth is set up in
Section 3. Section 4 clarifies his vision on economic
dynamics using the reconstructed model and presents
a more theoretical understanding of the maximum
wage depending on the land productivity and a moretheoretical explanation of his correction concerning
the growth rates of population and of capital. Section
5 presents some concluding remarks.
2. Wages
2.1 Actual wage
The term ‘wages’ is defined as the ‘articles[necessaries and comforts] of wealth which thelabourer receives, in exchange for his labour’. It is‘real wages’ or ‘commodity wages’ that ‘consist of thequantity of necessaries and comforts which thelabourer receives’ while it is ‘nominal wages’ or‘money wages’ that are ‘the sum of money in which heis paid’. An increase (a decrease) in ‘the quantity ofnecessaries and comforts which the labourer received’means a raise (fall) in real wages (Torrens, 1829, p.
454). All the discussions in the 1829 Appendix are
explained with the wage in terms of a commodity,‘corn’, that is, the real wage in terms of corn.
The real level of ‘actual wages’2 is determined by‘the ratio between population and capital, or, morecorrectly, between the quantity of labour and theamount of capital’ in the labour market (p. 468). 3

There, however, are the limits beyond which the actual
wage rate cannot raise and below which it cannot fall.
2.2 Maximum wage
The limit beyond which the actual wage rate cannot
rise is the ‘maximum’ wage rate. What Torrens calls‘the maximum of wages’ is defined as ‘that quantity ofthe products of industry which remains, after replacing
the advances, not consisting of wages, and paying thecapitalist the lowest rate of profit which will induce
him to continue the work of production’ (p. 461). His
numerical example in terms of corn illustrates that theamount of ‘maximum wages’ received by 100 workers(200 quarters) is the amount of output (428 quarters)
minus the amount of non-wages fund (200 quarters)
minus the amount of minimum profits (28 quarters)
which is calculated as the amount of capital (400quarters), composed of wages fund and non-wages
fund, times the lowest profit rate (7%):
Workers⏞2qrs.×100 men⏟Maxmum wages= 428qrs.
−⏟

200qrs.
⏟

OutputNon wages fund
Capital(2qrs.×−⏞100 men + 200qrs. ) × 7%.
⏟
Minimum profits

The maximum wage per capita thus can be calculatedby:
2qrs. per capita =428qrs.⁄100 men 200qrs.
−,
1 + 7%
100 menthat is,Maxmum wage rate=

Output per capita
1 + Minimum profit rate

− Non wages fund per capita.
The maximum wage rate depends on the following
three circumstances: ‘the quality of the soil, the skill
with which labour is applied, and the degree of
freedom which is allowed to trade’ (p. 462). Torrens
states that the maximum wage rate ‘may be raised,either by the cultivation of land of a better quality, or

2

The minimum wage is equal to what Torrens had called ‘themarket price of labour’ in the first edition of External Corn
Trade (Torrens, 1815).
3
Considering that capital composts of not only wages fund but

also of non-wages fund, Torrens later changed the phrase ‘theamount of capital’ into ‘the quantity of the ingredients of capital
destined for its maintenance’ (1834, p. 22).

Torrens and the Classical Theory of Growth3

by improvements in the effective powers of industry;
and that this maximum will be lowered, either byresorting to poorer soils, or by a falling off in theeffective powers of industry’ (p. 458). Our work on the
dynamics of a closed economy requires analyses of theeffects of land productivity and labour skill upon theeconomy.
According to Torrens, in an industrial country suchas England, which is densely populated, the maximum
wage rate tends to be depressed ‘by the necessity ofresorting to inferior soils’ for supplying food to large
numbers of people (p. 457). Hence, in order to ‘raise
the maximum of wages to the highest point’, Englandshould carry on ‘a perfectly free trade withthinly-peopled countries, in which none but soils offirst-rate quality are under tillage’ (p. 459). That is,
improving the degree of freedom allowed tointernational trade has the same effect as that ofincreasing the land productivity upon the maximum
wage.4 Torrens was dissatisfied with the government
policies:
England having acquired in manufacturing industry an efficacy
unexampled in the history of the world, and having the newcountries of the American continent open to her commerce, isplaced in that precise situation in which the maximum of wages
may be elevated to the highest attainable point. But the vast, theincalculable advantages of this situation are counteracted by the
restrictive system [of Corn Laws], which excludes cheap raw
produce from our markets. (pp. 459–60)

2.3 Minimum wage
The limit below which the actual wage rate cannot
permanently fall is the ‘minimum’ wage rate. What
Torrens calls ‘the minimum of wages’ is defined as ‘aquantity of the necessaries and conveniences of life
sufficient to preserve the labourer in working
condition, and to induce him to keep up the race oflabourers’. 5 The quantity and variety of thesecommodities is exogenously determined by ‘climate
4

According to O’Brien (1966, p. 338), in Torrens’s 1826 letterto Wilmot-Horton which is considered to be the draft of the1829 Appendix, emigration, instead of free trade, had been
proposed as one of the means of raising the maximum wage.
5
The minimum wage is the same as what Torrens called ‘thenatural price of labour’ in the first edition of External Corn
Trade. In the Appendix, Torrens reuses some passages of thefirst edition with slight alternation and the phrase ‘the natural
price of labour’ is replaced with ‘the minimum of wages’.

and custom’, or ‘natural causes’ and ‘moral causes’(pp. 460–61):
A labourer in Hindostan may continue to work with perfectvigour, while receiving a supply of clothing which would beinsufficient to preserve a labourer in Russia from perishing.
Even in countries situated in the same climate, different habitsof living will often occasion variations in the minimum ofwages, as considerable as those which are produced by naturalcauses. The labourer in Ireland will rear a family undercircumstances which would not only deter an English workman
from marriage, but would force him on the parish for personal
support. (p. 460)6

