# Understanding Interest Rates

Ever since Scott Sumner suggested that someone needs to put together a model for market monetarism it is something I have been thinking about. My background is physics, and I have had only a very limited exposure to the world of building economic models, but it seems that most macroeconomic models are some form of equilibrium analysis. Market monetarism is fundamentally about disequilibrium.

This post introduces the concept of the natural rate of interest, and shows how changing the interest rates allows one to introduce disequilibrium and hence violate some of the most fundamental accounting identities, that much of equilibrium analysis 101 depends on.

Let us make a few assumptions about how the world works. First, actors have a savings preference, $S\equiv S(I)$, where $I$ is the real interest rate, and the savings preference is expressed as a percentage. All we are saying here is that saving preferences respond to the real interest rate, i.e. people save more when interest rates are higher. Secondly, workers have some function that describes their productivity. One of the inputs to this is $k$ which is the capital invested in said worker. Let us say that $F\equiv F(k, ...)$, where the ellipsis stands for all the other inputs which matter. Now, if I can borrow money at interest rate $I$, and use it to increase $k$ and hence $F$, then I should do that as long as the marginal rate of return on $F$ is higher than the interest rate. In other words,

$\frac{\partial F}{\partial k} = I.$

Let us assume that this expression is invertible for $k$. Now capital is persistent, up to some depreciation, so I should consider that if $K = \sum k$, then the capital spending is $\delta K = K_{Y}(I_{Y}, ....)-K_{Y-1}(I_{Y-1}, ...)*D$, where the subscript denotes the year/time period, and the $D$ stands for depreciation.

By construction then, the total amount of saving is $GNI*S(I)$, where $GNI$ is national income, and the total capital spending is $\delta K$. Now, we have two terms, saving increases with $I$, and capital spending decreases, so we can always choose an interest rate such that these two terms match exactly. We call this the natural rate of interest.

We can now write down the national expenditure identity:

$GDP = GNI*(1 - S- T) + \delta K + G$

Now the three terms are clearly understood. The first is consumer spending,  which is income less savings and taxes, the second is capital spending, and the final one is government spending. Since everyone’s spending is some one else’s income, we should have GDP=GNI, However, this type of equilibrium analysis misses the point, as my spending is based on my previous income. If we imagine that everyone is paid once a year on a particular day, then we see that our years expenditure, saving etc, its based on our previous income. Let us assume that the real production of the economy is fixed. Then if the interest rate is at the `natural rate’, and the government runs no deficit, then we get zero expenditure growth – the price level is fixed. If the right hand side is greater than GNI, then you get price inflation. This can happen in two ways, either $GNI*T \not= G$ or $GNI*S \not= \delta K$.

Now we can draw three very important conclusions from this simple analysis.

(1) Firstly, the natural rate of interest is not stable, it depends on those real factors which determine the marginal rate of return on capital. My own preferred narrative is that every so often a disruptive technological shock comes along, like mass production, and suddenly there is lots of very productive capital spending opportunities. In order to get more savings to meet demand for capital spending, interest rates rise. Slowly the economy fills the available capital opportunities through innovation, and so we get a slow and steady decline in the natural rate of interest, until we get the next technology shock. Demographics are also an issue, as saving behaviour is not constant for individuals over their lifetime. In short, high interest rates signal high demand for capital.

(2) The accounting identities like GNI=GDP, and S=I, so familiar from classical economics, are equilibrium conditions which do not need to hold in the real world. They implicitly assume that expenditure and income occur at the same time. The insight that there is a time lag between income and expenditure, creates the conditions for the possibility of disequilibrium. We can also frame this as an expectations issue. If I foresee declining income, I can reduce expenditure now, either way, it is the difference in time between expenditure and income which creates the possibilities.

(3) Now, if we hold potential production constant, then we get inflation, essentially, if GDP now exceeded the previous time period’s income. The paradox of thrift is now easily explained. We set the interest rate at the natural rate, such that S=I, and we have no deficit so GNI*T=G. All of a sudden there is a real shock causing a fall in income. If we allowed perfect elasticity in the price level, we get instant deflation, and say’s law is enforced. In practice, we get a stable price level and falling real output. This results in unemployment. Those workers who are still employed are now over capitalised. Thus capital spending falls. Of course, due to unemployment, saving also falls, but barring an absurdly fortune cancellation, savings will no longer equal investment, at the current interest rate. In other words, real shocks shift the natural rate of interest, and produce monetary disequilibrium. This is what Keynes called the paradox of thrift: People save too much in a recession, however, what really happened was that the natural rate of interest is lowered, and hence, money is much too tight. This means that monetary policy essentially causes the recession by not being omniscient. If we want to return GDP to potential, we must shift the interest rate, or increase G. These are basically equivalent strategies. (As economist Hulk tweets:HULK CONFUSED BY @EDBALLSMP. WANTS FISCAL STIMULUS BUT TO KEEP 2% INFLATION TARGET. HULK NOT AWARE OF MACRO MODEL WHERE THIS MAKES SENSE. ) Both will raise GDP compared to GNI. Monetary dominance is also a feature here. If we assume that the central bank will manipulate the interest rate so as to produce two percent inflation, that means that the central bank chooses an interest rate such that GDP exceeds GNI by two percent, by insuring that the interest rate is chosen such that $GNI*(1-S-T)+\delta K + G = 1.02*GNI$. This expresses monetary dominance, whatever deficit a government runs, the central bank can choose an interest rate such that this equation is enforced (ignoring the ZLB).

When a student asked Richard Feynman for advice to graduate students, he wrote a single word on a piece of paper:

Disregard.

His point being that if you want to advance understanding, you should pretty much ignore the consensus. Of course, that is a bit easier to do if when you hand in your thesis you have already produced seminal contributions to Quantum Field Theory and helped build the atomic bomb. Perhaps not great career advice for us lesser mortals. Anyway, my point is that I have really no idea if the idea’s laid out above are common knowledge in economics or not. I like economics, but I have limited time, and its not my only hobby. 🙂 I would like to build an economic model that is tractable and comprehensible and demonstrates monetarism in action. This post is hopefully a first step in that direction. Its certainly the first collection of thoughts that I have managed to gather up into a reasonably coherent framework. I had fun thinking about this stuff, so I hope you guys enjoyed reading about it.