Cambridge capital controversy

Cambridge capital controversy

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Encyclopedia
The Cambridge capital controversy – sometimes simply called "the capital controversy" – refers to a theoretical and mathematical debate during the 1960s among economists
Economics
Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...

 concerning the nature and role of capital
Capital (economics)
In economics, capital, capital goods, or real capital refers to already-produced durable goods used in production of goods or services. The capital goods are not significantly consumed, though they may depreciate in the production process...

 goods (or means of production
Means of production
Means of production refers to physical, non-human inputs used in production—the factories, machines, and tools used to produce wealth — along with both infrastructural capital and natural capital. This includes the classical factors of production minus financial capital and minus human capital...

) and the critique of the dominant neoclassical
Neoclassical economics
Neoclassical economics is a term variously used for approaches to economics focusing on the determination of prices, outputs, and income distributions in markets through supply and demand, often mediated through a hypothesized maximization of utility by income-constrained individuals and of profits...

 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
Joan Robinson
Joan Violet Robinson FBA was a post-Keynesian economist who was well known for her knowledge of monetary economics and wide-ranging contributions to economic theory...

 and Piero Sraffa
Piero Sraffa
Piero Sraffa was an influential Italian economist whose book Production of Commodities by Means of Commodities is taken as founding the Neo-Ricardian school of Economics.- Early life :...

 at the University of Cambridge
University of Cambridge
The University of Cambridge is a public research university located in Cambridge, United Kingdom. It is the second-oldest university in both the United Kingdom and the English-speaking world , and the seventh-oldest globally...

 in England and economists such as Paul Samuelson
Paul Samuelson
Paul Anthony Samuelson was an American economist, and the first American to win the Nobel Memorial Prize in Economic Sciences. The Swedish Royal Academies stated, when awarding the prize, that he "has done more than any other contemporary economist to raise the level of scientific analysis in...

 and Robert Solow
Robert Solow
Robert Merton Solow is an American economist particularly known for his work on the theory of economic growth that culminated in the exogenous growth model named after him...

 at the Massachusetts Institute of Technology
Massachusetts Institute of Technology
The Massachusetts Institute of Technology is a private research university located in Cambridge, Massachusetts. MIT has five schools and one college, containing a total of 32 academic departments, with a strong emphasis on scientific and technological education and research.Founded in 1861 in...

, in Cambridge, Massachusetts. The two schools are often labeled "Sraffian" or "neo-Ricardian" and "neoclassical
Neoclassical economics
Neoclassical economics is a term variously used for approaches to economics focusing on the determination of prices, outputs, and income distributions in markets through supply and demand, often mediated through a hypothesized maximization of utility by income-constrained individuals and of profits...

", respectively.

Most of the debate is mathematical, but some major elements can be explained in simple terms and as part of the 'aggregation problem
Aggregation problem
An aggregate in economics is a summary measure describing a market or economy. The aggregation problem refers to the difficulty of treating an empirical or theoretical aggregate as if it reacted like a less-aggregated measure, say, about behavior of an individual agent as described in general...

'. That is, the critique of neoclassical capital theory might be summed up as saying that it suffers from the fallacy of composition
Fallacy of composition
The fallacy of composition arises when one infers that something is true of the whole from the fact that it is true of some part of the whole...

, 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.

Ethical issues


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 product
Marginal product
In economics and in particular neoclassical economics, the marginal product or marginal physical product of an input is the extra output that can be produced by using one more unit of the input , assuming that the quantities of no other inputs to production...

ivity 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
Time preference
In economics, time preference pertains to how large a premium a consumer places on enjoyment nearer in time over more remote enjoyment....

 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
Karl Marx
Karl Heinrich Marx was a German philosopher, economist, sociologist, historian, journalist, and revolutionary socialist. His ideas played a significant role in the development of social science and the socialist political movement...

 and other radical 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 capitalist
Capitalism
Capitalism is an economic system that became dominant in the Western world following the demise of feudalism. There is no consensus on the precise definition nor on how the term should be used as a historical category...

s) 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
Rate of profit
In economics and finance, the profit rate is the relative profitability of an investment project, of a capitalist enterprise, or of the capitalist economy as a whole...

