3. But they still couldn't explain the rate of interest. Because it's hard to explain the rate of interest if you don't want to talk about time preferences. And all the other prices depend on the rate of interest, as well as on technology. So they assumed the rate of interest was exogenous;Before we get to why Sraffa argued that the rate of interest is exogenously determined by the monetary authority, let me discuss the neoclassical assumptions behind Nick’s proposition. I would argue that point 3 is exactly in reverse, that is, it is impossible (not hard) to explain the rate of interest on the basis of subjective preferences.
4. Some economists in Cambridge US made a very special assumption that let them explain the rate of interest without talking about time preferences. They assumed that there was only one good, and it could be converted back and forth between the consumption good and the capital good by waving a wand.
Böhm-Bawerk famously argued that there are three conditions for the determination of the rate of interest, namely: (1) the differences between wants and provision in different periods of time; (2) the systematic underestimation of future wants and the means available to satisfy them; and (3) the technical superiority of present compared with future goods of the same quality and quantity. The first two are related to subjective preferences, and are behind the supply of savings or abstinence from consumption, while the third is related to productivity. With both thriftiness and productivity one gets a version of the neoclassical loanable funds theory of the natural rate of interest.*
Note that the subjective basis for the determination of the rate of interest is incredibly shaky. The marginalist approach suggests that there is a positive rate of time preference, that is people prefer to consume now rather than latter, and are willing to part with consumption now in order to get more at some future date. That is why there must be a positive rate of interest to convince consumers to postpone the immediate fruition of pleasure. Yet it is far from clear that the positive time preference precedes the positive rate of interest. It seems rather more logical to assume that given a positive rate of interest some people might be willing to postpone consumption. The neoclassical subjective analysis is no more than a tautology with very dubious assumptions about causality, to say the least. It is hard to see why one could base a theory of interest on such uncertain foundations.
Remember that classical authors were very skeptical of subjective individual behavior. They actually referred to social utility when they talked about preferences. In that sense, Sraffa, not only thought that the foundations for subjective theories were unsound, but also from a methodological point of view were not particularly relevant. Interest rates were not positive because some individual preferred things now rather than latter, but they had an institutional foundation, associated to the fact that certain social groups could extract a surplus from society as a whole.
What about the productivity part of the marginalist/neoclassical argument? That’s the part that the capital debates disqualified, as was accepted by no other than Paul Samuelson. I’m not going to discuss the whole issue again, but it suffices to say that there is no logical way to determine the quantity of capital independently of the rate of interest, which implies circular reasoning.
Sraffa had determined very early in his investigation of the determination of relative prices, as early as his first equations in 1927 (with the help of Ramsey) that he could solve the system of simultaneous equations simply with the technical coefficients of production and an exogenous rate of interest (see DeVivo, 2003; subscription required). Sraffa after several changes and developments of his basic equations eventually settled (by the 1940s) on the notion that the rate of profit was determined exogenously by the monetary rate of interest (a proposition not unlike that of certain classical authors, in particular Thomas Tooke, and similar to Keynes idea of a conventional normal rate of interest in the General Theory), in the famous paragraph 44 of PCMC.
Note that classical authors for the most part assumed that the real wage was the exogenously determined distributive variable. The reasons for why Sraffa settled with a monetary theory of distribution require a different post. However, it should be clear that the exogenous rate of interest is not an arbitrary assumption as Nick suggests, but is required for the logical solution of the system of simultaneous equations (which demand the rate of profit to be determined independently of relative prices, something that the marginalist theory is unable to do in a system with a uniform rate of profit). Finally, the important part of the exogeneity of the rate of interest, besides the fact that it fits the historical/institutional framework of the capitalist economies that we live in, where central banks actually do determine the rate of interest, is that institutions play a role in the classical-Keynesian theory of distribution. As noted above, it is class and power that are behind a positive rate of interest and not you aunt's preferences for chocolate cake tomorrow.
Regarding point 4, there is an incredible confusion in the comment by Nick. Sraffa’s system never assumes any aggregate production or a one good economy. There is a composite commodity in the construction of the standard commodity and system, but production is a circular process. Even if it has similar properties as the Ricardian corn model, it is actually composed of several commodities. It is in fact the neoclassical theory, including the disaggregate Walrasian (in its Arrow-Debreu version) model, that requires a one commodity world to bring about the equilibrium of investment (the demand for a quantity of capital) to full employment savings. It is the marginalist theory of the natural rate of interest that lacks any logical foundation.
* Irving Fisher was critical of the limitations of Böhm-Bawerk’s theory even within the neoclassical paradigm. For the debate between them see Avi Cohen (2011).