Thursday, March 1, 2012

The capital debates: A brief introduction

Teaching on the capital debates this and last week. So here are some thoughts, based on my class notes and the required readings (see below). The capital debates remain a puzzling chapter in the history of economic ideas. Nearly everyone accepts that the British (as opposed to the Massachusetts) Cambridge won the debate, something Paul Samuelson acknowledged early on.[1] Yet, no one seems to grasp the full implications and relevance of the debate itself. Typically it is assumed that the capital debates relate simply to problems of aggregation, and that the use of aggregate production functions and aggregative measures of capital are still justifiable, for simplicity’s sake. However, contrary to this viewpoint the capital debates did not rest upon the possibility of building aggregate measures.

The capital debates are associated with the very notion of capital. Classical political economy authors, from William Petty to Karl Marx, including Quesnay, Smith and Ricardo, treated the process of production as a circular one. In this context, capital is a produced means of production,[2] rather than a factor of production used in the process of obtaining final goods. The most important result of the capital debates is that, once capital is defined as produced means of production, there is no direct relation between the relative abundance or scarcity of the means of production and its remuneration. Distribution, in other words, is not governed by supply and demand.

Since the Marginalist Revolution, and the rise of the so-called neoclassical school, the notion that relative prices are determined by supply and demand, and that these reflect the relative abundance or scarcity of all goods and services – including factors of production – became consensual. As a result, the supply and demand for capital became the determination for the remuneration of capital. The more abundant is capital, the lower its remuneration, and vice versa if it is scarce. Conflict has no role to play in the determination of distribution, and social classes vanished entirely from analysis.

Additionally, substitution leads to the full utilization of resources and their optimal allocation. If capital is scarce and expensive, and labor abundant and cheap, economic agents substitute labor for capital and fully utilize labor. Thus, despite the abundance of labor, its relative cheapness, through the principle of substitution, leads to full employment. Indeed, unhampered markets do lead to the veritable best of all possible worlds.

It is the logic of the principle of substitution, based on relative scarcities that the capital debates shattered. Contrary to the neoclassical parable, the capital debates showed that it is not generally possible to obtain a univocal relation between remuneration and relative scarcity. For example, assume that we have two commodities produced with capital and labor, and that one can be said to be univocally more capital abundant than the other. In this case, as capital becomes more abundant the profit-to-real-wage ratio will fall, more capital will be used, and more of the capital-intensive good will be produced. However, it is possible that one good would be more capital intensive at high levels of the profit-to-real-wage ratio, and that the other becomes the capital intensive good at lower levels of the same ratio. That is, we would have factor intensity reversal. In the instance of factor intensity reversals, the conventional relation between factor scarcity and relative prices breaks down.

In this situation, it would be possible that as the profit-to-real-wage ratio falls, more labor will be used, and more of the labor-intensive good will be produced. In other words, there would be reverse capital deepening and a lower rate of profit associated with a reduction in the use of capital. Substitution moves in the wrong direction, so to speak, and more of the scarce factor is demanded. A simple algebraic exercise may help understand the point.

Let’s assume that there are two methods of production, associated to the manufacture of capital (iron) and consumption (corn) goods respectively. The prices are determined by:

(1) pc=wlc+rpkkc
(2) pk=wlk+rpkkk

where the subscripts refer to consumption and capital, l and k are the technical coefficients of production, and w and r are the real wage and rate of profit. Using pc as a numeraire and solving for pk we obtain:

(3) pk=(1-wlc)/rkc

From (3) into (2) we get:

(4) [(1-wlc)/rkc]=wlk+rkk[(1-wlc)/rkc]

Simplifying, and solving for w we find:

(5) w=(1-rkk)/[lc+(lkkc-lckk)r]

If the expression in the small parenthesis in the denominator is equalized to zero we obtain a wage-profit frontier that is linear. This assumption is what Samuelson (1962, p. 225, n. 7) refers to as the equi-proportional assumption. Figure 1 shows the choice of technique under this assumption.
Figure 1

The firm chooses the highest rate of profit for a given real wage, which implies that as the rate of profit falls the firm must choose the more capital intensive (b in this case). In this case, the neoclassical parable works; there is an inverse relation between factor intensity and its remuneration.

