### Murphy, Still Befuddled by Interest After All These Years

Y'all know I love Bob, and I am certain he's a very smart guy. But for ten years now, I believe he's been tying himself up in knots on interest, and things are simpler than he is making them out to be. For instance, he writes:

"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."

But, I think physical convertibility is not the relevant matter: it is the pricing of the capital good. And how is it priced? By the discounted value of the consumer good flows it will produce in the future. And how are those discounted? By the interest rate. And in equilibrium, that sets r = MPK.

And this formulation is not at all what Bohm-Bawerk was critiquing. It is not that productivity sets the interest rate, it is that we value future productivity according to our time preference, and thus price capital goods so that their marginal productivity will equal the interest rate. Anything else allows pure arbitrage opportunities, which, in equilibrium, can't exist. If r < MPK, you borrow money (or whatever can be borrowed in that model economy) and buy the capital good. If r > MPK, you lend money and short the capital good.

Then I think we get an indifference curve that touches the PPF just where r = MPK. Of course, there are many points on the PPF where r != MPK. But in equilibrium we won't find ourselves at those points, because of how the price of the machine gets set.

1. Gene, I left this comment at my blog, wanted to make sure you saw it.

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Gene, I will try to do a follow-up post about your other blog post on this. But let me ask you to clarify it. Here’s the problem:

You are saying stuff like “if r < MPK then…” Since r is a percentage, that means MPK must be a percentage. But how the heck do you look at a tractor and say, “What is the percentage of its output?” I can look at a worker and say, “How many extra apples per hour can I have picked on my orchard, if I hire this guy?” That is a *physical* technological question. But I can’t look at a drill press and ask, “How many percentage points per year do I get if I incorporate this into my production?” That is a market value question, that can’t be answered merely by physical or technological facts.

So clearly, there is something qualitatively different between saying in equilibrium “w=MPL” and “r=MPK.”

Last point: I one time sat in Boyan Jovanavich (sp)’s office and spelled this stuff out for like 15 minutes. He looked at me and said with almost disbelief, “Assume you can transform bananas into tractors one-for-one.” He wasn’t cracking a joke. That was how he dealt with my perspective on this stuff.

So while you may have a solution that rescues the statement “r=MPK just like w=MPL,” your solution is not at all what mainstream guys have in mind. That’s what I mean about this being like the OLG thing. You eventually settled on a position that was fine, but it was not at all what Krugman and Baker were saying originally.

2. Bob/Gene,

The more I look at it the more it seems to me that r=MPK is very similar to w=MPL. In both cases they are derived from the present value of future (marginal) revenue discounted by the interest rate (itself derived from time preference.)

Can you point to anything that explains what the "mainstream guys" say ? I'm curious how they derive this differently from the "Austrian" way ?

1. By "they" I meant the wage rate and the price of capital goods. In equilibrium they will end up at the point where r=MPK and w=MPL are both true

3. How are capital goods priced?

There are two equilibrium conditions:

r=MPK/Pk (that is oversimplified, because it assumes expected future Pk = current Pk, and ignores depreciation, but never mind that).

Pk = Marginal Cost of producing more capital goods.

The first equation defines the demand curve for capital goods, and the second equation defines the supply curve.

Physical convertibility (understood as the ability to switch resources out of producing consumption goods into producing capital goods, or vice versa) is what determines that MC curve.

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