Note that ‘moral causes’, unlike ‘natural causes’,can be artificially altered. Torrens argues that the
improvement of the habits of living in Ireland ‘would
raise the minimum of wages in that country to an
equality with their minimum in England’ (p. 460).
Torrens thus acknowledges the possible alterations in
the minimum wage, but he insists that the minimum‘may, in any given time and place, be regarded as very
nearly stationary’ because the alterations in it ‘cannot
be suddenly effected’:
So far as this minimum depends upon climate, it isunchangeable; and even so far as it is determined by the habitsof living, and [the established scale of comfort],7 it can beeffected only by those circumstances of prosperity or decay,and by those moral causes of instruction and civilization, which
are ever gradual in their operation. (p. 461)

The minimum wage, from the definition, does notnecessarily mean a physical subsistence level. Torrens,
however, mentions it as ‘the extreme minimum, below
which the merely animal wants of the labouring
population cannot be supplied’ (p. 470). The ‘extreme
minimum’ of wages would consist in a quantity of the
6

The similar sentence appeared in the first edition of ExternalCorn Trade (Torrens, 1815, p. 63), which Ricardo cited in the
Principles of Political Economy and Taxation (Ricardo, 1819, p.
97)
7There was originally the phrase ‘the prudential check which
may exist with respect to marriage’. However, the prudential
check should mean limiting the birthrate under a fixedminimum wage. Therefore, Torrens (1834, p. 12–13) later
corrected the text as shown with the square bracket. In a sameway, the phrase ‘a prudential check to marriage’ (1829, p. 460)
in a different part was later replaced with ‘a taste for the
comforts of life’ (1834, p. 12). Torrens’s notion of ‘scale ofcomfort’ is considered to stem from the Malthusian idea of
habit formation.

Torrens and the Classical Theory of Growth4

necessaries ‘sufficient for the support of animal life’ (p.
467) but in no quantity of the conveniences. Torrens’s
discussion of economic dynamics was developed in
the case of the ‘extreme minimum’ wage exogenously
fixed at the physical subsistence level.
3. Reconstructing Torrens’s theory of economic
growth3.1 Assumptions
The model is built on the following suppositions.
Supposition 1. The system is a closed economy.
Torrens himself describes the dynamics of a closed
economy.
Supposition 2. Corn is the only commodity producedin the system.
This is the assumption made by Torrens for the sake ofsimplification.
Supposition 3. Labour and land are the only inputs to
production, but ‘seed, material, and machinery’ are
not used in the production. Thus, capital consists of
the wages fund for employing labour.8
This is assumed in order to avoid complication of theanalysis.
Supposition 4. The maximum wage depends on theland productivity (diminishing returns inagricultural production) and the labour skill, theminimum wage is fixed at the physical subsistencelevel, and the actual wage is determined by theratio of capital to labour between both wages.
Based on Supposition 1, the effect of tradeliberalisation on the maximum wage must be
exempted from the consideration. As Section 2
mentioned, Torrens’s discussion was developed in thecase of the ‘extreme minimum’ wage exogenously
fixed at the physical subsistence level. The
determination of the actual wage rate follows thewages fund theory: ‘the one and only cause which can
8This differs from Rosell’s assumption of the inputs of cornand seed to production. He, however, removes the uses ofmaterial and machinery in the production from his suppositions(see Torrens, 1829, p. 464). Our simplification as well as
Rosell’s should be allowed in a single homogeneous corn
growth model.

determine where, between the maximum andminimum, the [actual] wages of … labourers, shall be
fixed, is, the proportion which the farmer’s capitalbears to the quantity of labour’ (p. 462).9
Supposition 5. There are diminishing returns to labourin agriculture.10
This assumption is necessary not only for his conceptof maximum wage (see above) but also for ‘theprinciples of rent’ (see below).
Supposition 6. The marginal product of labour, or thelabour product in the marginal land, is divided
wages as income of workers and profits as incomeof capitalists. Rent as income of landowners is thatsurplus of total production which remains once
workers and capitalists have obtained their income.
This assumption is based on the theory of differential
rent, which was developed in detail in Chapter 8, Part
1, of the 1829 edition of External Corn Trade.
Supposition 7. Landowners and workers consume all
their income, while capitalists save out of theirprofit income and invest it to employ labour only if
the rate of profit is higher than the minimum. The
growth rate of capital is an increasing function of
the difference between the profit rate and theminimum profit rate.
The assumption of capital accumulation can be
obviously driven from Torrens’s statement that ‘thelowest rate of profit … will induce him [capitalist] tocontinue the work of production’.11
Supposition 8. The term ‘population’ is used with thesame meaning as the number of workers, namely,‘labour population’. The growth rate of population
is an increasing function of the difference between
the actual and the minimum wage rate.
The assumption is based on the Malthusian law of
9Hollander (1968) and O’Brien (2004, pp. 131–35)
reconstructed the wages fund model incorporating non-wages
in Torrens’s Wages and Combination.
10
This differs from Rosell’s constant returns assumption.
11
In Chapter 2 of Wages and Combination, Torres, inillustrating the case in which the farmer obtains 24% of profits
above the minimum rate, 7%, states, ‘Increasing profits always
occasion a more rapid accumulation of capital’ (1834, pp. 33–
36).

