Must-Read: Nick Rowe provides a very brief masterclass in “capital theory”, which is really the theory of the price system not just at a point in time but over time. (Cf. (1975): Capital Theory and the Distribution of Income.) Needless to say, there is no presumption that there is only one equilibrium vector for the intertemporal price system. And there is no presumption that problems of aggregation for commodities called “capital” is any easier than problems of aggregation for commodities called “labor” or “services” or “nondurable goods”. (The question of whether the problems of aggregation for commodities called “capital” is any more difficult than for any other not-completely-unreasonable grouping is left as an exercise):
: Interest, Capital, MRScc=(1+r)=1+(MPK/MRTci)+(dMRTci/dt)/MRTci: “The slope of the indifference curve [is] the Marginal Rate of Intertemporal Substitution…
…between consumption this year and consumption next year. Call it MRScc…. The slope of the PPF [is] the Marginal Rate of Intertemporal Transformation, between consumption this year and consumption next year. Call it MRTcc. The equilibrium condition is: MRScc = (1+r) = MRTcc…. We don’t need ‘capital’, or its marginal product, to determine the rate of interest…. Where is ‘capital’ in this model? And where is the Marginal Product of Kapital? Does MPK determine r? Does MPK=r? ‘No’, is the answer to both those questions.
The equilibrium condition is MRScc=(1+r)=MRTcc. MPK is one of the things, but not the only thing, that affects MRTcc. And MRTcc is equal to (1+r), but it does not determine (1+r)…. MPK is defined as the extra apples produced per extra existing machine, holding technology and other resources constant, and holding the production of new machines constant. If we move along the PPF between consumption and investment this year, we will have a bigger stock of capital goods next year, which will shift out next year’s PPF. MPK tells us how much it shifts out, per extra machine….
If capital exists, the real rate of interest is equal to, but not determined by, the Marginal Product of Kapital divided by the real price of the machine, plus the capital gains from appreciation of the real price of machines…. Rather than saying ‘MPK determines r’, it would be more true to say ‘MRScc determines r, which determines the prices of capital goods’. And the only thing wrong with saying that is that is that MRScc… depends on the expected growth rate of consumption, which in turn depends on our ability to divert resources to producing extra capital goods instead of consumption goods, and the productivity of those extra capital goods…