This 2003 JEL provides a brief introduction to the famous Capital Cambridge Controversy and its aftermath. The controversy, essentially, is as follows: It is easy to imagine situations where a given combination of labor and capital is optimal under low and high rates of interests, and a different combination is optimal under medium rates (Samuelson (1966) gives the classic wine production example). The interest rate and the “value of capital” are also inherently linked when capital is heterogeneous: we assume the interest rate is determined by the amount of capital, but a change in the rate of interest changes the future return from a stock of capital, and hence changes the price of each type of capital (and therefore the aggregated total amount of capital). The combination of these two ideas suggests first that aggregating capital into one big K may be problematic, and further that statements such as “the distribution of Y to capital and labor is determined by their relative marginal scarcities,” since with multiple equilibria, nothing is determined at all. A side issue of this capital controversy was Joan Robinson’s arguments against equilibrium analysis, which I think is mistaken for reasons not discussed in Cohen/Harcourt; in particular, we cannot (really! Look up the mathematical research here…) know but little about complex dynamics such as how birds form flocks, but we can certainly say a lot about the nature of the birds in a flock, and since birds spend most of their time in flocks, a theory of birds ought study that equilibrium. Likewise, economies tend to be in equilibrium (we rarely see massive shortages or surpluses), so our time is best spent studying equilibrium.
The resolution of the CCC is less well-known. Cohen and Harcourt note that even Solow and Samuleson accept the basic problem identified by the “Anglo” side. They also, however, do not see such capital switching problems as particularly common – an analogy in demand would be Giffen Goods, which are theoretically possible but rare enough that we can feel safe assuming demand is downward sloping. The “American” side (and hence the “2010 economics” side) argue that models are inherently simplifications of reality, and assuming that economies act “as if” the problems of the CCC do not exist is the type of simplification that we make all the time when writing models of reality. Cohen and Harcourt provide a great quote from Solow that “if God had meant there to be more than two factors of production, He would have made it easier for us to draw three-dimensional diagrams.” Another reason that capital switching may have been ignored by modern economists is that Robinson and other leaders of the “Anglo” side were functionally communists (Robinson famously defended China’s Cultural Revolution, although she took back her praise before she died), and the capital controversy was essentially an argument that “this model of neoclassical economics has a flaw, therefore we should throw out neoclassical economics and use an alternative model where power of classes is important and hence communism is better justified.” Arguments on either side of the controversy about the importance of capital switching were therefore ideological arguments as much as mathematical arguments.
The resolution, then, is thus: continue writing Y=f(K,L), but know that such aggregation of capital may be misleading when modeling certain types of heterogeneous capital.
http://dept.econ.yorku.ca/~avicohen/Linked_Documents/JEP_Cohen_Harcourt.pdf. A related paper of interest in Mas-Collel (1989), “Capital Theory Paradoxes: Anything Goes,” which notes that the “problems” coming from aggregating capital are very much related to the Anything Goes theory of Sonmenschein-Mantel-Debreu, and about as problematic to neoclassical theory (meaning, not very problematic).
The CCC are not an issue for “neoclassical economics”, i.e. maximization + equilibrium, but rather for the (naive) theory of marginal product distributive justice, i.e. what J.B. Clark and, more recently Mankiw, seem to be arguing.