Torrens and the Classical Theory of Growth5

population.
3.2 The model
Let us now reconstruct a simple model of Torrens’s
economic dynamics, following the above assumptions.

of 𝑁 units of labour, 𝑁 × 𝑑𝑌⁄𝑑𝑁, must be divided
as workers’ wages and capitalists’ profits:
𝑁(𝑑𝑌⁄𝑑𝑁) = 𝑤𝑁 + 𝑟𝐾. Considering 𝑑𝑌⁄𝑑𝑁 =𝑎𝐴𝑁 𝑎−1 and 𝐾 = 𝑤𝑁, we obtain𝑟=

Production function. Denoting the amount of land
under cultivation by 𝐿, and an index of diminishingreturns by 𝑎 with 0 < 𝑎 < 1, let the physical output
of corn as the products of land be simply 𝑥𝐿𝑎 , where𝑥 means as follows. Denoting the quantity of labour
employed per unit of land under cultivation by 𝑙 with𝑙 > 0 , let 𝑥 define as 𝑥 = 𝐴𝑙 𝑎 , where 𝐴 is aconstant coefficient with 𝐴 > 0. Assuming that 𝑙 is
exogenously given, from such a definition of 𝑥, we
can obtain the following production function of corn
as the products of labour combined with land:
𝑌 = 𝐴𝑁 𝑎 .

(1)where 𝑌 is the physical output of corn and 𝑁 is thequantity of labour bestowed to land.12
Let us now consider the case in which the
improvement of labour skill enables a smaller numberof workers to produce the same quantity of corn as
before while the amount and quality of land remains
the same: that is, the case in which 𝑙 is exogenously
reduced while 𝑥 and 𝑎 remain constant. Then 𝐴
rises. The raised 𝐴 thus indicates the improvement oflabour skill when the land productivity remains thesame.
Actual wage rage. From Supposition 3, capital(denoted 𝐾) consists of the wage fund for employing
𝑁 units of labour under a rate of actual wage (denoted𝑤): 𝐾 = 𝑤𝑁. The actual wage rate thus is determinedby the capital-population ratio, following the wages
fund theory (Supposition 4):
𝑤=

𝐾
.
𝑁

(2)

𝑎𝐴𝑁 𝑎−1
− 1.
𝑤

(3)

Rent. Since, following the theory of differential rent,
rent (denoted 𝑅) can be defined as that surplus of total
production which remains once workers and capitalists
have obtained their income, 𝑌 − (𝑤𝑁 + 𝑟𝐾) = 𝑌 −𝑁(𝑑𝑌⁄𝑑𝑁) . Considering 𝑑𝑌⁄𝑑𝑁 = 𝑎𝐴𝑁 𝑎−1 , weobtain𝑅 = (1 − 𝑎)𝐴𝑁 𝑎 .

(4)

Torrens argues that it is necessary and clear, ‘fromthe principles of rent’, that ‘the high rent’ must ‘bepaid for all the more fertile soils’ with resorting to theinferior lands (pp. 466–67), which is easily shown by𝑑𝑅⁄𝑑𝑁 = 𝑎(1 − 𝑎)𝐴𝑁 𝑎−1 > 0.
Capital accumulation. From Supposition 7, the growthrate of capital is an increasing function of thedifference between the profit rate and the minimum
profit rate (denoted 𝑟min ). If the minimum is given as
𝑟min > 0, denoting a reaction coefficient by 𝑏, with
0 < 𝑏 < 1, the capital accumulation function can begiven as the following specified equation:𝑘 = 𝑏(𝑟 − 𝑟min ),(5)

where 𝑘 is the growth rate of capital: 𝑘 = 𝐾̇⁄𝐾.13
Population growth. From Supposition 8, the growthrate of population is an increasing function of thedifference between the actual and the minimum wage
rate (denoted 𝑤min ). If the minimum is given as
𝑤min > 0, denoting a reaction coefficient by 𝑐, with
0 < 𝑐 < 1, the population growth can be given as thefollowing specified equation:

Profit rate. From Supposition 6, the marginal products
Substituting 𝑥 = 𝐴𝑙 𝑎 with 𝑌 = 𝑥𝐿𝑎 , it follows that𝑌 = 𝐴(𝑙𝐿)𝑎 . Further substituting 𝑁 = 𝑙𝐿 , we can obtainequation (1).
12

13

A dot over a variable denotes a derivative with respect totime.

Torrens and the Classical Theory of Growth6𝑛 = 𝑐(𝑤 − 𝑤min ),(6)

where 𝑛 is the growth rate of population: 𝑛 = 𝑁̇⁄𝑁.
In addition, the system is assumed to satisfy theconditions that allow the economy to grow but not togrow indefinitely (see Casarosa, 1982, p. 230):
lim 𝐴𝑁 𝑎 > 𝑤min (1 + 𝑟min ),𝑁→0

andlim 𝐴𝑁 𝑎 < 𝑤min (1 + 𝑟min ).

𝑁→∞

If the initial values of 𝐾 and 𝑁 are exogenously
given, the system, composed of equations (1)–(6), is
determined.
4. Torrens vision on economic dynamics
4.1 The maximum wage curve and the minimum wageline
In the (𝑁, 𝑤) plane, the minimum wage is simply
described as a horizontal line, as Figure 1 shows. We
may call it the minimum wage line, 𝑊min .
Figure 1.
The maximum wage curve and the minimum wage line

𝑤

[II]

[I]

𝐸

𝑤min
[IV]

𝑊min
[III]
𝑊max

𝑂

𝑁∗

𝑁

On the other hand, on some suppositions, we haveto redefine the maximum wage which we explained in
Subsection 2.2. From Supposition 3, non-wages funds,
such as seed, material and machinery, are not inputtedto production. From Supposition 6, we also must
consider the labour products in the no-rent land―
namely, the marginal products of labour, instead of theoutput per capita. Therefore, the maximum wage canbe redefined as

Maximum wage rate =Marginal products of labour
1 + Minimum profit rate

From equation (3), we obtain𝑤=

𝑎𝐴𝑁 𝑎−1.
1+𝑟(7)