 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
Economic power
There is no agreed-upon definition of power in economics. At least five definitions of power have been used:*Purchasing power, i.e., the ability of any amount of money to buy goods and services. Those with more assets, or, more correctly, net worth, have more power of this sort...

 that owning the means of production gives this minority to exploit
Exploitation
This article discusses the term exploitation in the meaning of using something in an unjust or cruel manner.- As unjust benefit :In political economy, economics, and sociology, exploitation involves a persistent social relationship in which certain persons are being mistreated or unfairly used for...

 the majority of workers and to receive profit
Profit (economics)
In economics, the term profit has two related but distinct meanings. Normal profit represents the total opportunity costs of a venture to an entrepreneur or investor, whilst economic profit In economics, the term profit has two related but distinct meanings. Normal profit represents the total...

. 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
Neoclassical economics
Neoclassical economics is a term variously used for approaches to economics focusing on the determination of prices, outputs, and income distributions in markets through supply and demand, often mediated through a hypothesized maximization of utility by income-constrained individuals and of profits...

, a production function
Production function
In microeconomics and macroeconomics, a production function is a function that specifies the output of a firm, an industry, or an entire economy for all combinations of inputs...

 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
Numéraire
Numéraire is a basic standard by which values are measured. Acting as the numéraire is one of the functions of money, to serve as a unit of account: to measure the worth of different goods and services relative to one another, i.e. in same units...

, 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
Diminishing returns
In economics, diminishing returns is the decrease in the marginal output of a production process as the amount of a single factor of production is increased, while the amounts of all other factors of production stay constant.The law of diminishing returns In economics, diminishing returns (also...

.

In some more complicated general equilibrium
General equilibrium
General equilibrium theory is a branch of theoretical economics. It seeks to explain the behavior of supply, demand and prices in a whole economy with several or many interacting markets, by seeking to prove that a set of prices exists that will result in an overall equilibrium, hence general...

 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
Neoclassical economics
Neoclassical economics is a term variously used for approaches to economics focusing on the determination of prices, outputs, and income distributions in markets through supply and demand, often mediated through a hypothesized maximization of utility by income-constrained individuals and of profits...

, 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
Marginal product
In economics and in particular neoclassical economics, the marginal product or marginal physical product of an input is the extra output that can be produced by using one more unit of the input , assuming that the quantities of no other inputs to production...

 of labor. More importantly for the discussion here, the rate of profit
Rate of profit
In economics and finance, the profit rate is the relative profitability of an investment project, of a capitalist enterprise, or of the capitalist economy as a whole...

 (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 will lead to a change in the use of that factor – a fall in the rate of profit
Rate of profit
In economics and finance, the profit rate is the relative profitability of an investment project, of a capitalist enterprise, or of the capitalist economy as a whole...

 (associated with rising wages) will lead to more of that factor being used in production. The law of diminishing marginal returns
Diminishing returns
In economics, diminishing returns is the decrease in the marginal output of a production process as the amount of a single factor of production is increased, while the amounts of all other factors of production stay constant.The law of diminishing returns In economics, diminishing returns (also...

 implies that greater use of this input will imply a lower marginal product, all else equal
Ceteris paribus
or is a Latin phrase, literally translated as "with other things the same," or "all other things being equal or held constant." It is an example of an ablative absolute and is commonly rendered in English as "all other things being equal." A prediction, or a statement about causal or logical...

: 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
Piero Sraffa
Piero Sraffa was an influential Italian economist whose book Production of Commodities by Means of Commodities is taken as founding the Neo-Ricardian school of Economics.- Early life :...

 and Joan Robinson
Joan Robinson
Joan Violet Robinson FBA was a post-Keynesian economist who was well known for her knowledge of monetary economics and wide-ranging contributions to economic theory...

 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
Income distribution
In economics, income distribution is how a nation’s total economy is distributed amongst its population.Income distribution has always been a central concern of economic theory and economic policy...