Once, the assumption of equi-proportional capital to labor ratios in the machine and consumption sectors (which would mean in Marxist terminology the same organic composition of capital in both sectors) is dropped the wage-profit frontier is not linear anymore. If we assume that the capital goods sector is more capital intensive than the consumption sector then the wage-profit frontier will be concave (as shown in Figure 2).

Figure 2

In this case, we have two switches; at high levels of the rate of profit the firm choose technique a, which is more labor-intensive, and as the rate of profits falls it switches to b the capital-intensive one as prescribed by neoclassical economics. However, at even lower levels of the rate of profit, the firm switches back to the more labor intensive technique. Reswitching and reverse capital deepening, hence, result from the dismissal of the equi-proportionality assumption, which is what one would expect in a world with several goods.

The implications for neoclassical theory cannot be overstated. First and foremost, there is no relation between relative scarcity and the remuneration of factors of production, and, as a result, distribution is not simply the product of market forces. Further, there is no guarantee that all resources will be fully utilized.[3]

It must be noted that, even though the capital debates are fundamentally about the logical coherence of the neoclassical approach, the results of the capital debates have important empirical implications. Neoclassical theory makes strong predictions vis-à-vis substitution effects and the relation between relative scarcity and remuneration. Yet the capital debates suggest that some of those predictions might not be consistent, and, as a result, the absence of those relations might be expected in the real world.

The most obvious prediction is the inverse relation between investment (capital intensity) and the rate of interest (its remuneration). As it is well known, there is little evidence that investment is sensitive to variations in the real rate of interest. In a rare survey of the empirical literature on the determinants of investment Robert Chirinko (1993, p. 1906) argues, “[T]he response of investment to price variables tends to be small and unimportant relative to quantity variables.” In other words, interest rates have little effect on gross capital formation, and the substitution effects that imply that agents use the cheaper factor of production are not operative. Further, the empirical evidence suggests that investment reacts to quantity variables, meaning the level of activity. This suggests that the income effects tend to be larger tahn substitution effects and that a firm facing less demand will not buy capital goods, even if the interest rate is low. These results underscore the empirical relevance of the capital debates.[4]

Similarly, the capital debates highlighted the futility of using the aggregate production function to measure the growth and productivity performance of real economies. The theoretical problems with the aggregate production function, associated to the notion of capital as a scarce resource, are compounded by the impossibility of disentangling it from the identity of income with the structure of the functional distribution of income (Felipe and Fisher, 2003). In other words, if one runs a regression of income on capital and labor, as is often done by those using a production function, it necessarily follows that income grows because capital and labor grow. Furthermore, changes in income distribution also affect income growth, as total income (net of taxes) is by definition the wage multiplied by labor utilized in production plus capital multiplied by its remuneration.

In this way, the capital debates demolished the theoretical foundations of neoclassical economics, and provided significant empirical evidence that those neoclassical models and their resultant policy prescriptions should be viewed with a healthy measure of skepticism.

Faced with the impossibility of using both the notion of aggregate capital and the principle of substitution, neoclassical economics opted to apply the principle of substitution to each kind of capital good taken as a distinct factor of production, by using the Arrow-Debreu model of intertemporal general equilibrium (Garegnani, 1976; Milgate, 1979). Even though the idea of intertemporal equilibrium, in which capital is treated as a vector of heterogeneous capital goods, was first developed by Eric Lindahl and then popularized by John R. Hicks in the 1930s, and used by Arrow and Debreu in the 1950s, it was only after the capital debates that it came to be dominant within the mainstream.

The problem with the use of heterogeneous capital goods is that it implies a change in the traditional method of economics. Normal equilibrium positions are associated to a uniform rate of profit; however, when dealing with heterogeneous capital goods that are not substitutable between each other, it becomes necessary to discard the notion of long run equilibrium. In Arrow-Debreu models all prices are short run prices, associated to differential rentals for each capital good, and any change in the data of the system – preferences, technology, and information for given initial endowments – affects the direction to which the economy adjusts (Petri, 2003).