Since a worker is paid the maximum wages when acapitalist only obtains the minimum rate of profits,
𝑤max = 𝑤|𝑟=𝑟min , we can obtain𝑤max =𝑎𝐴𝑁 𝑎−1.
1 + 𝑟min(8)

This is described as a convex curve in Figure 1 (seeAppendix A). We may call it the maximum wage
curve, 𝑊max . It is evident from equation (8) that theland productivity lowers the maximum wage in theprocess of economic growth and that the exogenous
improvement of labour skill raises the maximum.
In Figure 1, the maximum wage curve and theminimum wage line intersect at a point, 𝐸, and divide
the space into four regions [I]–[VI]. Torrens discusses
the motions of actual wage between the maximum andthe minimum wages, or in other words, themovements of the system in Region [I] which exist
between the maximum wage carve and the minimum
wage line (see Appendix B). The system starting froma point in Region [I], according to Torrens (1829, p.
457), cannot morally shoot up through the maximum
wage curve, while it is impossible that the systemcannot move below the minimum wage line (seeSubsection 4.2).
From equations (3), (5) and (7), we obtain thefollowing dynamic relationship:𝑤 ⋛ 𝑤max ⟹ 𝑘 ⋚ 0.
Further, from (6), we obtain𝑤 ⋛ 𝑤min ⟹ 𝑛 ⋛ 0.
Therefore, both capital and population increase if thesystem exists in Region [I]: that is, 𝑘 > 0 and 𝑛 > 0
if 𝑤 ∈ (𝑤min , 𝑤max ). Torrens states,
When, in the progress of wealth and population, wages andprofits have fallen to their minimum, and when the next qualityof land to be taken in cannot be made to yield a reproduction
sufficient to pay these minimum wages and to replace advanceswith minimum profits, then that which is saved from revenue tobe added to capital cannot be employed at home … (p. 469)

Torrens and the Classical Theory of Growth7‘At this point’, Torrens says, ‘population remain[s]stationary’ (p. 469); more strictly, both population andcapital remain stationary in the domestic market. 14
The maximum wage carve and the minimum wage
line intersect at point 𝐸:

we obtain the following relationship:𝑘 ⋛ 𝑛 ⟹ 𝜔 ⋛ 0,
where 𝜔 is the increase rate of 𝑤: 𝜔 = 𝑤̇⁄𝑤. Torrens
obviously discusses the possibility of economic
growth that capital and population go on increasing inthe same ratio:

1

1−𝑎
𝑎𝐴
(𝑁 , 𝑤 ) = ([, 𝑤min ).
]𝑤min (1 + 𝑟min )∗∗As they intersect when 𝑎𝐴𝑁 𝑎−1 ⁄(1 + 𝑟min ) = 𝑤min ,it follows that both the rates of wage and of profit are
reduced to their minimum at the stationary state point.
The existence of such a point also is ensured by thecontinuity of the two loci (𝑊max and 𝑊min ) and thefollowing assumption: lim𝑁→0 𝐴𝑁 𝑎 > 𝑤min (1 +𝑟min ) and lim𝑁→∞ 𝐴𝑁 𝑎 < 𝑤min (1 + 𝑟min ) (seeCasarosa, 1982, p. 231).
Torrens’s notions of the maximum and theminimum wages are close to what Hicks andHollander (1977), in their reconstruction of the
Ricardian growth model, call ‘the roof’ and ‘the flour’,respectively. It is evident that there is the idea of thelimit below which the wage cannot fall in both
Torrens’s and Ricardo’s texts. However, it should be
noted that Torrens obviously presents the concept ofthe maximum wage, while Ricardo, in his text, neverclearly proposes what Hicks and Hollander call ‘theroof’, which is a necessary tool for their rational
reconstruction. 15 In sum, Torrens gives a clearer
explanation of the upper limit of wage than Ricardo.
4.2 The dynamic equilibrium wage curve
Since both capital and population increase in Region[I], the motions of actual wage depend on the
difference between the growth rates of capital andpopulation. Taking the natural log of both sides ofequation (4), and differentiating with respect to time,

But though labour and capital should go on increasing in thesame proportion, and though they should constantly preserve
the same ratio to each other, yet the necessity of resorting toinferior soils might gradually reduce the maximum of wages
until it coincided with the extreme minimum, below which
labour cannot be sustained. At this point, the supply of labour
could be no further increased … (1829, pp. 468–69; emphasis
added).16

From equations (5) and (6), the balance between thegrowth rates of capital and of population is𝑏(𝑟 − 𝑟min ) = 𝑐(𝑤 − 𝑤min ).

(9)

Substituting in equation (3), we can obtain the
dynamic equilibrium wage rate (denoted 𝑤
̂ ) asfollows.
𝑤̂=+

𝑐𝑤min 𝑏(1 + 𝑟min )−2𝑐
2𝑐√[𝑐𝑤min − 𝑏(1 + 𝑟min )]2 + 4𝑎𝑏𝑐𝐴𝑁 𝑎−1
.
2𝑐

(10)

This is described as a convex curve between the
maximum wage curve and the minimum wage line, as
Figure 2 shows (see Appendix C). We may call it thê.
dynamic equilibrium wage curve, 𝑊We can now describe the phase diagram for the
dynamics of 𝑤 and 𝑁 in Figure 2. In this figure, themaximum wage curve ( 𝑊max ), the dynamiĉ ) and the minimum wage line
equilibrium curve (𝑊(𝑊min ) divide the space into six regions. The arrows
show the directions of motion in each region (seeAppendix D).

14

Torrens mentions the possibility that the surplus capital ‘will
be invested in foreign loans and foreign adventures’ (1829, p.
469). Further research requires the analyses of his theories ofinternational trade and systematic colonisation, which are out of
the scope of the present work.
15
Hicks and Hollander (1977, p. 353), in setting up the
Ricardian model, state, ‘We shall allow ourselves… to describe
the model in terms that, admittedly, are not Ricardo’s, but that
we think will quickly make it more intelligible to the modern
economist’.

