Cambridge capital controversy

The Cambridge capital controversy – sometimes simply called "the capital controversy" – refers to a theoretical and mathematical debate during the 1960s among economists concerning the nature and role of capital goods (or means of production) and the critique of the dominant neoclassical vision of aggregate production and distribution. The name arises because of 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. The two schools are often labeled "Sraffian" or "neo-Ricardian" and "neoclassical", respectively.

Most of the debate is mathematical, but some major elements can be explained in simple terms and as part of the 'aggregation problem' faced by neoclassical economics. Alternatively, this controversy might be seen as being parallel to the recent rejection of representative agent models of aggregate activity, based on neoclassical general equilibrium theory. That is, the critique of neoclassical capital theory might be summed up as saying that it suffers from the fallacy of composition, i.e., that we cannot simply jump from microeconomic conceptions to an understanding of production by society as a whole. The resolution of the debate, particularly how broad its implications are, has not been agreed upon by economists.

Despite the highly technical nature of most of the discussion, in many cases it generated more heat than light. This is partly because some see the technical criticisms of marginal productivity theory as connected to wider arguments with ideological implications. In neoclassical economics, the equilibrium rate of profit (and the income of the owners of capital goods) is seen as a price determined by technology, endowments of resources, and tastes (including intertemporal preferences on the part of investors and savers). Profits are a reward for saving and investment, so that the normal operations of the system pay them. Strictly speaking, the neoclassical theory does not say that capital's or labor's income is "deserved" in some moral or normative sense. But in many cases, the normative tone appeared anyway, partly because the neoclassical theory was originally developed (during the late 19th century) in opposition to the views of Karl Marx and other radical non-neoclassical economists during a period in which the legitimacy of property income and capitalism itself was being questioned. Thus, its practitioners used the newly-developed vision of economics to defend property rights and the existing economic system.

In contrast, 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. Though this point is entirely separate from the Sraffian critique of neoclassical economics, it dovetails with it. 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 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 a 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.

The body of this article concerns only technical issues.

The Aggregation Problem

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

q = A ƒ(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, 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 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 – say, a fall in the rate of profit – (associated with rising wages) will lead to more of that factor being used in production. A fall in this rate (seen as a price) means that more will be used. 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 fall to encourage the employment of that extra unit (assuming profit maximization).

Piero Sraffa and Joan Robinson before him, 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.

A 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; if the neoclassical story does not work under this assumption, it has been argued, it does not work at all. (Indeed, constant returns is a standard neoclassical assumption, as is perfect competition.)

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,

P_T = W + $20,000*(1 + r)

P_L = W + $10,000*(1 + r)

If W = $10,000 and r = 1 = 100% (an extreme case used to make the calculations obvious), then P_T = $50,000 and P_L = $30,000, as assumed. As above, K = $650,000.

Now, suppose that r falls to zero (another extreme case). Then P_T = $30,000 and P_L = $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.

A 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 Sonnenshein-Mantel-Debreu results (e.g., by Mas-Colell 1989) in general equilibrium theory, which show that representative agent models can be theoretically unjustified, except under restrictive conditions (see Kirman, 1992 for an explanation of the Sonnenshein-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.

A 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:

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

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).

Two production techniques

time period	input or output	technique A	technique B
–3	labor input	0	2
–2		7	0
–1		0	6
0	output	1	1

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 an 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:

cost = (1 + i)w×L_–1 + (1 + i)²*w×L_–2 + (1 + i)³*w×L_–3

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

Reswitching

interest rate	technique A	technique B
150%	$43.75	$46.25
75%	$21.44	$21.22
0%	$7.00	$8.00

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.

Conclusions?

