James Mill’s Annuity Approach to Capital Theory – Why was it Forgotten?

A hypothesis. Either the Labour theory of value is true or much less wrong than we are led to believe but we have forgotten the key theory which might demonstrate the theory’s veracity under a capitalist economy.

The period between 1820 and 1830 is widely accepted as the apogee of the Ricardian School of English Political Economy. Ricardo had erected a grand theoretical structure which fundamentally modified the classical economics of Adam Smith. In place of Smith’s ‘adding up’ approach to price where costs of production were added to profits to determine price, wages and profits were inversely related to each other. This necessarily led to the rejection of Smith’s ‘labour commanded’ theory of value – as the factors which determined price were unchanged by changes in wages – instead Ricardo argued for a labour embodied approach to value theory.

The theories outlines in ‘Principles of Political Economy’ were incomplete. The theory worked well for labour and variable capital consumed in production (as my recent series demonstrated) but when it came to unequal fixed capitals, unequal in terms of turnover period and/or ratio between fixed and variable capital, the theory had limitations. Some challenged the labour theory from this perspective, notably Torrens and Malthus, other sought to extend Ricardo’s theory to give a fuller description of fixed capital.

James Mill, De Quicncy and McCulloch were the great popularisers of Ricardo’s ideas, especially after his death. James Mill also had a close working relationship with Ricardo during his lifetime, encouraging him to publish, and especially to simplify and clarify his arguments. James Mill was also by far the most important of these three thinkers as he did much more than the simply popularise and systematise Ricardo’s ideas; he filled in many of the missing theoretical gaps. To James Mill we can credit the not altogether benign influence of the ‘wages fund’ theory and the ‘Theory of Markets’ (also known as Says Law though properly originating in Mill). Though both ideas have an element of truth in them it took over 100 years to disentangle both arguments satisfactorily. Ironically Mill’s main work includes a clear statement of how recessions can arise through excessive savings/demand for money – though it lacked the conceptual link between demand and investment that Keynes added.

With regards to arguably James Mills most important contribution in his own ‘Elements of Political Economy’ (second edition 1825) – his ‘perpetuity theory’ of fixed capital, it has taken nearly 200 and to my mind still has not been fully disentangled. His approach was to try to define precisely the contribution to value of fixed capital. Though framed within the context of the labour theory of value, and how to value unequal capitals on an equal basis the valuation issues this raises apply equally to any objective theory of value including cost of production theories.

His definition of capital was as follows:

Whatever is consumed productively becomes capital (OLL P115)


For Ricardo fixed capital remained a puzzle and he struggled with it until his death. Though he knew of James Mills’s theories he considered


‘What I call exceptions and modifications of the general rule you appear to me to say come under the general rule itself. Works, vol. IX, p. 127


Ricardo neither embraced or explicitly rejected Mill’s Perpetuity approach to capital, indeed I can find no direct mention of it. Ricardo only once mentions the application of the theory to the wine in casks example in his final paper Absolute Value and Exchangable Value. Here he expresses the view that if one takes another commodity as the measure of value its value will be constantly changing against the value of the wine even though no new labour has been expended. From a modern perspective we can say that only a standard commodity can be an invariable measure and with the depreciation of fixed capital a new standard system is being created at every moment in time. I speculate that Ricardo did not tackle directly the perpetuity issue as he had direct experience in this aspect of finance and did not understand its underlying mathematics. Ricardo did fully appreciate the complications arising from time to value theory but did not live long enough to hear the ideas of Senior on Cairns on how time, or rather waiting, can be a cost of production in itself.


Marx also was well aware of Mill’s theories over fixed capital and though he endorsed one element of them (the joint production method), and criticised Mills very loose definitions of profit and wages he again never directly addressed Mill’s most interesting contributions. His Paris notebooks and Theories of Surplus Value simply skip over them. Marx often spent little time on ideas he agreed with so a possibility is that he treated much of Mill’s approach of fixed capital as given. As far as I can see the only important economists to properly grapple with Mills ideas were Bohm-Bawerk and Sraffa. Much of Sraffa’s project involved mathematising many of Mill’s ideas. Arguably though Sraffa lost his way over some critical areas of Mill’s ideas, replacing Mills dynamic and monetary view with one based on timeless and moneyless infinitely small slices in time, and hence his theories were incomplete.