16

O’Brien argues that Torrens here ‘rejects a simple
Wage-Fund analysis’ (1966, p. 337, original emphasis).
However, his argument, as Vint properly criticises, ‘isincomplete’ because ‘Torrens is here making use of a long run
analysis and not the short run wages fund doctrine’ (1994, p.
105).

Torrens and the Classical Theory of Growth8

Figure 2.
Phase diagram: the dynamics of wage and population

𝑤

[iii]

[ii]
[i]
𝑤min

𝐸
[iv]
[v]

[iv]

𝑊min
̂
𝑊
𝑊max

𝑂

𝑁∗

𝑁

In his view of economic dynamics, Torrens
discusses the system starting from a point in Region[i]. In the progress of capital and population, theeconomy necessarily enters Region [ii]. Once arriving
at the region, the economy grows until it reaches astationary state point, 𝐸, while both the wage and the
profit rate keep falling to their minimum level.
It is Casarosa (1982) who first strictly incorporated
the idea of ‘the dynamic equilibrium path of the wage
rate’ (p. 234) into the Ricardian growth model built byHicks and Hollander. However, Casarosa gave no
textual evidence of Ricardian dynamic equilibrium
growth, and we cannot find Ricardo’s own obvious
statement of such a growth in his text. On the other
hand, Torrens clearly mentioned the possibility of suchgrowth of an economy. His view on economic
dynamics therefore can be given a more appropriate
explanation by applying the Ricardian growth model
which Hicks, Hollander and Casarosa rationallyreconstructed.
4.3 The tendency in population to increase faster thancapitalTorrens’s early perspective had indicated that
population tends to increase faster than the means ofsubsistence so that the wage keeps decreasing. For
example, in the 1817 paper on the Poor Laws, Torrens

stated that in a country such as the America of his time
where ‘an abundance of new and fertile land’ remain,‘capital and subsistence may be made to increase as
fast as man can multiply his kind’, but in a country,
such as England here ‘all the good and well-situatedlands of a country have already been appropriated andoccupied… capital and subsistence can no longer be
kept up to the population’ (Torrens, 1817, pp. 517–18,
original emphasis). 17 If capital is continuously
accumulated as a ratio less than that of population
growth, the economy tends to be reduced to ‘thestarving point’ (p. 511) or ‘the extreme limit ofsubsistence’ (p. 512), in which all the people must be
confined ‘to the business of providing for merely
animal wants’ (p. 513).
Now note that capital increases faster than
population in Region [i], while population increases
faster than capital in Region [ii] in our model.
Torrens’s early view can be explained in that theeconomy has a tendency to move in Region [ii], or atleast, the system departing from a point in Region [i]has a tendency to immediately enter Region [ii].
In 1829, Torrens revised his previous view andinsisted ‘that, in almost every society, the tendency is
not to increase population faster than capital; but, on
the contrary, to increase capital more rapidly than
population’ (1829, pp. 474–75).
‘A few weeks’ (Blaug, 1958, p. 113) after the
publication of Torrens’s text, Senior’s letter to Thomas
Robert Malthus (1766–1834) dated 15 March was
brought out, criticising:
You would still say, that, in the absence of disturbing causes,
population has a tendency to increase faster than food… I
should still say, that, in the absence of disturbing causes, food17

Consistent with Malthus (1798), Torrens (1817, p. 523)
states that ‘population… must have a tendency to increase in ageometrical ratio’. However, he also argues, ‘If we were toextend tillage over inferior tracts, or to heighten the cultivationof our fertile lands, each additional quantity of capital applied
to the soil, would yield a less proportional return; and while thegenerative powers (unless reduced by the deterioration of thehuman constitution) remained unimpaired, the productive
powers which supply subsistence would be perpetually
decreasing’ (pp. 523–24). According to Blaug (1958, p. 104),‘Malthus never emphasized the law of diminishing returns in
the Essay on Population’, but the ‘Ricardian economists…
were quick to relate the arithmetic ratio explicitly to theassumption of diminishing returns in agriculture’. In such asense, Torrens was one of the Ricardian economists.

Torrens and the Classical Theory of Growth9
has a tendency to increase faster than population… (Senior,
1829, p. 58)

Torrens later claimed his anticipation of Senior’s
criticism of Malthus:
It was maintained that there is a tendency in population toincrease faster than the capital which gives employment tolabour; and that this tendency keeps down wages to the starving
point. I believe I may venture to claim the credit of being thefirst to refute this doctrine… In the appendix to the last editionof my Essay on the Corn Trade, I proved that there was notendency in population to increase faster than capital; but that,on the contrary, the tendency is for capital to increase fasterthan population. This correct and consolatory view of thequestion was subsequently taken, I believe, in a correspondence
which was published, between Mr. Malthus and Mr. Senior…
The doctrine, therefore, that population has a tendency toincrease faster than capital, may be considered as exploded.
(Torrens, 1833, pp. 43–44)

However, Torrens’s criticism of Malthus was not
appropriate. Malthus (1798, pp. 9, 11) only stated that
population, ‘when unchecked’, has a tendency toincrease faster than the means of subsistence, while
Torrens (1829, pp. 476–77) considers that there is not
such a tendency because, ‘in every civilized
community’ or ‘in every country of Europe except
Ireland’, the people’s ‘power “to increase andmultiply”… is checked and controuled [sic] by theprevailing efficacy’ of the ‘instruction’ of ‘prudence
and precaution’.
However, even though Torrens’s criticism wasnever proper, his new position has an importance inthe analysis of his dynamic motions of the system. The‘only possible means of eradicating pauperism’,Torrens emphasised in 1817, ‘is by keeping down the
population to a level with capital and subsistence’.
One of ways of doing so, which he had devised, was‘a prudential or moral restraint for preventing the birthof superfluous numbers’ (1817, p. 518).18 At the time,
he believed that a ‘moral instruction’ (p. 519) for
limiting birth had not prevailed among the working
classes.
Torrens, in 1829, however, acknowledged that theworking classes sufficiently acquire prudential habits
18

