Cambridge capital controversy

In classical, orthodox economic theory,[4] economic growth is assumed to be exogenously given: Growth is dependent on exogenous variables, such as population growth, technological improvement, and growth in natural resources. Classical theory claims that an increase in either of the factors of production, i.e. labor or capital, while holding the other constant and assuming no technological change, will increase output but at a diminishing rate that will eventually approach zero.[5]

The so-called natural rate of economic growth is defined as the sum of the growth of the labor force and the growth of labor productivity.[6][note 1] The concept of the natural rate of growth first appeared in Roy Harrod’s 1939 article where it is defined as the "maximum rate of growth allowed by the increase of population, accumulation of capital, technological improvement and the work/ leisure preference schedule, supposing that there is always full employment in some sense."[7][note 2] If the actual economic growth-rate falls below the natural rate, then the unemployment rate will rise; if it rises above it, the unemployment rate will fall. Consequently, the natural rate of growth must be the rate of growth that keeps the rate of unemployment constant.

If the natural rate of growth is not exogenously given, but is endogenous to demand, or to the actual rate of growth, this has two implications.[6] At the theoretical level, there are implications for the efficiency and speed of the adjustment process between the warranted and the natural rates of growth in Harrod's growth model. Also, there are implications for the way the growth process should be viewed, and for understanding why growth rates differ between countries: whether growth is viewed as supply determined; or whether growth is viewed as demand determined; or determined by constraints on demand before supply constraints begin to operate.[6]

Harrod produced a mathematical model of growth whereby the natural rate of growth fulfills two important functions. First, it sets the ceiling to the divergence between the actual growth rate and warranted growth rate[note 3] and turns cyclical growth into slumps. Consequently, it is important for generating cyclical behavior in trade-cycle models that rely on first-order difference equations. Second, it ostensibly provides the maximum attainable long-run rate of growth.[note 4] The natural rate is treated as strictly exogenous; it is shaped by the growth of the labor force and the growth of labor productivity, without recognition nor assumption that both might be endogenous to demand.[note 5] Additionally, there was no fiscal or other economic mechanism in the theory that could bring the warranted rate of growth in line with the natural rate of growth, i.e. for society to achieve full or fuller utilization of its resources.

The question of whether the natural growth rate is exogenous, or endogenous to demand (and whether it is input growth that causes output growth, or vice versa), lies at the heart of the debate between neoclassical economists and Keynesian/post-Keynesian economists. The latter group argues that growth is primarily demand-driven because growth in the labor force as well as in labor productivity both respond to the pressure of demand, both domestic and foreign. Their view does not mean, post-Keynesians state, that demand growth determines supply growth without limit; rather, they claim that there is not one, single, full-employment growth path, and that, in many countries, demand constraints (related to excessive inflation and balance of payments difficulties) tend to arise long before supply constraints are ever reached.[6]

The Harrod–Domar model's lack of a mechanism that could bring the warranted rate of growth into line with the natural rate of growth triggered the growth debate in the mid-1950s, a debate that "engaged some of the greatest minds in the economics profession for over two decades."[6] The neoclassical and Neo-Keynesian sides were represented by Paul Samuelson, Robert Solow, and Franco Modigliani, who taught at the MIT, in Cambridge, Massachusetts, USA, while the Keynesian and Post-Keynesian sides were represented by Nicholas Kaldor, Joan Robinson, Luigi Pasinetti, Piero Sraffa, and Richard Kahn, who mostly taught at the University of Cambridge in England. The common name of the two places gave rise to the terms "the two Cambridges debate" or "the Cambridge capital controversy."

Both camps generally treated the natural rate of growth as given. Virtually all the focus of the debate centered on the potential mechanisms by which the warranted growth rate might be made to converge on the natural rate, giving a long-run, equilibrium growth-path. The American Cambridge side focused on adjustments to the capital/output ratio through capital-labour substitution if capital and labour were growing at different rates. The English Cambridge side concentrated on adjustments to the saving ratio through changes in the distribution of income between wages and profits, on the assumption that the propensity to save out of profits is higher than out of wages.[6]

Much of the emotion behind the debate arose because the technical criticisms of marginal productivity theory were connected to wider arguments with ideological implications. The famous neoclassical economist John Bates Clark saw the equilibrium rate of profit (which helps to determine the income of the owners of capital goods) as a market price determined by technology and the relative proportions in which the "factors of production" are used in production. Just as wages are the reward for the labor that workers do, profits are the reward for the productive contributions of capital: thus, the normal operations of the system under competitive conditions pay profits to the owners of capital. Responding to the "indictment that hangs over society" that it involves "exploiting labor," Clark wrote:

