Oh man, here I’m trying to really be productive. I have even come up with strict limits on my Facebook time. And then Nick Rowe goes and starts posting on capital & interest theory!
Here’s the situation in a nutshell. In mainstream economics, it is commonplace for people to say that in a competitive equilibrium, the interest rate equals the “marginal product of capital.” This is considered to be analogous to the claim that in a competitive equilibrium, the wage rate equals the marginal product of labor. After all, the thinking goes, interest is the price (or opportunity cost) of using another unit of capital, and the marginal return to capital is diminishing, so firms use more and more capital until the point at which r*=MPK (in standard notation). Just like, firms keep hiring workers until w*=MPL.
The only problem with all of this, is that it’s totally wrong. Besides all of the critiques you might raise about formal modeling etc., there is a qualitative sense in which interest does not have anything to do with the “marginal product of capital,” in the way that we can plausibly say that wages are intimately related to the marginal product of labor.
Now the Austrian School has had a bee in its bonnet over this issue since at least Bohm-Bawerk, and his famous critique of “the naive productivity theory of interest.” (I give the details for a layperson here.)
And yet, if you go to the trouble of learning modern techniques for mathematical models of the economy, it seems that r=MPK just pops out of the model. There’s nothing wrong with the math. So what the heck is going on here? Was there a flaw in the Austrian verbal logic, that the precise mainstream guys blew up?
Nope, not at all. What’s happening is that the standard r=MPK result–where MPK is defined as the increment in physical output from an additional input of capital into the production function–crucially assumes that the capital and consumption good are the same physical things, or at least, that they are always physically convertible into each other in a constant ratio.
If you can read the notation, I think I showed very clearly in the Appendix of my dissertation. I had a model with a distinct capital and consumption good, and then derived the expression for the equilibrium interest rate. It wasn’t at all r=MPK, because you had to worry about the changing market value of the capital good vis-a-vis the consumption good. But, if you assumed that they always traded at par against each other (which would of course be true if they were the same thing, as one-good models assume), then my expression reduced to r=MPK.
Nick Rowe is groping towards the same type of result. (I’m not saying “groping” to be disparaging; he is admitting upfront that his modeling is a bit off.) Here are the crucial passages from his post:
Here is the simple aggregate technology macroeconomists often assume:
C + I = F(K,L) where I = dK/dt (I have ignored depreciation for simplicity).
Some economists object to the right hand side of that equation. They complain that it aggregates all labour into one type of labour L. And they complain that it aggregates all capital goods into one type of capital good K.
But I object more to the left hand side of that equation.
It aggregates newly-produced consumption goods C with newly-produced capital goods I. It assumes they are perfect substitutes in production. It assumes the Production Possibilities Frontier between C and I is a straight line with a slope of minus one. It assumes the opportunity cost of producing one more capital good is always and everywhere one less consumption good. It means that the price of the capital good will be always one consumption good. And that means that the (real) rate of interest will always equal the marginal product of capital.
We don’t assume a straight line PPF between two different consumption goods. Why should we assume a straight line PPF between consumption goods and capital goods?
…[This approach] also shows what’s wrong with “r = MPK”, in a simple model.
You could add in a second capital good if you like. Just add K2 to F( ), and I2 to H( ), then you get a second equation for Pk2, for R2, and for r as a function of Pk2 and R2. But I don’t think it makes as much difference. The problem is not aggregating capital goods. The problem is aggregating the capital good with the consumption good. [Bold added.]
When I’m not daydreaming about becoming today’s Bobby Darin, I imagine I have tenure down the hall from Nick. We go to lunch, and talk about overlapping generations and then robots that can build copies of themselves.