In other words, the forces of competition that lead capitalists to those sectors with higher remuneration and establish a uniform rate of profit do not operate in the Walrasian world.[5] Hence, the Walrasian models are incapable of ascertaining tendencies in real economies, a defect that is not mitigated with the introduction of imperfections (Stiglitz, 1993, p. 109), which Stiglitz calls the post-Walrasian and post-Marxist paradigm. Far from increasing the realism of the model, the casting about of such lifelines only complicates the results of an exceptionally unrealistic one.

Information imperfections, and other related imperfections like price rigidities or lack of rationality, once introduced leave the Arrow-Debreu model unable to produce Pareto efficient solutions, or even market equilibrium, since some markets may not exist. Additionally, the introduction of imperfections renders the aggregative model prone to suboptimal outcomes. Suboptimal results in the presence of imperfections suggest that in their absence markets would still produce optimal outcomes.[6]

Some authors tend to confuse the imperfectionist arguments, and the implicit support that they provide for policy intervention, as a break with orthodoxy. While it is clear that they provide space for flexibility in policy advice, they remain firmly based on orthodox grounds.[7] The capital debates, in contrast, showed that unhindered markets, free of imperfections of any type, do not lead to market efficiency in general.

Faced with the logical problems that neoclassical aggregative models are riddled with on the one hand, and the irrelevance of general equilibrium models on the other, neoclassical economists did what any rational agent would do: disregard the critiques and in so doing, their deleterious results, and proceed as if nothing had happened. However, an innovative, if not peculiar, development generated a curious division of labor within neoclassical economics. Aggregative models were deployed for the purposes of teaching and policymaking, while the Arrow-Debreu model became the retreat of neoclassical authors when questioned about the logical consistency of their models. In this response, a harsh tradeoff between logical consistency and relevance was cultivated in the very core of mainstream economics.

The degree of fragmentation – as Roncaglia (2005, p. 468) so aptly expresses it – and confusion in the mainstream today is the result of such inconsistency at the core of economics, and not uniquely, as is frequently asserted, because of the demise of the Keynesian consensus. The collapse of the certainties provided by the old aggregative neoclassical model has brought about an often cynical defense of market-oriented policies for their own sake. The return of Vulgar Economics, which “sticks to appearances … [and] believes that ‘ignorance is a sufficient reason’” (Marx, 1867, p. 307) is complete.

[1] See Samuelson (1966). A typical position is that of Robert Lucas (1988, p. 36) who notes the victory of the British argument, and yet remains oblivious to the problems of using the aggregate production function in the same paper.
[2] For Marx (1867, p. 189) capital also involved a social relation between the owners of the means of production and those forced to sell their labor power. For him, capital “can spring to life, only when the owner of the means of production and subsistence meets in the market with the free laborer selling his labor-power.”
[3] Both results are important, for example, for the Keynesian possibility of unemployment equilibrium. Keynes’ (1936, p. 243) emphasis on the unimportance of the natural rate of interest not only implies that the supply and demand for capital (loanable funds) do not determine the equilibrium rate of interest, but also that the conventional rate of interest may be set at such a level that brings about persistent unemployment.
[4] In the same way, the empirical evidence seems to contradict the notion that higher wages would lead to substitution of cheaper factors of production for labor. The exemplary case is the well-know study of the fast food industry in New Jersey, which found a positive correlation between the minimum wage and employment (Card and Krueger, 1995).
[5] That GE models do not support the notion that the abundance of a factor of production will be associated with lower remuneration has been pointed out by a survey of those models (Bliss, 1975).
[6] The same is valid for Bowles and Gintis’ (1993, p. 84) notion that market exchanges are usually contested and endogenous enforcement costs are not zero, and, as a result, there are conflicts of interest among exchanging parties. Therefore, if enforcement costs were nonexistent the Arrow-Debreu results would prevail. It must be noted that all the literature on post-Walrasian economics presumes continuity between the classical political economy authors and the post-marginalist revolution economics, which would mean that there are no significant distinctions between Smith, Marx, Walras and Arrow.
[7] Colander et al. (2004), for example, seems to suggest that several of the post-Walrasian developments can be seen as breaking up with orthodoxy. For a critique see Vernengo (2010).

Additional Readings:

Bliss, Christopher (1975), Capital Theory and the Distribution of Income, Amsterdam and New York: Elsevier North-Holland.