It is the purpose of this work [his 1899 'Distribution of Wealth'] to show that the distribution of the income of society is controlled by a natural law, and that this law, if it worked without friction, would give to every agent of production the amount of wealth which that agent creates. However wages may be adjusted by bargains freely made between individual men [i.e., without labor unions and other "market imperfections"], the rates of pay that result from such transactions tend, it is here claimed, to equal that part of the product of industry which is traceable to the labor itself; and however interest [i.e., profit] may be adjusted by similarly free bargaining, it naturally tends to equal the fractional product that is separately traceable to capital.[15]

These profits are in turn seen as rewards for saving, i.e., abstinence from current consumption, which leads to the creation of the capital goods. (Later, John Maynard Keynes and his school argued that saving does not automatically lead to investment in tangible capital goods.) Thus, in this view, profit income is a reward for those who value future income highly and are thus willing to sacrifice current enjoyment. Strictly speaking, however, modern neoclassical theory does not say that capital's or labor's income is "deserved" in some moral or normative sense.

Some members of the Marxian school argue that even if the means of production "earned" a return based on their marginal product, that does not imply that their owners (i.e., the capitalists) created the marginal product and should be rewarded. In the Sraffian view, the rate of profit is not a price, and it is not clear that it is determined in a market. In particular, it only partially reflects the scarcity of the means of production relative to their demand. While the prices of different types of means of production are prices, the rate of profit can be seen in Marxian terms, as reflecting the social and economic power that owning the means of production gives this minority to exploit the majority of workers and to receive profit. But not all followers of Sraffa interpret his theory of production and capital in this Marxian way. Nor do all Marxists embrace the Sraffian model: in fact, such authors as Michael Lebowitz and Frank Roosevelt are highly critical of Sraffian interpretations, except as a narrow technical critique of the neoclassical view. There are also Marxian economists, like Michael Albert and Robin Hahnel, who consider the Sraffian theory of prices, wages and profit to be superior to Marx's own theory.[16]

In neoclassical economics, a production function is often assumed, for example,

${\displaystyle Q=Af(K,L)}$

where Q is output, A is factor representing technology, K is the sum of the value of capital goods, and L is the labor input. The price of the homogeneous output is taken as the numéraire, so that the value of each capital good is taken as homogeneous with output. Different types of labor are assumed reduced to a common unit, usually unskilled labor. Both inputs have a positive impact on output, with diminishing marginal returns.

In some more complicated general equilibrium models developed by the neoclassical school, labor and capital are assumed to be heterogeneous and measured in physical units. In most versions of neoclassical growth theory (for example, in the Solow growth model), however, the function is assumed to apply to the entire economy. This view portrays an economy as one big factory rather than as a collection of a large number of heterogeneous workplaces.

This vision produces a core proposition in textbook neoclassical economics, i.e., that the income earned by each "factor of production" (essentially, labor and "capital") is equal to its marginal product. Thus, with perfect product and input markets, the wage (divided by the price of the product) is alleged to equal the marginal physical product of labor. More importantly for the discussion here, the rate of profit (sometimes confused with the rate of interest, i.e., the cost of borrowing funds) is supposed to equal the marginal physical product of capital. (For simplicity, abbreviate "capital goods" as "capital.") A second core proposition is that a change in the price of a factor of production will lead to a change in the use of that factor – an increase in the rate of profit (associated with falling wages) will lead to more of that factor being used in production. The law of diminishing marginal returns implies that greater use of this input will imply a lower marginal product, all else equal: since a firm is getting less from adding a unit of capital goods than is received from the previous one, the rate of profit must increase to encourage the employment of that extra unit, assuming profit maximization.

Piero Sraffa and Joan Robinson, whose work set off the Cambridge controversy, pointed out that there was an inherent measurement problem in applying this model of income distribution to capital. Capitalist income (total profit or property income) is defined as the rate of profit multiplied by the amount of capital, but the measurement of the "amount of capital" involves adding up quite incomparable physical objects – adding the number of trucks to the number of lasers, for example. That is, just as one cannot add heterogeneous "apples and oranges," we cannot simply add up simple units of "capital." As Robinson argued, there is no such thing as "leets," an inherent element of each capital good that can be added up independent of the prices of those goods.

