I've read various things recently on the theory of loanable funds and the natural rate of interest, so I thought I'd say something on it.
I want to start by looking at how we might illustrate a market of loanable funds in a typical demand and supply graph, with quantity on the horizontal axis and some benchmark rate of interest on the vertical axis. The supply curve then shows the amount that people wish to save at each interest rate, other things being equal. The demand curve shows how much people wish to borrow at each interest rate, other things being equal.
I think this graph makes a kind of sense. However, we have to be careful. Normally, we might use such a graph to show how a change in either demand or supply would lead to a change in price (the interest rate) so as to restore equilibrium. Let's suppose that this graph showed demand and supply for apples instead. Then, if the demand curve shifts to the right say, we might expect a rise in the price of apples, changing quantities supplied and demanded until they were equal again.
Bear in mind here the assumption that all other things are equal. In fact this is an assumption that is almost certainly incorrect. An increase in demand for apples requires changes in demand and supply in some other market. You can't just demand more apples; you have to demand more apples instead of something else - bananas say (or leisure time). So the demand curve for bananas will also move, with implications for the price of bananas. That in turn will impact on the demand for apples. So clearing in the apple market is not brought about exclusively by a change in the price of apples, but by changes in all markets.
This is particularly important when thinking about the loanable funds market. Critically, one of the things assumed constant is the level of income. So the supply curve shows the amount people wish to save at any given level of income. Now we need to consider what happens if people wish to save more. On the face of it, this would lead to a shift in the supply curve to the right. However, like with the apples, you can't just save more - there has to be some counterpart, which in this case must mean spending less. And spending less results in lower incomes.
This means that if we want to consider an increased desire to save, we cannot simply represent this as a rightward shift in the supply curve leading to a fall in the rate of interest. In fact, the income implications of an increased desire to save may result in both the demand and supply curve shifting leftward.
So we need to be careful about applying the standard story to an interest rate clearing the market for loanable funds.
However, if that were all, there would still be some value in drawing a demand and supply graph for loanable funds. We might show curves that represented demand and supply on the assumption that all other markets were clearing, including in particular the employment market. This would then tell us what level of interest would prevail in that situation. It would not tell us how that level of interest came about (and we know to be suspicious of the idea that it arises from supply and demand pressures in that market), but we would know that if every market was clearing, then that must be the rate that would hold.
This rate would then be the natural rate of interest - the rate of interest at which planned savings equals planned borrowing, under conditions of full employment. You might want to invoke rates of time preference or marginal efficiency of capital, but this is not really necessary for a natural rate to exist.
Nevertheless, it is not clear that such a rate does exist. If we make certain assumptions about household preferences or production functions, we can certainly show that there is a rate which meets this condition. But these assumptions are completely ad hoc, and there is no reason to believe they reflect the real world better than some different assumptions. It is entirely possible that when we draw out the demand and supply curves for loanable funds, that we find that there is no rate for which demand equals supply under full employment. For example, the graph may look like this.
If that were the case, then full employment would be impossible.
Much of the current consensus approach in economics relies on the idea that full employment (and price stability) can be achieved by setting the market interest at the natural rate. It may not matter how such a rate is determined, but is clearly essential for this that such a rate does in fact exist. It is far from clear to me that it does.
On the whole, I find objections to loanable funds and the natural rate of interest overdone. As theoretical concepts, I think they have their place, if used correctly. However, I am very sceptical of the idea that the monetary policy is all about matching the market rate to the natural rate. This is mainly because I have doubts about the stability of the relationships involved. But at a more basic level, it's also because I don't think we can take it for granted that there is in fact any interest rate that can achieve clearing in all markets.