I think this is the technology that Paul Krugman has in mind:
1. C + Kdot = A.La.K(1-a)
C is consumption, K is the capital stock, Kdot is investment, L is employment, A is a parameter that represents productivity, a is a parameter that (in competitive profit-maximising equilibrium) will equal labour's share of national income, so (1-a) is capital's share. (I've ignored physical depreciation of capital for simplicity).
Macroeconomists commonly assume this technology, mostly because it's easy to work with. But this technology creates a problem for Paul. That's because with this technology (in competitive profit-maximising equilibrium) the (real) rate of interest equals the rental rate on capital goods (equals the Marginal Product of Kapital), and Paul wants to explain why capital income has gone up while the rate of interest has gone down. (So he has to assume that monopoly power has increased, and that what looks like an increase in capital income is really an increase in monopoly rents.)
Let me make one small change. Change 1 to:
2. C + Kdot/A = La.K(1-a)
(I've divided both sides of 1 by 'A', then deleted the 'A' underneath 'C'.)
In the first, standard model, an improvement in productivity (an increase in A) affects the production of both consumption goods and capital goods equally. In the second model, an improvement in productivity affects the production of capital goods but not consumption goods.
That small change in the assumptions makes a big difference to the model.
In the first model, r = MPK
In the second model, r = MPK.A - Adot/A
In the first model, consumption and investment goods are perfect substitutes in production, with a marginal rate of transformation always equal to one, so that (under competition) the price of the capital good will always equal one (taking the consumption good as numeraire). This means that the rate of interest will always be equal to MPK. A decrease in 'a' will reduce labour's share, MPL and wages for labour, increase capital's share, MPK and rentals on capital, and raise the rate of interest. An increase in 'A' will raise both wages and capital rentals, and raise the rate of interest.
In the second model, consumption and investment goods are still perfect substitutes in production, but the marginal rate of transformation now equals 1/A, so that (under competition) the price of the capital good will always equal 1/A (taking the consumption good as numeraire). As A increases over time, capital goods become cheaper in terms of consumption goods. The rate of interest must equal the rate of return on owning capital goods, but that rate of return is lowered by the fact that the price of those capital goods is falling over time.
Here is an explanation of a higher capital share but lower real rate of interest: (1-a) has increased; but Adot/A has increased too.
Do I have any evidence to support my explanation? No. That's not what I am trying to do here.
What I am trying to do here is to show that the technology commonly assumed by many macroeconomists, mostly because it is easy to work with, is a very special technology. It distorts our thinking. It makes us think that an increase in capital rentals and an increase in the real interest rate are much the same thing. They aren't. My model is just an example that illustrates the possibility that one could go up while the other goes down. And my model is just the same as Paul's, except for one small change.
And people think that all the action in equation 1 comes from the right hand side, while the left hand side is just a boring national income accounting identity. They are wrong. I could forget the "Cobb-Douglas" bit, and replace the right hand side with "A.F(L,K)" and nothing important would change in this post. It's the left hand side that pins down the price of capital goods. And knowing MPK tells you nothing about r unless you also know the price of capital goods and how it is changing over time.
When all is said and done, Cambridge UK were wrong. Because "C+Kdot = F(L,K)" is not the same as "Neoclassical Capital Theory". But sometimes I think that they did have a point. Because "C+Kdot = F(L,K)" does play a bigger role in our thinking than it should.
[Addendum: when national income accountants measure the stock of capital they measure it in value units rather than in physical units. An increase in A will generally cause an increase in the physical units of capital (because they are now cheaper to produce), but will also cause a lower price of those units, so the effect on the measured stock of capital will be ambiguous.
If you want to solve "my model", just add a saving function "r = r* + B.Cdot/C" and you should be good to go, provided your math is better than mine.]
Update: Hmmm. Maybe "my model" sorta does work empirically? Suppose somebody invents computers. Adot/A is a lot higher for computers than for other capital goods (or consumption goods). And so Adot/A slowly rises over time as more and more stuff gets computerised. Dunno.