Simple mathematical presentation

A third way to look this problem is to remember that many neoclassical economists assume that both individual firms (or sectors) and the entire economy fit the Cobb–Douglas production function with constant returns to scale. That is, output of each sector i is determined by the equation:

${\displaystyle Y_{i}=A_{i}.K_{i}^{a}.L_{i}^{1-a}}$

Here, A is a constant (representing technology and the like), K is supposed to represent the stock of capital goods (assumed to be measurable), and L is the amount of labor input. The coefficient a is supposed to represent the technology for this sector i. (Its subscript is left out for convenience.)

The problem is that unless we impose very strong mathematical restrictions, we cannot say that this Cobb–Douglas production function for sector i plus one for sector j (plus that for sector k, etc.) adds up to a Cobb–Douglas production function for the economy as a whole (with K and L being the sum of all of the different sectoral values). In short, for the sum of Cobb–Douglas production functions to equal a Cobb–Douglas, the production functions for all of the different sectors have to have the same values of A and a.

Reswitching means that there is no simple (monotonic) relationship between the nature of the techniques of production used and the rate of profit. For example, we may see a situation in which a technique of production is cost-minimizing at low and high rates of profits, but another technique is cost-minimizing at intermediate rates.

Reswitching implies the possibility of capital reversing, an association between high interest rates (or rates of profit) and more capital-intensive techniques. Thus, reswitching implies the rejection of a simple (monotonic) non-increasing relationship between capital intensity and either the rate of profit, sometimes confusingly referred to as the rate of interest. As rates fall, for example, profit-seeking businesses can switch from using one set of techniques (A) to another (B) and then back to A. This problem arises for either a macroeconomic or a microeconomic production process and so goes beyond the aggregation problems discussed above.

In a 1966 article, the famous neoclassical economist Paul A. Samuelson summarizes the reswitching debate:

"The phenomenon of switching back at a very low interest rate to a set of techniques that had seemed viable only at a very high interest rate involves more than esoteric difficulties. It shows that the simple tale told by Jevons, Böhm-Bawerk, Wicksell and other neoclassical writers — alleging that, as the interest rate falls in consequence of abstention from present consumption in favor of future, technology must become in some sense more 'roundabout,' more 'mechanized' and 'more productive' — cannot be universally valid." ("A Summing Up," Quarterly Journal of Economics vol. 80, 1966, p. 568.)

Samuelson gives an example involving both the Sraffian concept of new products made with labor employing capital goods represented by dead or "dated labor" (rather than machines having an independent role) and the "Austrian" concept of "roundaboutness" — supposedly a physical measure of capital intensity.

Instead of simply taking a neoclassical production function for granted, Samuelson follows the Sraffian tradition of constructing a production function from positing alternative methods to produce a product. The posited methods exhibit different mixes of inputs. Samuelson shows how profit maximizing (cost minimizing) indicates the best way of producing the output, given an externally specified wage or profit rate. Samuelson ends up rejecting his previously held view that heterogeneous capital could be treated as a single capital good, homogeneous with the consumption good, through a "surrogate production function".

Consider Samuelson's "Austrian" approach. In his example, there are two techniques, A and B, that use labor at different times (–1, –2, and –3, representing years in the past) to produce output of 1 unit at the later time 0 (the present).

Then, using this example (and further discussion), Samuelson demonstrates that it is impossible to define the relative "roundaboutness" of the two techniques as in this example, contrary to Austrian assertions. He shows that at a profit rate above 100 percent technique A will be used by a profit-maximizing business; between 50 and 100 percent, technique B will be used; while at an interest rate below 50 percent, technique A will be used again. The interest-rate numbers are extreme, but this phenomenon of reswitching can be shown to occur in other examples using more moderate interest rates.

The second table shows three possible interest rates and the resulting accumulated total labor costs for the two techniques. Since the benefits of each of the two processes is the same, we can simply compare costs. The costs in time 0 are calculated in the standard economic way, assuming that each unit of labor costs $w to hire:

${\displaystyle Cost=(1+i)w.L_{-1}+(1+i)^{2}w.L_{-2}+(1+i)^{3}w.L_{-3}+...+(1+i)^{n}w.L_{-n}}$

where L_–n is the amount of labor input in time n previous to time 0.

The results in bold-face indicate which technique is less expensive, showing reswitching. There is no simple (monotonic) relationship between the interest rate and the "capital intensity" or roundaboutness of production, either at the macro- or the microeconomic level of aggregation.