The other way Torrens (1817, p. 518) thought of was aself-supporting emigration for ‘removing’ the superfluous
numbers who had already come into the world.

so that they can keep the growth rate of population not
to exceed that of capital. That is, the system, which
departed from a point in Region [i], tends to remain in
the same region by the diffusion of prudential habits
among workers. In this region, capital increases fasterthan population so that wages enjoyed by workers
continue to increase. Torrens states, ‘The fact that thecondition of the labouring classed has improved withthe progress of wealth and civilization, demonstrates
that population has not a tendency to increase fasterthan capital’ (1829, p. 473). Torrens (1829, pp. 473–74) here cites Adam Smith’s mention that a nation’s
conditions improved with an increase in skill as aresult of the division of labour (Smith, 1776, pp. 23–
24). An improvement of labour skill and thus anexogenous raise in 𝐴 enlarges the area of Region [i],in which capital increases faster than population, as
Figure 3 shows (see Appendix 5). Torrens might have
had a theoretical perspective of the positive effects ofimproved skill on the working classes.
Figure 3.
The effect of the improvement of labour skill

𝑤

[ii]

[i]
𝑤min

𝐸

𝐸′

𝑊min
̂′
𝑊
′
𝑊max

𝑂

𝑁∗

𝑁∗∗

𝑁

5. Concluding remarks
In his 1829 text, Torrens put forward not only thestatics theories of wage and profit reformulated byRosell, but also the dynamics theory of economic
growth based on the notions of the maximum,
minimum and actual wage. His view on economic
dynamics can be explicitly explained by the growthmodel, which incorporates a dynamic equilibrium

Torrens and the Classical Theory of Growth10

between the growth rates of population and of capital.
Such a theory has been attributed to Ricardo by Hicks,
Hollander and Casarosa, but Torrens presented a moreingenious theory than Ricardo’s.
The actual wage rate is determined by thecapital-population ratio between the maximum and theminimum wage rate. In the accumulation of capitaland the increase in population, the maximum wage
rate keeps being lowered by the decreasing marginal
productivity of labour in agriculture, while theminimum wage rate is exogenously fixed and thus isconstant through time. Based on these notions ofwages, Torrens’s economic dynamics is generally
described as follows.
In the early stage of economic growth, capitalincreases faster than population so that the actual wage
rate keeps rising above the minimum. Thereafter, once
arriving at the dynamic equilibrium wage path, inwhich population and capital grow at the same ratio,
the economy sets about growing with a tendency for
population to increase faster than capital so that theactual wage rate keeps falling below the decreasing
maximum and profit rate simultaneously goes onfalling, until the system converges at the stationary
state where the both the actual wage and the profit rateequals their minimum.
In Torrens’s 1829 view, workers in every civilised
country, sufficiently acquire prudential habits so thatthey can keep the growth rate of population not toexceed that of capital. Such habits, with thecooperation of a temporary effect of the improved
labour skill, enable the actual system to grow while
maintaining the condition in which capital increases
faster than population and workers enjoy higher
wages.
In concluding this work, we should briefly treat
Torrens’s late positon of the relationship between
population and capital. In 1857, he re-published his
early pamphlet entitled The Economists Refuted(Torrens 1857). The re-published pamphlet has not
only some trivial corrections and the deletion ofChapter 6 on the ‘observations on the expediency ofmaking peace with France’ which appeared in the1808 edition, but also great revisions to Chapter 4. In
particular, it should be noted that the following
sentence was added:[F]rom the relative proportions according to which populationand capital have, in all old countries, been hitherto found toincrease, the supply of labour has such a tendency to exceed thedemand for it, that the labouring classes, even when there is no
extraordinary stagnation or revulsion in the channels of industry,
are commonly reduced to a degree of distress… (Torrens, 1857,
p. 35)

Considering that an improvement of labour skill has
no permanent effect in the enlargement of Region [i],in 1857, Torrens might have been about to consider
that the system had already gone out the region andcontinued progressing in Region [ii], in which
population increases faster than capital. At any rate,
this should be observed in future research.
References
Blaug, M. 1958. Ricardian Economists: A Historical
Study, New Haven, Yale University Press
Casarosa, C. 1982. The New View of the Ricardian
Theory of Distribution and Economic Growth, pp.
227-39 in Baranzini, M. (ed.), Advances in
Economic Theory, Oxford, Blackwell
De Vivo, G. 2000. Bibliographical List of Torrens’s
Works, Collected Works of Robert Torrens, vol. 8,
pp. ix-xlviii, Bristol, Thoemmes
Dome, T. 1992. Kotenkeizaigaku no Mokeibunseki[Classical Economics: Modeling Anaysis], Tokyo,
Yuhikaku (In Japanese)
Hicks, J. R. and Hollander, S. 1977. Mr. Ricardo and theModerns, Quarterly Journal of Economics, vol. 91,
no. 3, pp. 351-69
Hisamatsu, T. 2015. Robert Torrens and Malthusian
Population Theory, The Annual Bulletin of theMalthus Society, no. 24, pp. 67-106 (In Japanese)
Hollander, S. 1968. The Role of Fixed Technical
Coefficients in the Evolution of the Wage-FundControversy, Oxford Economic Papers, new series,
vol. 20, no. 3, pp. 320-41
Malthus, T. R. 1798. An Essay on the Principle ofPopulation, in Wrigley, E. A. and Souden, D. (eds),
The Works of Thomas Robert Malthus, vol. 1,
London, Pickering, 1986
O’Brien, D. P. 1966. Torrens on Wages and Emigration,
Economica, new series, vol. 33, no. 131, pp. 336-40
O’Brien, D. P. 2004. The Classical Economists Revisited,
Princeton, Princeton University Press.
Ricardo, D. 1819. On the Principles of Political Economy,
and Taxation, second edition, in Sraffa, P. and Dobb.
M. H. (eds), The Works and Correspondence ofDavid Ricardo, vol. 1, Cambridge: Cambridge