 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
Classical economics
Classical economics is widely regarded as the first modern school of economic thought. Its major developers include Adam Smith, Jean-Baptiste Say, David Ricardo, Thomas Malthus and John Stuart Mill....

 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
Classical economics
Classical economics is widely regarded as the first modern school of economic thought. Its major developers include Adam Smith, Jean-Baptiste Say, David Ricardo, Thomas Malthus and John Stuart Mill....

 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
Neoclassical economics
Neoclassical economics is a term variously used for approaches to economics focusing on the determination of prices, outputs, and income distributions in markets through supply and demand, often mediated through a hypothesized maximization of utility by income-constrained individuals and of profits...

 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 exogenous
Exogenous
Exogenous refers to an action or object coming from outside a system. It is the opposite of endogenous, something generated from within the system....

ly 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
Static analysis
Static analysis, static projection, and static scoring are terms for simplified analysis wherein the effect of an immediate change to a system is calculated without respect to the longer term response of the system to that change...

), they cannot handle the more complicated issues of dynamics
System dynamics
System dynamics is an approach to understanding the behaviour of complex systems over time. It deals with internal feedback loops and time delays that affect the behaviour of the entire system. What makes using system dynamics different from other approaches to studying complex systems is the use...

. 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
Financial capital
Financial capital can refer to money used by entrepreneurs and businesses to buy what they need to make their products or provide their services or to that sector of the economy based on its operation, i.e. retail, corporate, investment banking, etc....

 can. For the latter, all units are measured in money
Money
Money is any object or record that is generally accepted as payment for goods and services and repayment of debts in a given country or socio-economic context. The main functions of money are distinguished as: a medium of exchange; a unit of account; a store of value; and, occasionally in the past,...

 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 price
Price
-Definition:In ordinary usage, price is the quantity of payment or compensation given by one party to another in return for goods or services.In modern economies, prices are generally expressed in units of some form of currency...

s as indices of scarcity
Scarcity
Scarcity is the fundamental economic problem of having humans who have unlimited wants and needs in a world of limited resources. It states that society has insufficient productive resources to fulfill all human wants and needs. Alternatively, scarcity implies that not all of society's goals can be...

 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 Sonnenschein-Mantel-Debreu results (e.g., by Mas-Colell 1989) in general equilibrium theory, which show that representative agent
Representative agent
Economists use the term representative agent to refer to the typical decision-maker of a certain type ....

 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.

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:
Yi = Ai*Kia*Li1-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
Capital intensity
Capital intensity is the term in economics for the amount of fixed or real capital present in relation to other factors of production, especially labor...

 and either the rate of profit
Rate of profit
In economics and finance, the profit rate is the relative profitability of an investment project, of a capitalist enterprise, or of the capitalist economy as a whole...

, sometimes confusingly referred to as the rate of interest
Interest rate
An interest rate is the rate at which interest is paid by a borrower for the use of money that they borrow from a lender. For example, a small company borrows capital from a bank to buy new assets for their business, and in return the lender receives interest at a predetermined interest rate for...

. 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
Neoclassical economics
Neoclassical economics is a term variously used for approaches to economics focusing on the determination of prices, outputs, and income distributions in markets through supply and demand, often mediated through a hypothesized maximization of utility by income-constrained individuals and of profits...

 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
Eugen von Böhm-Bawerk
Eugen Ritter von Böhm-Bawerk was an Austrian economist who made important contributions to the development of the Austrian School of economics.-Biography:...

, Wicksell
Knut Wicksell
Johan Gustaf Knut Wicksell was a leading Swedish economist of the Stockholm school. His economic contributions would influence both the Keynesian and Austrian schools of economic thought....

 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
Roundaboutness
Roundaboutness, or roundabout methods of production, is the process whereby capital goods are produced first and then, with the help of the capital goods, the desired consumer goods are produced....

" — supposedly a physical measure of capital intensity
Capital intensity
Capital intensity is the term in economics for the amount of fixed or real capital present in relation to other factors of production, especially labor...

.