Why was Mill’s quite sophisticated capital theory misunderstood and forgotten? Firstly the English empirical approach to political economy distrusted mathematicisation and applied the Ricardian vice of logically deriving ideas in causal sequence rather than try to develop a schema where certain coefficients were mutually co-determining. Both Torren’s and Mill’s approaches to Capital theory implied use of simultaneous equations. Lack of knowledge of this technique hobbled the development of classical economics and led to its advancement shifting to the mathematically trained in Germany and Russia. Had Marx fully understood Mills he would not have needed to go down the bland ally of transforming values to prices and the problems this created.

The other reason it was forgotten is that the theoretical and mathematical tools necessary to formalise it were not available for another 80 + years. Only towards the end of the 19th Century did we get satisfactory treatments of depreciation, net present value and the flow of services from investments in the accounting and finance literature, ten introduced to Economics by Fisher. By the time they had arrived many of the theoretical problems relating to capital in value theory had been forgotten.

A reason perhaps is Mill’s sloppy use of terminology and definitions, quite unlike Ricardo. Mill’s confused wages with profits, so his writings require some formalisation to understand. Another is because as McCulloch said of the elements

‘…it is of too abstract a character to be either popular or of much utility’

Indeed the Elements is a precise but rather difficult book to those without a good grounding in the subject it purports to introduce.

What was Mill’s annuity approach to fixed capital? It was one plank in a capital theory which has eight elements all of which are necessary to give a full picture. Underpinning his approach was his simple doctrine that

either … cost of production consists in labour an capital combined; or that one of these may be resolved into the other. If one of them can be resolved into the other, it follows that cost of production does not consist in both combined (page 52)

To say, therefore, that the value of a product is determined by the value of the capital, is of no use, when you have to go beyond the value of the capital, and ask, what it is by which that value is itself determined.(page 54)

Mill’s project involved exploring the capital element, as follows:


  1. Reviving the Concept that it is the Combination of Labour and Nature that Create Value – This idea was central to the origins of classical economics in Petty and Cantillon, but in Smith and Ricardo had been deemphasised and confined solely to consideration of rent. In Mill it is brought again to the fore. Mill however noted that free goods from nature don’t create value, and rent from differential natural products transfer’s value, only work can create value. For Mill labour transformed nature and wealth and civilisation was the result of that transformation.
  2.  The use of Joint Production to Describe Fixed Capital – this approachwas borrowed from Malthus and Torrens (its originator). Marx approvingly quoted Mills application of this approach but sadly there is no evidence of him actually adopting it. There is a mode of viewing the gross return to the capitalist, which has a tendency to simplify our language, and, so far, has a great advantage to recommend it. The case of fixed and of circulating capitalmay be treated as the same, by merely considering the fixed capital as a product, which is regularly consumed and replaced, by every course of productive operations. The capital, not consumed,may be always taken, as anadditional commodity, the result of the productive process.According to this supposition, the share of the capitalist is always equal to the whole of his capital, together with its profits. (page 47 OLLL Edition)

    The early discussions on Ricardo’s theory’s quickly threw up the problem of how to calculate profits when fixed capital depreciates. You cannot then know profits unless you calculate the funding necessary to secure retention of capital (depreciation). The joint production solution was a brilliant solution. It enabled fixed capital to be treated as a stream in time of value equivalent, if the valuation is correct, to variable capital, potentially aiding the solution to other problems in value theory. The problem was not realised, with economic writings as opposed to accountancy theory, until Sraffa’s and Hotelling’s writings. Many of the simple relationships in single production did not hold in joint production, including raising the potential of negative labour values. Perhaps too much has been made of this as Sraffa himself indicated to Schefold late in life. With fixed capital depreciating at steady interest rates negative labour values simply indicates that it has reached its economic life and is a no longer viable technique, a raise in interest rates can accelerate depreciation bringing this forward. There are a variety of vector based optimising methods available which are capable of showing only positive labour values in viable and economically reproducing techniques treating the general joint production case – which is beyond the scope of this article to explore; however given that a viable fixed capital technique will always have positive labour values up to the point of its economic depreciation, and if it becomes viables after that point (for example due to a fall in interest rates) then it can simply be treated as a free good attracting rent. Therefore it is always possible to compare systems of production of different outputs and calculate the positive ‘reduced’ dated labour values.

