From here, parsed with LaTeX to help me see what's going on.

*I wrote this post. Then I realised it was wrong. I really wish my math were better. So I'm turning it into a sort of bleg. I should have written the technology in implicit form as \(F(C,I,K,L)=0\) rather than \(H(C,I)=F(K,L)\). Because the way I wrote it makes \(P_k\) depend only on \(I/C\), when it should also depend on \(K/L\) as well. I can't think of any plausible underlying story that would make \(H(C,I)=F(K,L)\) legitimate and reasonably general. But \(F(C,I,K,L)=0\) is ugly and unintuitive and unteachable, even though it works fine theoretically, and is just a little bit more complicated.*

*Maybe someone has some ideas?*

*Here's what I originally wrote:*

Macroeconomists like to aggregate things. To keep it simple. Especially for teaching. But we don't want it

*too*simple, so we have to wave our hands when we want to talk about things that*can't*happen in the model.
Here is the simple aggregate technology macroeconomists often assume:

$$C + I = F(K,L) \mbox{ where } I = \frac{dK}{dt} \mbox{ (I have ignored depreciation for simplicity).}$$

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?

Let's relax that left hand side assumption. Let's instead assume:

$$H(C,I) = F(K,L)$$

Where \(H( )\) is some convex function, so the PPF between \(C\) and \(I\) is bowed out. That means that the marginal cost of investment (in terms of foregone consumption) will be an increasing function of investment. So the price of the capital good \(P_k\) (in terms of the consumption good) will also be an increasing function of investment.

Let's continue to assume, as macroeconomists usually do, constant returns to scale. We assume that for both \(H(\cdot,\cdot)\) and \(F(\cdot,\cdot)\). So if we double both \(K\) and \(L\) we can also double both \(C\) and \(I\). So the derivatives of \(F\) with respect to \(K\) and \(L\) depend only on the \(K/L\) ratio. And the derivatives of \(H\) with respect to \(C\) and \(I\) depend only on the \(I/C\) ratio.

The price of the capital good \(P_k\) (in terms of the consumption good) will equal the marginal cost of producing one more capital good (in terms of consumption goods foregone):

$$P_k = -\frac{dC}{dI} = \frac{H_c}{H_i},$$ which is an increasing function of \(I/C\).

The real wage \(W\) (in terms of the consumption good) will equal the marginal product of labour (the extra consumption goods produced):

$$W = \frac{dC}{dL} = \frac{F_L}{H_c},$$ which is an increasing function of \(I/C\) and an increasing function of \(K/L\).

The real capital rental \(R\) (in terms of the consumption good) will equal the marginal product of capital (the extra consumption goods produced):

$$R = \frac{dC}{dK} = \frac{F_K}{H_C},$$ which is an increasing function of \(I/C\) and an decreasing function of \(K/L\).

In equilibrium, the real rate of interest \(r\) (in terms of the consumption good) must equal the rate of return from owning one unit of the capital good. That rate of return will equal \(R/P_k\), plus the annual percentage rate at which \(P_k\) is rising. (If you pay $100 to buy the machine, rent it out for $5 per year, and the price of machines rises by 2% per year, your rate of return will be 5%+2%=7%, and if the rate of interest is also 7% you will be just indifferent between buying and not buying that machine.)

$$r = \frac{R}{P_k} + \bigg(\frac{dP_k}{dt}\bigg)\frac{1}{P_k}$$

Substituting for \(R\) and \(P_k\) we get:

$$r = \frac{F_K}{H_i} + \bigg(\frac{d(H_C/H_i)}{dt}\bigg)\frac{1}{H_C/H_i}$$

So that \(r\) will be a decreasing function of \(K/L\), a decreasing function of \(I/C\), and an increasing function of the rate at which \(I/C\) is rising over time. (In steady state the \(C/I\) ratio will be constant over time, so that second term will be zero.)

In the standard model, \(r\) is a decreasing function of \(K/L\) only.

In the standard model we get a perfectly elastic investment demand curve. An increase in desired saving and hence investment has no immediate effect on the rate of interest; it reduces the rate of interest slowly over time as the capital stock grows over time. \(K\) cannot jump, so \(r\) cannot jump (unless \(L\) jumps).

In the revised model we get a downward-sloping investment demand curve. An increase in desired saving and hence investment causes \(P_k\) to increase immediately and \(r\) to fall immediately.

I think that's a lot cleaner than the "adjustment costs" approach to getting a downward-sloping investment demand curve.

And it lets us talk about how changes in desired savings and the rate of interest will affect the price of capital goods.

It 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 \(K_2\) to \(F(\cdots)\), and \(I_2\) to \(H(\cdots )\), then you get a second equation for \(P_{k2}\), for \(R_2\), and for \(r\) as a function of \(P_{k2}\) and \(R_2\). 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.

To complete the model we need to add a labour supply function and a savings function. One simple savings function would be a consumption-Euler equation where \(r\) is an increasing function of the growth rate in consumption, and so is an increasing function of \(I/C\).

But is it simple enough to teach? I need to think up some diagrams, and a good name for the \(H(\cdots)\) function, so students can understand it.

I don't know if anyone else has done it like this before. They may have.

I don't know if I got any of the math wrong. I may have. By the way, what am I implicitly assuming when I write \(H(C,I)=F(K,L)\) instead of \(G(C,I,K,L)=0\)? I originally planned to write the technology that second way, but thought the first way was a bit more intuitive.

(I thank Bob Murphy for sending me a copy of one of his papers, that inspired me to do this. (Got a link, Bob?). I think Bob and I are saying at least roughly the same thing. I'm just leaving out all the "what Samuelson said wrong" and "what Bohm-Bawerk said right" stuff that Bob goes into. I'm trying to keep it simple.)