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
Neoclassical economics
Neoclassical economics is a term variously used for approaches to economics focusing on the determination of prices, outputs, and income distributions in markets through supply and demand, often mediated through a hypothesized maximization of utility by income-constrained individuals and of profits...

 concepts of input substitution and capital scarcity
Capital (economics)
In economics, capital, capital goods, or real capital refers to already-produced durable goods used in production of goods or services. The capital goods are not significantly consumed, though they may depreciate in the production process...

 or labor scarcity. It puts in jeopardy the neoclassical theory of capital
Capital (economics)
In economics, capital, capital goods, or real capital refers to already-produced durable goods used in production of goods or services. The capital goods are not significantly consumed, though they may depreciate in the production process...

 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
Knut Wicksell
Johan Gustaf Knut Wicksell was a leading Swedish economist of the Stockholm school. His economic contributions would influence both the Keynesian and Austrian schools of economic thought....

 monetary theories
Monetary theory
Monetary economics is a branch of economics that historically prefigured and remains integrally linked to macroeconomics. Monetary economics provides a framework for analyzing money in its functions as a medium of exchange, store of value, and unit of account. It considers how money, for example...

, 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
Macroeconomics
Macroeconomics is a branch of economics dealing with the performance, structure, behavior, and decision-making of the whole economy. This includes a national, regional, or global economy...

, based on aggregate production functions — is affected by capital reversing. It has been pointed out, however, that when neoclassical general equilibrium model
General equilibrium
General equilibrium theory is a branch of theoretical economics. It seeks to explain the behavior of supply, demand and prices in a whole economy with several or many interacting markets, by seeking to prove that a set of prices exists that will result in an overall equilibrium, hence general...

s 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
Heckscher-Ohlin model
The Heckscher–Ohlin model is a general equilibrium mathematical model of international trade, developed by Eli Heckscher and Bertil Ohlin at the Stockholm School of Economics. It builds on David Ricardo's theory of comparative advantage by predicting patterns of commerce and production based...

 (as authors such as Sergio Parrinello, Stanley Metcalfe, Ian Steedman
Ian Steedman
Ian Steedman was for many years a Professor of economics at the University of Manchester before moving down the road to Manchester Metropolitan University...

, and Lynn Mainwaring have demonstrated), of the Hicksian neutrality of technical progress
Hicks-neutral technical change
A Hicks-neutral technical change is a change in the production function of a business or industry which satifies certain economic neutrality conditions. The concept of Hicks neutrality was first put forth in 1932 by John Hicks in his book The Theory of Wages...

 concept (as Steedman has shown), of neoclassical tax incidence theory (as Steedman and Metcalfe have shown), and of the Pigouvian taxation theory
Pigovian tax
A Pigovian tax is a tax levied on a market activity that generates negative externalities. The tax is intended to correct the market outcome. In the presence of negative externalities, the social cost of a market activity is not covered by the private cost of the activity...

 applied in environmental economics
Environmental economics
Environmental economics is a subfield of economics concerned with environmental issues. Quoting from the National Bureau of Economic Research Environmental Economics program:...

 (as Gehrke and Lager have shown)." (Gehrke and Lager 2000)

Neoclassical Views


The neoclassical economist Christopher Bliss comments:

"...what one might call the existential aspect of capital theory has not attracted much interest in the past 25 years. A small band
of ‘true believers’ has kept up the assault on capital theory orthodoxy until today, and from their company comes at least one of my co-editers[sic]. I shall call that loosely connected school the Anglo-Italian theorists. No
simple name is ideal, but the one I have chosen indicates at least that the influences of Piero
Sraffa and Joan Robinson, in particular, are of central importance. Even in that case, there is a
flavour of necrophilia in the air. If one asks the question: what new idea has come out of
Anglo-Italian thinking in the past 20 years?, one creates an embarrassing social situation. This
is because it is not clear that anything new has come out of the old, bitter debates.