Bowles, Samuel and Herbert Gintis (1993), ‘The revenge of homo economicus: Contested exchange and the revival of political economy’ The Journal of Economic Perspectives, 7(1), 83-102.

Card, David and Krueger, Alan (1995), Myth and Measurement: The New Economics of the Minimum Wage, Princeton: Princeton University Press.

Chirinko, Robert (1993), ‘Business fixed investment spending: Modeling strategies, empirical results, and policy implications’, Journal of Economic Literature, 31(4), 1875-1911.

Colander, David, Rick Holt, and J. Barkley Rosser Jr. (2004), “The changing face of mainstream economics,” Review of Political Economy, 16, pp. 485-99.

Felipe, Jesus and Franklin Fisher (2003), ‘Aggregation in production functions: what applied economists should know’, Metroeconomica, 54(2-3), 208-262.

Garegnani, Pierangelo (1976), ‘On a change in the notion of equilibrium in recent work on value: a comment on Samuelson’, in M. Brown, K. Sato and P. Zarembka (eds), Essays in Modern Capital Theory, Amsterdam: North-Holland.

Heim, John J. (2009), ‘Which Interest Rate Seems Most Related to Business Investment?’, American Society of Business and Behavioral Sciences E-Journal, 5(1), February.

Keynes, John M. (1936), The General Theory of Employment, Interest and Money, New York: Harcourt Brace.

Lucas, Robert (1988), 'On the Mechanics of Economic Development,' Journal of Monetary Economics 22 (1), pp. 3–42.

Marx, Karl (1867), Capital, NY: International Publishers.Capital, NY: International Publishers.

Milgate, Murray (1979), 'On the origin of the notion of ‘intertemporal equilibrium’,' Economica, 46(181), 1-10.Economica, 46(181), 1-10.

Petri, Fabio (2003), ‘A ‘Sraffian’ critique of general equilibrium theory, and the classical Keynesian alternative’, in F. Petri and F. Hahn (eds), General Equilbrium: Problems and Prospects, London and New York: Routledge, pp. 387-421.

Samuelson, Paul (1962), “Parable and Realism in Capital Theory: The Surrogate Production Function,” Review of Economic Studies, 29(3), pp. 193-206.

Samuelson, Paul (1966), ‘A summing up,’ Quarterly Journal of Economics, 80(4), 568-583.Quarterly Journal of Economics, 80(4), 568-583.

Vernengo, M. (2010), 'Conversation or Monologue? On Advising Heterodox Economists,' Journal of Post Keynesian Economics, 32(3), pp. 389-96.


  1. Thanks, I look forward to any further commentary you have to make on the capital debates.

  2. Hi NK,

    Very interesting lesson!

    I hope not to be making a completely stupid question.

    Let me explain my doubt at length (it may help identify where my error is).

    In relation to Figure 1: each of the two linear wage-profit frontiers have an equation of the form r = (1 - l[c]*w)/k[k] (the square brackets denote subscript). In reality, I should add another subscript, to differentiate the two equations, but I trust you get my gist.

    Let's suppose the two frontiers cross at (w[*], r[*]). If w[t] > w[*], a capitalist would be better off producing over the beta frontier. If w[t] < w[*], the opposite: she would be better off over alpha. Up to here, I think I am okay.

    So, in reality how better or worse off a capitalist is is determined by the slope of the frontier: we always choose the upper most frontier. But this is determined by the slope.

    Note that each of the two frontiers have slopes of the form -l[c]/k[k].

    But l[c] is the labour required to produce corn in the corn industry, while k[k] is the capital required to produce iron, in the iron industry.

    So, we are not really talking about a single industry capital intensity. Wouldn't that be of the form -l[c]/k[c] or -l[k]/k[k]? (That is, labour and capital within the same industry)

    Hope the question makes some sense! :-)

    1. Well Nate replied below. But yes you are correct. The point, that Samuelson was trying to make was that if you have several of this wgae-profit forntiers, you end up having a smooth relation that approximates the neoclassical story (as a simple parable) in which you can choose the technique that maximizes profits, and that would imply an inverse relation between capital intensity and the rate of interest. In the original paper (Samuelson, 1962 here, in a footonte on the conclusion he notes that this post-doc guy (Piero A. Garegnani, by the way) suggested that his results hinged on the assumption of linearity, that is the fact that labor-capital ratios are the same in all sectors, and he sarcastically said that he expected him to publish something explaining why that was important. Oops!