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University Press, 1951
Robbins, L. C. 1958. Robert Torrens and the Evolution ofClassical Economics, London, Macmillan
Rosell, O. ‘Wages, competition and the surplus of labour:a classical contribution to explaining profit’,Cambridge Journal of Economics Advance Accesspublished April 5, 2016, doi:10.1093/cje/bei005
Senior, N. W. 1829. Two Lectures on Population,
Delivered before the University of Oxford in Easter
Term, 1828. To which is Added, a Correspondence
between the Author and the Rev. T. R. Malthus,
London, Saunders and Otley.
Smith, A. 1776. An Inquiry into the Nature and Causes ofthe Wealth of Nations, in Campbell, R. H. andSkinner, A. S. (eds), vols. 1 and 2, Oxford,
Clarendon Press, 1976
Torrens, R. 1808. The Economists Refuted; or, an Inquiry
into the Nature and Extent of the Advantages
Derived from Trade: with Observations on the
Expediency of Making Peace with France, in
Groenewegen, P. D. (ed.), The Economists Refutedand Other Early Economic Writings, New York,
Kelley, 1993
Torrens, R. 1815. An Essay on the External Corn Trade,
London, Hatchard.
Torrens, R. 1817. A Paper on the Means of Reducing thePoors [sic] Rates and of Affording Effectual andPermanent Relief to the Labouring Classes, The
Pamphleteer, vol. 10, no. 20, pp. 509-28
Torrens, R. 1829. An Essay on the External Corn Trade,with an Appendix on the Means of Improving theCondition of the Labouring Classes, A New Edition,
London, Longman, Rees, Orme, Brown and Green
Torrens, R. 1833. Letters on Commercial Policy, London,
Longman and Co.
Torrens, R. 1834. On Wage and Combination, London,
Longman, Rees, Orme, Brown, Green and Longman
Torrens, R. 1837. A Letter to the Right Honourable Lord
Russell, on the Ministerial Measure for Establishing
Poor Laws in Ireland, and on the Auxiliary Means
which it will be Necessary to Employ in Carryingthat Measure into Effect, London, Longman, Rees,
Orme, Brown and Green
Torrens, R. 1844. A Letter to Lord Ashley, on the
Principles which Regulated Wages and on the
Manner and Degree in which Wages would be
Reduced, by the Passing of a Ten Hours Bill,
London, Smith, Elder and Co
Torrens, R. 1857. The Economists Refuted; or, an Inquiry
into the Nature and Extent of the Advantages
Derived from Trade, in De Vivo, G. (ed.), Collected
Works of Robert Torrens, vol. 6, Bristol, Thoemmes
Vint, J. 1994. Capital and Wages: A Lakatosian Historyof the Wages Fund Doctrine, Aldershot, Elgar

Appendix A
In the (𝑁, 𝑤) plane, the maximum wage is described as
the convex curve by satisfying𝑑𝑤max𝑎(1 − 𝑎)𝐴𝑁 𝑎−2
=−
< 0,
𝑑𝑁
1 + 𝑟minand𝑑2 𝑤max 𝑎(1 − 𝑎)(2 − 𝑎)𝐴𝑁 𝑎−3
=
> 0.
𝑑𝑁 2
1 + 𝑟minAppendix B
As a necessary and sufficient condition for the system
not to shoot up through the maximum wage curve, theelasticity of the labour population with respect to the
maximum wage rate is equal to or strictly more thanminus one (see Dome, 1993, pp. 32–3):
𝑑𝑤max 𝑁≥ −1.
𝑑𝑁 𝑤max

The elasticity of the labour population with respect to the
maximum wage rate is
𝑑𝑤max 𝑁𝑎(1 − 𝑎)𝐴𝑁 𝑎−2
𝑁=−𝑎−1⁄(1 + 𝑟min )𝑑𝑁 𝑤max
1 + 𝑟min𝑎𝐴𝑁= −(1 − 𝑎).

From −(1 − 𝑎) ≥ −1, the condition must be 𝑎 ≥ 0.
The diminishing returns assumption, 0 < 𝑎 < 1 ,satisfies it and allows the system, as Torrens describes, tomove between the maximum and the minimum wages
and to converge at the stationary state point.
Appendix C
In the (𝑁, 𝑤) plane, the dynamic equilibrium wage is
described as the convex curve by satisfying𝑑𝑤̂
𝑎(1 − 𝑎)𝑏𝐴𝑁 𝑎−2
=−
< 0,
𝑑𝑁
√[𝑐𝑤min − 𝑏(1 + 𝑟min )]2 + 4𝑎𝑏𝑐𝐴𝑁 𝑎−1

and
𝑑2 𝑤
̂
= [(2 − 𝑎){𝑐𝑤min − 𝑏(1 + 𝑟min )}2 𝑁 𝑎−3
𝑑𝑁 2+ 2𝑎(3 − 𝑎)𝑏𝑐𝐴𝑁 2(𝑎−2) ]×𝑎(1 − 𝑎)𝑏𝐴
3[{𝑐𝑤min − 𝑏(1 + 𝑟min )}2 + 4𝑎𝑏𝑐𝐴𝑁 𝑎−1 ]2> 0.