    Again Mill was unfortunate in that this correct method of depreciating capital by treating each ‘age’ of a machine as a separate commodity was not mathematically formalised until the end of the 19Th C by Labelle, and then later Hotelling extended the approach

  3. The use of an annuity approach to value fixed capital –

    If two commodities are produced, a bale of silk, for example, for immediate consumption, and a machine, which is an article of fixed capital; it is certain, that if the bale of silk and the machine were produced by the same quantity of labour, and in the same time, they would exactly exchange for one another: quantity of labour would clearly be the regulator of their value. But suppose that the owner of the machine, instead of selling it, is disposed to use it, for the sake of the profits which it brings; what is the real character and nature of his action? Instead of receiving the price of his machine all at once, he takes a deferred payment, so much per annum: he receives, in fact, an annuity, in lieu of the capital sum; an annuity, fixed by the competition of the market, and which is therefore an exact equivalent for the capital sum. Whatever the proportion which the capital sum bears to the annuity, whether it be ten years’ purchase, or twenty years’ purchase, such a proportion is each year’s annuity of the original value of the machine.

    Make now a different supposition: that the machine is an article of
    fixed capital, and not worn out, and let us trace the consequences. It was correctly supposed, in the former case, that 100 days’ labour were expended by wearing out the machine; but 100 days’ labour have not been expended in the second, because the machine is not worn out. Some labour, however, has been expended, because 100 days’ labour in a mass has been applied. How much of it shall we say has been expended? We have an exact measure of it in the equivalent which is paid. If the equivalent which was obtained when the machine was worn out, was a measure of 100 days’ labour, whatever proportion of such equivalent is received as a year’s use of the machine when not worn out, must represent a corresponding proportion of the labour expended upon the machine.

    ‘Capital is allowed to be correctly described under the title of hoarded labour. A portion of capital produced by 100 days’ labour, is 100 days’ hoarded labour. But the whole of the 100 days’ hoarded labour is not expended, when the article constituting the capital is not worn out. A part is expended, and what part? Of this we have no direct, we have only an indirect measure. If capital, paid for by an annuity, is paid for at the rate of 10 per cent, one-tenth of the hoarded labour may be correctly considered as expended in one year. (p56-57)’

    A logical step Mill’s failed to make from this was to treated land as an example of fully durable joint production whose value can be calculated through a perpetuity, which could have led to a comprehensive approach to dealing with land and capital together rather than as separate systems.

    Annuities were known as early as Roman times when they were used to pay retired legionnaires salaries. There cost was calculated using early accurist tables of life expectancy. They were widely used in early 19th C and sometimes even sold to raise government debt.

    A perpetuity is  ‘ a terminating “stream” of fixed payments, i.e., a collection of payments to be periodically received over a specified period of time’ A perpetuity is an annuity with no end date, it has a non infinite present value providing there is a positive rate of interest. What Mill was arguing is that the value of a the output of an item of fixed capital at a fixed point in future time could be precisely and indirectly calculated by calculating the value of an annuity producing precisely that value during the same time period.

    The theory, if correct, would redefine price as being equal to labour to one of price being equal to discounted marginal labour. Let’s take each element in turn.

    If one considers a case of constant returns to scale then the marginal contribution of labour does not factor we need only be concerned with average contributions. If we have declining or increasing returns then production will continue so long as the rate of profit remains above the average rate of profit, the additional contribution to value from the additional does of labour being the marginal contribution of labour. At the optimal quantity demanded in terms of maximisation of profits this will also equal the marginal quantity demanded. The marginal quantity demanded being the labour commanded. Indeed Jevons was explicit that providing labour was treated in such marginal terms he had no objection.

    The second issue relates to discounting. If a piece of fixed capital releases accumulated labour over 2 years then one needs to discount the labour released over that period. The cost of doing so is an opportunity cost – as set out by Senior. What rate to apply? If one compares two capital assets you need only compare the two own rates. In a monetary economy however where returns are capitalised and presented as prices of traded stocks and shares we have a monetary single rate of interest for any term.

    This produces an interesting question – did Marx realise that this method requires treatment of marginal discounted labour values rather than average non-discounted values?

  4. Treatment of goods which gain in value over time as fixed capital – The most famous example being wine is a cask, a argument misunderstood and later poorly presented by McCulloch. 