Naturally enough, the two contending schools arrive at different conclusions concerning this debate. It is useful to quote some of these.

Part of the problem in this debate revolved around the high level of abstraction and idealization that occurs in economic model-building on topics such as capital and economic growth. The original neoclassical models of aggregate growth presented by Robert Solow and Trevor Swan were straightforward, with simple results and uncomplicated conclusions which implied predictions about the real, empirical, world. The followers of Robinson and Sraffa argued that more sophisticated and complicated mathematical models implied that for the Solow–Swan model to say anything about the world, crucial unrealistic assumptions (that Solow and Swan had ignored) must be true.

To choose an example that did not get much attention in the debate (because it was shared by both sides), the Solow–Swan model assumes a continuously-attained equilibrium with 'full employment' of all resources. Contrary to Keynesian economics, saving determines investment in these models (rather than vice versa). The fact that the critique was also stated entirely using exactly the same kind of unrealistic assumptions meant that it was very difficult to do anything but 'criticize' Solow and Swan. That is, Sraffian models were explicitly divorced from empirical reality. And, as is very common in debates, it was much easier to destroy neoclassical theory than to develop a full-scale alternative that can help us understand the world.

In short, the progress produced by the Cambridge Controversy was from the unrealistic reliance on unstated or unknown assumptions to a clear consciousness about the need to make such assumptions. But this left the Sraffians in a situation where the unreal assumptions prevented most empirical applications, along with further developments of the theory. Thus it is not surprising that Bliss asks: "what new idea has come out of Anglo-Italian thinking in the past 20 years?"

Even though Sraffa, Robinson, and others had argued that its foundations were unfounded, the Solow–Swan growth model based on a single-valued aggregate stock of capital goods has remained a centerpiece of neoclassical macroeconomics and growth theory. It is also the basis for the "new growth theory." In some cases, the use of an aggregate production function is justified with an appeal to a instrumentalist methodology and a need for simplicity in empirical work.

Neoclassical theorists, such as Bliss, (quoted above) have generally accepted the "Anglo-Italian" critique of the simple neoclassical model and have moved on, applying the 'more general' political-economic vision of neoclassical economics to new questions. Some theorists, such as Bliss, Edwin Burmeister, and Frank Hahn, argued that rigorous neoclassical theory is most appropriately set forth in terms of microeconomics and intertemporal general equilibrium models.

The critics, such as Pierangelo Garegnani (2008), Fabio Petri (2009), and Bertram Schefold (2005), have repeatedly argued that such models are not empirically applicable and that, in any case, the capital-theoretical problems reappear in such models in a different form. The abstract nature of such models has made it more difficult to clearly reveal such problems in as clear a form as they appear in long-period models.

Since Samuelson had been one of the main neoclassical defenders of the idea that heterogeneous capital could be treated as a single capital good, his article (discussed above) conclusively showed that results from simplified models with one capital good do not necessarily hold in more general models. He thus mostly uses multi-sectoral models of the Leontief-Sraffian tradition instead of the neoclassical aggregate model.

Most often, neoclassicals simply ignore the controversy, while many do not even know about it. Indeed, the vast majority of economics graduate schools in the United States do not teach their students about it:

"It is important, for the record, to recognize that key participants in the debate openly admitted their mistakes. Samuelson's seventh edition of Economics was purged of errors. Levhari and Samuelson published a paper which began, 'We wish to make it clear for the record that the nonreswitching theorem associated with us is definitely false. We are grateful to Dr. Pasinetti...' (Levhari and Samuelson 1966). Leland Yeager and I jointly published a note acknowledging his earlier error and attempting to resolve the conflict between our theoretical perspectives. (Burmeister and Yeager, 1978).

However, the damage had been done, and Cambridge, UK, 'declared victory': Levhari was wrong, Samuelson was wrong, Solow was wrong, MIT was wrong and therefore neoclassical economics was wrong. As a result there are some groups of economists who have abandoned neoclassical economics for their own refinements of classical economics. In the United States, on the other hand, mainstream economics goes on as if the controversy had never occurred. Macroeconomics textbooks discuss 'capital' as if it were a well-defined concept — which it is not, except in a very special one-capital-good world (or under other unrealistically restrictive conditions). The problems of heterogeneous capital goods have also been ignored in the 'rational expectations revolution' and in virtually all econometric work." (Burmeister 2000)

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