  3. Magpie-

    Capital can move between industries in response to profit rates. What you have written is exactly the point. If we are in a one good world the neoclassical story has a chance, otherwise it makes little sense.

  4. Hi Nathaniel and NK

    Sorry I did not reply earlier.

    I think I understand what you both said.

    NK: because we have a series of linear frontiers with different negative slopes we end up having a figure that can be approximated with a convex function.

    Indeed, that's a standard assumption of classical economists, as well: you can move to industries where profit is higher.
    But my question was directed to the meaning of the slope of the frontier itself: it mixes information from both industries.
    After some more thought, I suppose this is due to the fact that industries provide inputs to each other.
    For some reason, however, at a first glance this seemed odd. Maybe not!

    Thanks guys!

  5. NK,

    I am most impressed with this article. It merits considerate and continuous reading.

    But at least some implications appear to be immediate, too.

    Say, for instance, here:

    "In other words, interest rates have little effect on gross capital formation, and the substitution effects that imply that agents use the cheaper factor of production are not operative. (...) [And] that a firm facing less demand will not buy capital goods, even if the interest rate is low."

    This I'd say, shows that monetary stimulus, in the shape of lower interest rates, have marginal stimulatory effect, at best. Which explains the US case, for instance. No need for confidence fairies to explain the lack of investment.

    This also fits the MMT paradigm wonderfully.

    I consider myself an unorthodox Marxist and the passage:

    "Hence, the Walrasian models are incapable of ascertaining tendencies in real economies, a defect that is not mitigated with the introduction of imperfections (Stiglitz, 1993, p. 109), which Stiglitz calls the post-Walrasian and post-Marxist paradigm."

    Really aroused a lot of interest in me. If you ever decide to write another article on this, it would be terrific.

    Thanks for an excellent article! And congratulations!

  6. Thanks. On Sraffa and MMT (or endogenous money as it used to be called), note that in the famous paragraph 44 of his Production of Commodities by Means of Commodities, says that it is the money rate of interest that determines the rate of profits. This monetary theory of distribution was develped by Massimo Pivetti and Carlo Panico in several works worth reading.

  7. Great post. You've read Keen's Debunking Economics, right? What you said, at greater length and with even more detailed support.

    1. Hi Steve. THanks. Actually, I have a copy, but didn't have the time to read, and it sits in the to do pile. Certainly most of what he says is perfectly in line with the ideas in the blog. I'm less keen on the notion that chaos is a relevant element for understanding real economies though.

  8. Great introduction to the capital debates and a fascinating view of how mainstream economics persists despite its substantial flaws. In discussing the PK and Austrian views of heterogeneous capital I suggest people read your introduction:

  9. Why do firms have to choose higher rate of profit? According to neoclassical theory, firms maximize profit, not rate of profit, as far as I know.
    I am new to this topic, so can you tell me where I went astray if I am wrong?

    1. Yes, firms maximize profits when deciding to produce, meaning in the conventional model that profits are zero, at the point which the marginal cost curve crosses the average cost at its minimum. However, given the choice of two techniques, the firm would choose the one that would, given the same wage cost, provide higher profits.

  10. So what I meant is, if price ever becomes lower, by low rate of interest, and induced more demand so that total profit exceeds the case when higher rate of interest is there, then wouldn't firms choose lower rate of interest according to neoclassical theory? After all, total amount of capital used would not be invariant on rate of interest.

  11. Prof.° Vernengo, you forgot to put the Stiglitz (1993) reference in the additional readings. May you do it? I've got a bit interested on it!

    1. Só agora vi. Acho que é uma do JEP, post-Walrasian, post-Marxist economics, ou algo assim

  12. Professor Vernengo, what other examples/papers can you refer to about evidence regarding capital re-switching?

  13. I recommend the following paper by Petri and if you want something more empirical (not a big fan of this literature on the linearity of wage curves) this one by Schefold


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