    ‘It is said that the exchangeable value of commodities is affected by time, without the intervention of labour; because, when profits of stock must be included, so much must be added for every portion of time which the production of one commodity requires beyond that of another. For example, if the same quantity of labour has produced in the same season a cask of wine, and 20 sacks of flour, they will exchange against one another at the end of the season: but if the owner of the wine places the wine in his cellar, and keeps it for a couple of years, it will be worth more than the 20 sacks of flour, because the profits of stock for the two years must be added to the original price. Here is an addition of value, but here it is affirmed, there has been no new application of labour; quantity of labour, therefore, is not the principle by which exchangeable value is regulated. (p 57)

    …The case of the wine in the cellar coincides exactly with that of a machine worn out in a year, which works by itself without additional labour. The new wine, which is one machine, is replaced by its produce, the old wine, with that addition of value which corresponds with the return to capital employed upon the land; and the account which is to be rendered of the one return, is also the true account of the other. (p59)’

Ricardo in the aforementioned quote clearly did not fully understand Mills original argument, as it was not a case of new value being created without new labour but rather that new labour being released slowly over time.

  1. Treatment of money capital as ‘accumulated labour’ – These elements comprised a fairly comprehensive approach to value theory. It supplies a framework which is easily capable of explaining rent as well as value of produced commodities, and via rent to explain the influence on value of monopoly. His perspective on money – seeing it as accumulated value – offers a potential approach (though by him unexplored – this was taken up by Senior and Cairne’s) towards treatment of interest as the cost of waiting and the opportunities forgone when purchasing goods with an inelastic supply – the point of criticism by the early marginalist writers over the generality of objective theories of value.

    The power of Mill’s arguments convinced many of the veracity of the objective theory. Arguably he was too convincing as his theories were not developed and formalised. It lead to the easy consensus that problems concerning capital of different compositions posed no threat to value theory and his son’s premature statement that all issues of value theory had been settled.

    we have no practical means of ascertaining before hand the exact quantity of hoarded labour which goes to production, since the only measure we have of its quantity is the price which it brings.(p65)

    The terms though ‘accumulated labour’ (I have not quite got to the bottom of the issue over whether it was Torrens or Mill who first used this term) or ‘hoarded labour’ are imprecise. It is not that thye fixed capital stores labour like some kind of battery, rather by running they are able to leverage value, alongside living labour (to use Marx’s term) which is equivalent to the discounted amount of labour ’embodied’ that would be used without that leverage at the point in time that the fixed capital was created. At that precise singular instant in time there is no difference between the embodies and commanded measures of value as no monetary, price or value shifts can have taken place to take apart these measures.

    Money can also be seen as the crystallisation of this ability to leverage labour, providing it is circulated. If this is not circulated, or if excessively spent on unproductive purposes, the process of reproduction can be harmed. As such the LTV can be extended to also to cover prices of assets and non-produced goods using ‘proxy’ labour values.

Some Implications – I am not nailing my colours to the mast here and saying that Mills approach solves the outstanding issues of value theory or provides a proof of a (modified) labour theory of value. I am currently building a MATHCAD model to test this. However with Mill’s approach many of the traditional lines of attack on the LTV simply fall away they are not applicable. Also there is no transformation problem if one instead uses temporal discounting to bring everything back to NPV of embodies marginal labour. This also implies that transformation back of output prices to input prices, as used in the TSS or ‘New Solution’ approaches is unnecessary as one consistently uses input prices with discounting used for all future anticipated prices. The key difference being in Mill’s approach the reasons and causations for difference between output and input prices are explained. In the ‘without error’ theories simply one set out values is rubbed out and the corrected values written in Also there is no contradiction between any of Marx’s aggregates. I find this interesting as I did not approach the issue from a Marxian perspective of proving Marx without error (note: though I do think Marx made mistakes in his treatment of depreciation). Some may justifiably claim that by use of marginal measures there is no fundamental disagreement with marginal analysis. However if there is a deeper common value theory basis for both the marginal and LTV approaches then understanding this can be useful, especially as the classical approach adds insights on how pricing is dictated by the structure of property ownership, distribution and access to technology. Finally we should not underestimate how central the LTV was to the Political Economist schema, it made some issues such as trade theory, rent theory, tax theory etc. very simple that today seem opaque and difficult. It was used as a tool by the political economists to make many discoveries. Therefore whether or not the LTV theory fully holds we should be prepared to explore theories which narrow or eliminate those errors and see what results they bring.


One thought on “James Mill’s Annuity Approach to Capital Theory – Why was it Forgotten?

  1. Pingback: Human Capital is Labour (and Labour makes Capital) | Decisions, Decisions, Decisions

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