On the other hand, since the maximum wage curve andthe minimum wage curve intersect when𝑤min =𝑎𝐴𝑁 𝑎−1,
1 + 𝑟minTorrens and the Classical Theory of Growth12

it follows that𝑎𝐴𝑁 𝑎−1− (1 + 𝑟min ) = 0.
𝑤minMultiplying both sides of this equation by 4𝑏𝑐𝑤min(> 0), we obtain4𝑏𝑐𝑤min [𝑎𝐴𝑁 𝑎−1− (1 + 𝑟min )] = 0.
𝑤minAdding (𝑐𝑤min )2 + 2𝑏𝑐𝑤min (1 + 𝑟min ) ( > 0 ) to theleft side, we obtain the following inequality.
(𝑐𝑤min )2 + 2𝑏𝑐𝑤min (1 + 𝑟min )+ 4𝑏𝑐𝑤min [𝑎𝐴𝑁 𝑎−1− (1 + 𝑟min )] > 0,𝑤min

(𝑐𝑤min )2 − 2𝑏𝑐𝑤min (1 + 𝑟min ) + 4𝑎𝑏𝑐𝐴𝑁 𝑎−1 > 0.
Further, adding [𝑏(1 + 𝑟min )]2 (> 0) to both sides, we
obtain the following inequality(𝑐𝑤min )2 − 2𝑏𝑐𝑤min (1 + 𝑟min ) + [𝑏(1 + 𝑟min )]2+ 4𝑎𝑏𝑐𝐴𝑁 𝑎−1 > [𝑏(1 + 𝑟min )]2 ,[𝑐𝑤min − 𝑏(1 + 𝑟min )]2 + 4𝑎𝑏𝑐𝐴𝑁 𝑎−1 > [𝑏(1 + 𝑟min )]2 ,1𝑏>.
1 + 𝑟min √[𝑐𝑤min − 𝑏(1 + 𝑟min )]2 + 4𝑎𝑏𝑐𝐴𝑁 𝑎−1

Multiplying both sides by 𝑎(1 − 𝑎)𝐴𝑁 𝑎−2 (> 0), we
obtain𝑎(1 − 𝑎)𝐴𝑁 𝑎−2𝑎(1 − 𝑎)𝑏𝐴𝑁 𝑎−2>,
1 + 𝑟min√[𝑐𝑤min − 𝑏(1 + 𝑟min )]2 + 4𝑎𝑏𝑐𝐴𝑁 𝑎−1
𝑑𝑤max 𝑑𝑤
̂<(< 0).
𝑑𝑁
𝑑𝑁

Therefore, considering that the dynamic equilibrium
curve intersects the maximum wage curve at the point(𝑁 ∗ , 𝑤 ∗ ) and that 𝑑 2 𝑤max ⁄𝑑𝑁 2 > 0 and 𝑑 2 𝑤
̂ ⁄𝑑𝑁 2 >0 , we can find that in the range 𝑁 ∈ (0, 𝑁 ∗ ) , thedynamic equilibrium curve exists below the maximum
wage curve, while in the range 𝑁 ∈ (𝑁 ∗ , +∞) , theformer is above the latter.
Appendix D
Let us consider the case of the actual wage rate above(below) the dynamic equilibrium wage:𝑤≷

𝑐𝑤min 𝑏(1 + 𝑟min )
−
2𝑐
2𝑐
+

√[𝑐𝑤min − 𝑏(1 + 𝑟min )]2 + 4𝑎𝑏𝑐𝐴𝑁 𝑎−1.
2𝑐

From this, we obtain𝑎𝐴𝑁 𝑎−1𝑐(𝑤 − 𝑤min ) ≷ 𝑏 (− 1 − 𝑟min ),𝑤𝑐(𝑤 − 𝑤min ) ≷ 𝑏(𝑟 − 𝑟min ),𝑛 ≷ 𝑘.

That is, 𝑛 ≷ 𝑘 if 𝑤 ≷ 𝑤
̃ . This means that population(capital) increases faster than capital (population) in thesystem which exists above (below) the dynamic
equilibrium curve. Considering 𝜔 ≶ 0 if 𝑛 ≷ 𝑘, we canfind that the actual wage rate rises when the system starts
from the lower regions, [i], [v] and [vi], than the dynamic
equilibrium curve while it falls when the system from theupper region, [ii], [iii] and [iv]. It is evident that theactual wage rate remains stationary when the system is in
the dynamic equilibrium curve.
On the other hand, it is obvious from equation (6) that
population decreases when the system starts from theupper regions, [i], [ii] and [iii], above the minimum wageline―that is, if the actual wage rate is above theminimum―while it decreases when the system is fromthe lower regions, [iv], [v] and [vi]―that is, if the actual
wage rate below the minimum.
Therefore, the motion in each region is shown as thearrows in Figure 2.
Appendix E
The following relations ensure that an exogenous rise inlabour productivity, as described in Figure 4, enlarges
region II, where capital increases faster than population.
𝜕𝑤max
𝑎𝑁 𝑎−1=> 0,
𝜕𝐴
1 + 𝑟min𝜕𝑤̂
𝑎𝑏𝑁 𝑎−1=> 0,
𝜕𝐴 √[𝑐𝑤min − 𝑏(1 + 𝑟min )]2 + 4𝑎𝑏𝑐𝐴𝑁 𝑎−1
𝜕𝑤 ∗
= 0,
𝜕𝐴
𝑎

1−𝑎
𝜕𝑁 ∗
𝑎
𝑎𝐴
=
[
] > 0.
𝜕𝐴𝑤min (1 − 𝑎)(1 + 𝑟min ) 𝑤min (1 + 𝑟min )