Cambridge capital controversy

Overview

The Cambridge capital controversy, sometimes called "the capital controversy" or "the two Cambridges debate", was a dispute between proponents of two differing theoretical and mathematical positions in economics that started in the 1950s and lasted well into the 1960s. The debate concerned the nature and role of capital goods and a critique of the neoclassical vision of aggregate production and distribution. The name arises from the location of the principals involved in the controversy: the debate was largely between economists such as Joan Robinson and Piero Sraffa at the University of Cambridge in England and economists such as Paul Samuelson and Robert Solow at the Massachusetts Institute of Technology, in Cambridge, Massachusetts, United States. The English side is most often labeled "post-Keynesian", while some call it "neo-Ricardian", and the Massachusetts side "neoclassical". Most of the debate is mathematical, while some major elements can be explained as part of the aggregation problem. The critique of neoclassical capital theory might be summed up as saying that the theory suffers from the fallacy of composition; specifically, that we cannot extend microeconomic concepts to production by society as a whole. The resolution of the debate, particularly how broad its implications are, has not been agreed upon by economists.

Background

In classical, orthodox economic theory, 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. 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. 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." 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. 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. 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 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. 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. 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.

Central issue

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.

The Harrod–Domar model

Roy Harrod, in his seminal 1933 paper, developed a model, subsequently refined by Russian-born Evsey Domar, that aims to explain an economy's growth rate in terms of the level of saving and of the productivity of capital. Despite its progenitors' ostensibly Keynesian viewpoint, the Harrod–Domar model was actually the precursor to the exogenous growth model. According to the Harrod–Domar model there are three kinds of growth: the rate of warranted growth; the rate of actual growth; and the natural rate of growth. The warranted growth-rate is the rate of growth at which the economy does not expand indefinitely or go into recession. The actual growth rate is the real rate-increase in a country's yearly GDP. The natural rate of growth is the rate that the growth of an economy requires so that full employment is maintained. For example, if the labor force grows at 3 percent per year, with everything else being equal, then to maintain full employment, the economy's annual growth rate must be 3 percent. Neoclassical economists claimed shortcomings in the Harrod–Domar model, in particular pointing out instability in its solution, and, by the late 1950s, they started an academic dialogue that led to the development of the Solow–Swan model.

The Solow–Swan model

The model was developed separately and independently by Robert Solow and Trevor Swan in 1956, in response to the supposedly Keynesian Harrod–Domar model. Solow and Swan proposed an economic model of long-run economic growth set within the framework of neoclassical economics. They attempt to explain long-run economic growth by looking at capital accumulation; labor growth or population growth; and increases in productivity, commonly referred to as technological progress. At its core, the model offers a neoclassical (aggregate) production function, often specified to be of Cobb–Douglas type, which enables the model "to make contact with microeconomics".

The post-Keynesian theory of growth

Post-Keynesian economists, such as Nicholas Kaldor, Luigi Pasinetti, Richard Kahn, and Joan Robinson, proposed a different model of growth. In their approach, the warranted rate of growth is brought into equality with the natural rate of growth by adjustments to income distribution. Although Kaldor and Pasinetti, for example, differed in how to justify this, the rate of profits is the quotient of the rate of growth and the ratio of the savings rate out of profits. This equation is known as the Cambridge equation. Investment, as in Keynes, is taken as an independent variable, and savings adjust to investment.

The debate

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." The neoclassical and Neo-Keynesian sides were represented by Paul Samuelson, Robert Solow, and Franco Modigliani, who taught at the MIT, in Cambridge, Massachusetts, US, 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.

Ideological issues

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 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. 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.

The aggregation problem

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

Q = A f ( K , L )

{\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. Robinson developed her contribution to this critique in her 1953 paper “The Production Function and the Theory of Capital,” where she addressed the question of what units of capital could be measured in. She argued that valuing a stock of capital goods cannot be done without first knowing the rate of profit, since capital goods are valued by the income they are expected to generate. However, the production function is supposed to determine the rate of profit. As she put it, a student of economics is taught to write output as a function of labor and capital but is then “hurried on to the next question, in the hope that he will forget to ask in what units C is measured.” Robinson identified that this was a methodological error, since valuing capital requires the rate of interest to be taken as given, rather than determined by the theory itself. She insisted that this problem could not be resolved by only adding up monetary values of capital goods, since those values shift as the rate of profit changes. She concluded that this rendered “the major part of neo-classical doctrine spurious.” 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.

Sraffian presentation

Neoclassical economists assumed that there was no real problem here. They said: just add up the money value of all these different capital items to get an aggregate amount of capital (while correcting for inflation's effects). But Sraffa pointed out that this financial measure of the amount of capital is determined partly by the rate of profit. This is a problem because neoclassical theory tells us that this rate of profit is itself supposed to be determined by the amount of capital being used. There is circularity in the argument. A falling profit rate has a direct effect on the amount of capital; it does not simply cause greater employment of it. In very simple terms, suppose that capital currently consists of 10 trucks and 5 lasers. Trucks are produced and sold for $50,000 each, while each laser goes for $30,000. Thus, the value of our capital equals the sum of (price)*(quantity) = 10*$50,000 + 5*$30,000 = $650,000 = K. As noted, this K can change if the rate of profit rises. To see this, define the price of production for the two types of capital goods. For each item, follow the type of pricing rule used by Classical economics for produced items, where price is determined by explicit costs of production:

P = (labor cost per unit) + (capital cost per unit)*(1 + r) Here, P is the price of an item and r is the rate of profit. Assume that the owners of the factories are rewarded by receiving income proportional to the capital that they advanced for production (with the proportion being determined by the profit rate). Assume that the labor cost per unit equals W in each sector (and does not change). Both r and W are assumed to be equalized between sectors due to competition, i.e., the mobility of capital and labor between sectors. Note that this classical conception of pricing is different from the standard neoclassical "supply and demand" vision. It refers to long-run price determination. It can be reconciled with neoclassical economics by assuming that production follows constant returns to scale. Further, this formulation does not treat the rate of profit as a price determined by supply and demand. Rather, it fits more with neoclassical conceptions of "normal" profits. These refer to the basic profits that the owners of capital must receive in order to stay in business in their sector. Third, while neoclassical economics assumes that the "normal" rate of profit is determined by aggregate production (as discussed above), this formulation takes the rate of profit as exogenously given. That is because the whole neoclassical theory of profit-rate determination is being questioned: if we can go from the marginal product of capital to the profit rate, we should be able to go from the profit rate to the marginal product. In any event, few if any participants in the Cambridge Controversy attacked the Sraffian critique on these grounds. Go back to the pricing formula above. As in the real world, the capital intensity of production (capital cost per unit) differs between the sectors producing the different types of capital goods. Suppose that it takes twice as much capital per unit of output to produce trucks than it does to produce lasers, so that the capital cost per unit equals $20,000 for trucks (T) and $10,000 for lasers (L), where these coefficients are initially assumed not to change. Then,

PT = W + $20,000*(1 + r) PL = W + $10,000*(1 + r) If W = $10,000 and r = 1 = 100% (an extreme case used to make the calculations obvious), then PT = $50,000 and PL = $30,000, as assumed. As above, K = $650,000. Now, suppose that r falls to zero (another extreme case). Then PT = $30,000 and PL = $20,000, so that the value of the capital equals 10*$30,000 + 5*$20,000 = $400,000. The value of K thus varies with the rate of profit. Note that it does not vary in proportion as with a general inflation or deflation that changes both prices by the same percentage: the exact result depends on the relative "capital intensity" of the two sectors. This result is not changed by the fact that for both items, the capital cost per unit would change as the two prices change (contrary to the assumption made above). Nor does it change if the wage rate and labor cost per unit (W) change. Also, an obvious riposte is that we can aggregate capital simply by using the first set of prices and ignoring the second, as with many inflation corrections. This does not work, however, because the variation of the rate of profit is theorized as happening at a specific point in time in purely mathematical terms rather than as part of an historical process. The point is that if neoclassical conceptions do not work at a specific time (statics), they cannot handle the more complicated issues of dynamics. This critique of the neoclassical conception is more of a matter of pointing out its major technical flaws in the theory than of presenting an alternative. In general, this discussion says that the distribution of income (and r) helps determine the measured amount of capital rather than being solely determined by that amount. It also says that physical capital is heterogeneous and cannot be added up the way that financial capital can. For the latter, all units are measured in money terms and can thus be easily summed. Even then, of course, the price of a sum of financial capital varies with interest rates. Sraffa suggested an aggregation technique (stemming in part from Marxian economics) by which a measure of the amount of capital could be produced: by reducing all machines to a sum of dated labor from different years. A machine produced in the year 2000 can then be treated as the labor and commodity inputs used to produce it in 1999 (multiplied by the rate of profit); and the commodity inputs in 1999 can be further reduced to the labor inputs that made them in 1998 plus the commodity inputs (multiplied by the rate of profit again); and so on until the non-labor component was reduced to a negligible (but non-zero) amount. Then you could add up the dated labor value of a truck to the dated labor value of a laser. However, Sraffa then pointed out that this accurate measuring technique still involved the rate of profit: the amount of capital depended on the rate of profit. This reversed the direction of causality that neoclassical economics assumed between the rate of profit and the amount of capital. Further, Sraffa showed that a change in the rate of profit would change the measured amount of capital, and in highly nonlinear ways: an increase in the rate of profit might initially increase the perceived value of the truck more than the laser, but then reverse the effect at still higher rates of profit. See "Reswitching" below. The analysis further implies that a more intensive use of a factor of production, including other factors than capital, may be associated with a higher, not lower price, of that factor. According to the Cambridge, England, critics, this analysis is thus a serious challenge, particularly in factor markets, to the neoclassical vision of prices as indices of scarcity and the simple neoclassical version of the principle of substitution.

General equilibrium presentation

A different way to understand the aggregation problem does not involve the Classical pricing equations. Think about a decrease in the r, the return on capital (corresponding to a rise in w, the wage rate, given that initial levels of capital and technology stay constant). This causes a change in the distribution of income, the nature of the various capital goods demanded, and thus a change in their prices. This causes a change in the value of K (as discussed above). So, again, the rate of return on K (i.e., r) is not independent of the measure of K, as assumed in the neoclassical model of growth and distribution. Causation goes both ways, from K to r and from r to K. This problem is sometimes seen as analogous to the Sonnenschein–Mantel–Debreu results (e.g., by Mas-Colell 1989) in general equilibrium theory, which shows that representative agent models cannot be theoretically justified, except under restrictive conditions (see Kirman, 1992 for an explanation of the Sonnenschein–Mantel–Debreu results as an aggregation problem). Note that this says that it's not simply K that is subject to aggregation problems: so is L.

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:

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.