Meanwhile mainstream theorizing has taken different directions. Interest has shifted from general equilibrium style (high-dimension) models to simple, mainly one-good models. Ramsey-style dynamic-optimization models
Ramsey growth model
The Ramsey–Cass–Koopmans model or the Ramsey growth model is a neo-classical model of economic growth based primarily on the work of the economist and mathematician Frank P. Ramsey, with significant extensions by David Cass and Tjalling Koopmans...

 have largely displaced the fixed-saving coefficient approach. The many consumers that Stiglitz implanted into neoclassical growth modelling did not flourish there. Instead the representative agent is usually now the model's driver. Finally, the exogenous technical progress of Harrod
Roy Harrod
Sir Henry Roy Forbes Harrod was an English economist. He is best known for his biography of John Maynard Keynes and the development of the Harrod–Domar model, which he and Evsey Domar developed independently...

, and most writers on growth from whatever school in the 1960s and later, has been joined by numerous models which make technical progress endogenous in one of the several possible ways...

...Can the old concerns about capital be taken out, dusted down and addressed to contemporary models? If that could be done, one would hope that its contribution could be more constructive than the mutually assured destruction approach that marred some of the 1960s debates. It is evident that richer models yield richer possibilities. They do not do that in proportion when optimization drives model solutions. However, we know that many-agent models can have multiple equilibria when all agents optimize. There may be fruitful paths forward in that direction.

Old contributions should best be left buried when they involve using capital as a stick to beat marginal theory. All optima imply marginal conditions in some form. These conditions are part of an overall solution. Neither they nor the quantities involved in them are prior to the overall solution. It reflects badly on economists and their keenness of intellect that this was not always obvious to everyone." (Bliss 2005)

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
Robert Solow
Robert Merton Solow is an American economist particularly known for his work on the theory of economic growth that culminated in the exogenous growth model named after him...

 and Trevor Swan
Trevor Swan
Trevor Winchester Swan was an Australian economist. He is best known for his work on the neoclassical model of economic growth, published simultaneously with that of Robert Solow, for his work on integrating internal and external balance, represented by the Swan diagram and for pioneering work in...

 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
Keynesian economics
Keynesian economics is a school of macroeconomic thought based on the ideas of 20th-century English economist John Maynard Keynes.Keynesian economics argues that private sector decisions sometimes lead to inefficient macroeconomic outcomes and, therefore, advocates active policy responses by the...

, 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
Macroeconomics
Macroeconomics is a branch of economics dealing with the performance, structure, behavior, and decision-making of the whole economy. This includes a national, regional, or global economy...

 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
Frank Hahn
Frank Horace Hahn is a British economist whose work has focused on general equilibrium theory, monetary theory, Keynesian economics and monetarism...

, argued that rigorous neoclassical theory is most appropriately set forth in terms of microeconomics
Microeconomics
Microeconomics is a branch of economics that studies the behavior of how the individual modern household and firms make decisions to allocate limited resources. Typically, it applies to markets where goods or services are being bought and sold...

 and intertemporal general equilibrium
General equilibrium
General equilibrium theory is a branch of theoretical economics. It seeks to explain the behavior of supply, demand and prices in a whole economy with several or many interacting markets, by seeking to prove that a set of prices exists that will result in an overall equilibrium, hence general...

 models.

The critics, such as Pierangelo Garegnani
Pierangelo Garegnani
Pierangelo Garegnani was an Italian economist and professor of the Universitá degli Studi Roma Tre...

 (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
Wassily Leontief
Wassily Wassilyovich Leontief , was a Russian-American economist notable for his research on how changes in one economic sector may have an effect on other sectors. Leontief won the Nobel Committee's Nobel Memorial Prize in Economic Sciences in 1973, and three of his doctoral students have also...

-Sraffian tradition instead of the neoclassical
Neoclassical economics
Neoclassical economics is a term variously used for approaches to economics focusing on the determination of prices, outputs, and income distributions in markets through supply and demand, often mediated through a hypothesized maximization of utility by income-constrained individuals and of profits...

 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
Mainstream economics
Mainstream economics is a loose term used to refer to widely-accepted economics as taught in prominent universities and in contrast to heterodox 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
Rational expectations
Rational expectations is a hypothesis in economics which states that agents' predictions of the future value of economically relevant variables are not systematically wrong in that all errors are random. An alternative formulation is that rational expectations are model-consistent expectations, in...

 revolution' and in virtually all econometric work."
(Burmeister 2000)