LaTeXed WCI post

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.)

The loanable funds market

I've been thinking about the Austrian model, as described in Garrison's book, and its relationship with monetary economics.  Here is the Austrian model in a nutshell:

The economy is somewhere along the production possibilities frontier in the upper right: \(Y = C+I\).  The level of investment is by definition equal to the level of saving in the economy - that is, it is determined by the loanable funds market, where the saving (= supply) and investment (= demand) curves interact.  When the loanable funds market is in equilibrium, \(S = I\) at some equilibrium interest rate \(i_{eq}\).  Consumption, meanwhile, is equal to the value of all final goods and services, which have been produced over some period of time.

The interest rate pressures the intertemporal structure of production toward a particular slope.  That is, the rate at which value is added to goods as they move through stages of production (remember Macro 101, when you discuss why \(Y\) is the sum of final goods and services?) is pushed toward the interest rate.  If the value-added of a good compared to its production time is higher than the interest rate, firms will take advantage of arbitrage and invest in producing more of the good, increasing demand for loanable funds and decreasing the price of the good.  If the value-added of a good compared to its production time is less than the interest rate, investors will discontinue investment in firms producing the good and push the relevant funds into other investments, decreasing the interest rate and increasing the price of the good as marginal firms go out of business.

(There is an interesting discussion to be had at some other time as to the impact of interest rates on the internal structure of firms which encompass multiple steps of the production process.  The interest rate should "flex" such firms.)

While the chief contribution of the Austrian model is in disaggregating the capital structure of the economy and thus providing a mechanism by which the long-run can arise over many periods, here I want to think about the loanable funds market and the insight its particular abstraction gives into the broader structure of the economy.

Returning to the circular flow of Macro 101, firms purchase labor from households and households purchase final goods and services from firms.  Of course, this is a lie - in fact, the economy is a giant tangled mess of firms purchasing goods and services from each other, households selling their labor all over the place, firms purchasing labor from other firms, households selling goods and services to firms and other households, and we haven't even thrown the financial industry or the government into the mix - firms and households saving money through the financial industry, which fishes for investment opportunities, and of course the government with its sticky fingers and regulators in most transactions in the economy.

What's the function of the financial industry?  Steve Waldman opines that the financial industry is a morass of opaque risk-taking. Households don't want to know what sort of risks they're taking - they just want some return on their money.  Investments are repackaged, sliced, diced -- risk is packaged and maneuvered and spread out.

Let's run with the image of finance as a morass - literally.  Think of finance as a swamp through which water drains, from a lake on one side to a lake on the other.  At the end of the marsh, there's a steady current into the ocean, and on the one end, there's a steady current into the marsh.  But in the marsh, the water slowly wends this way and that, following aimless little streams and stopping in stagnant ponds, before it eventually outs into the lower lake.

But as far as the macroeconomy is concerned, the financial system is a black box.  Water goes into the swamp from one side and comes out the other.  Investors put money into one side, firms bid for investments on the other side, and the financial system equilibrates the two.  But., much like hot dogs and Project Mayhem, the first rule of finance is you do not ask questions about how it works.

Perhaps more to the point, you don't need to know how it works to know that it functions as a clearinghouse between investors and firms.  That's the beauty of the loanable funds model: it abstracts away the financial system so that details of financial interactions don't distract from the larger picture of the macroeconomy. As far as the macroeconomy is concerned, the details of how money moves from savers to investing firms is not relevant.  What matters is the capital stock accumulated via investment, the level of consumption deferred, and the impact of these decisions on growth and employment.

In order to place the financial system inc ontext, observe that the loanable funds market presents a savings (supply) curve and an investment (demand) curve relating dollars to a (single, risk-adjusted) interest rate.  We can interpret the savings curve as the relationship between the interest rate offered by the financial system and the willingness of savers to inject funds into the financial system.  Likewise, the investment curve is the relationship between the interest rate offered by the financial system and the willingness of firms to borrow funds from the financial system.

These curves and their interaction contain the macroeconomically relevant information in the financial system. Any macroeconomically relevant event in the financial system will manifest itself as a movement of curves in the loanable funds model.  Any macroeconomically irrelevant event in the financial system will be completely invisible in the loanable funds model - as it should be!