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

Sraffian Views

Here are some of the Cambridge critics' views:

"Capital reversing renders meaningless the neoclassical concepts of input substitution and capital scarcity or labor scarcity. It puts in jeopardy the neoclassical theory of capital and the notion of input demand curves, both at the economy and industry levels. It also puts in jeopardy the neoclassical theories of output and employment determination, as well as Wicksellian monetary theories, since they are all deprived of stability. The consequences for neoclassical analysis are thus quite devastating. It is usually asserted that only aggregate neoclassical theory of the textbook variety — and hence macroeconomic theory, based on aggregate production functions — is affected by capital reversing. It has been pointed out, however, that when neoclassical general equilibrium models are extended to long-run equilibria, stability proofs require the exclusion of capital reversing (Schefold 1997). In that sense, all neoclassical production models would be affected by capital reversing." (Lavoie 2000)

"These findings destroy, for example, the general validity of Heckscher-Ohlin-Samuelson international trade theory (as authors such as Sergio Parrinello, Stanley Metcalfe, Ian Steedman, and Lynn Mainwaring have demonstrated), of the Hicksian neutrality of technical progress concept (as Steedman has shown), of neoclassical tax incidence theory (as Steedman and Metcalfe have shown), and of the Pigouvian taxation theory applied in environmental economics (as Gehrke and Lager have shown)." (Gehrke and Lager 2000)

Conclusion

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, Fabio Petri, and Bertram Schefold, have argued 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)

References

• Christopher Bliss, "Introduction, The Theory of Capital: A Personal Overview", in C. Bliss, A. Cohen and G.C. Harcourt (eds.) Capital Theory, (Cheltenham, UK: Edward Elgar, 2005). Vol. I, pp. xxvii–lx.
• Edwin Burmeister, "The Capital Theory Controversy", in Critical Essays on Piero Sraffa's Legacy in Economics (edited by Heinz D. Kurz), Cambridge: Cambridge University Press, 2000.
• Avi J. Cohen, G. C. Harcourt, "Whatever Happened to the Cambridge Capital Theory Controversies?," Journal of Economic Perspectives, 17(1), Winter 2003, 199–214.
• Christian Gehrke and Christian Lager, "Sraffian Political Economy", Encyclopedia of Political Economy, Routledge 2000.
• G. C. Harcourt, Some Cambridge Controversies in the Theory of Capital. Cambridge: Cambridge University Press 1972
• G.C. Harcourt and N.F. Laing, Capital and Growth, Harmondsworth, UK: Penguin, 1971. (This book includes the Samuelson article cited above and many other relevant articles.)
• John R. Hicks . Value and Capital, Oxford: Clarendon Press, 1939, 2nd ed. 1946.
• Alan P. Kirman, "Whom or What does the Representative Individual Represent?" Journal of Economic Perspectives 6(2), Spring 1992: 117–136.
• Heinz D. Kurz, "capital theory: paradoxes, The New Palgrave: A Dictionary of Economics, London and New York: Macmillan and Stockton, 1987, pp. 359–363.
• Michael Lebowitz, 2009, "Another Crisis of Economic Theory: The Neo-Ricardian Critique", In his Following Marx: Method, Critique and Crisis, Leiden & Boston: Brill, 2009.
• Marc Lavoie, "Capital Reversing", Encyclopedia of Political Economy, Routledge, 2000.
• Andreu Mas-Colell, "Capital Theory Paradoxes: Anything Goes", in "Joan Robinson and Modern Economic Theory" (ed. by G. R. Feiwel), New York University Press, 1989
• Luigi L. Pasinetti and Roberto Scazzieri, "capital theory: paradoxes, The New Palgrave: A Dictionary of Economics, London and New York: Macmillan and Stockton, 1987, pp. 363–68. .
• Paul A. Samuelson (1987). "Sraffian economics," The New Palgrave: A Dictionary of Economics, v. 3, pp. 452–60.
• Bertram Schefold, Normal Prices, Technical Change and Accumulation. London: Macmillan, 1997.
• Joseph E. Stiglitz, "The Cambridge-Cambridge Controversy in the Theory of Capital; A View from New Haven: A Review Article," Journal of Political Economy, 82(4), Jul.-Aug. 1974: 893-903.