By His Bootstraps

1.0 Introduction
One can hold savings in various forms of assets. In effect, savings is a time machine for transferring purchasing power into the future. A debt, when purchased - that is, a bond - is one such asset in which one can store savings. The relationships between bonds of various maturities and the existence of well-developed markets in which to trade bonds allows the determination of interest rates without relying on the theory of time preference, a theory which is a lot of utter hogwash anyhow.

The point of this post is to explain how, under certain institutions for selling second-hand debt, a relatively stable long-term interest rate can be maintained by beliefs in its stability. No need arises to call on the forces of thrift and productivity. This post might even be relevant to current events in the USA.

2.0 Institutions Providing a Setting in Which a Second Decision Must be Made
The model I outline here is based on Keynes' account of the two decisions a saver must make:

"The psychological time-preferences of an individual require two distinct sets of decisions to carry them out completely. The first ... determines for each individual how much of his income he will consume and how much he will reserve in some form of command over future consumption. But this decision having been made, there is a further decision which awaits him, namely, in what form he will hold the command over future consumption which he has reserved, whether out of his current income or from previous savings." -- J. M. Keynes, The General Theory of Employment, Interest and Money (1936): p. 166

Assume that the debts of the best quality available for purchase consist of Treasury bills (T-bills) that mature in three months, T-bills that mature in a year, and Treasury notes (T-notes) that mature in 10 years. These are all available in the U.S.A., along with T-bills, T-notes, and T-bonds of other maturities. In this exposition, I abstract from the existence of these other maturities. By including debts of these three maturities, the model incorporates the decision to hold money, assets that pay the short-term interest rate, or assets that pay the long-term interest rate.

In describing three-month T-bills as money, I again follow Keynes:

"...we can draw the line between 'money' and 'debts' at whatever point is most convenient for handling a particular problem. For example, we can treat as money any command over general purchasing power which the owner has not parted with for a period in excess of three months, and as debt what cannot be recovered for a longer period than this; or we can substitute for 'three months' one month or three days or three hours or any other period; or we can exclude from money whatever is not legal tender on the spot. It is often convenient to include in money time-deposits with banks and, occasionally, even such instruments as (e.g.) treasury bills." -- J. M. Keynes, The General Theory of Employment, Interest and Money (1936): p. 167

Suppose, contrary to fact, that the short term interest rate, r, was known to be constant for the next ten years, where 100 r is stated as an annual percentage. Then the long term interest rate would be established in the market at the start of the year as 100 [(1 + r)¹⁰ - 1] percent for 10 years, and the interest rate on money would be 100 [(1 + r)^1/4 - 1] percent for three months. A higher price on a bond corresponds to a lower interest rate. For example, the price of a T-bill with a face value of $1000 to be paid in a year is 1000/(1 + r) dollars.

3.0 The Individual
In this model, federal authorities set the interest rate on money. The short term interest rate provides a market consensus on monetary policy is likely to be over the next year. If the annual interest rate embodied in the price of one-year T-bills is higher than the annualized interest rate on money, the market price of T-bills is predicting a tightening of monetary policy. The individual allocates his savings partly on his opinion of this consensus. If he thinks, for example, that the monetary authority is not going to tighten that much, he would sell three-month T-bills and buy one-year T-bills, so as to make a profit from speculation when the price of the latter rises.

The individual, one assumes, has some idea of what is a normal long-term interest rate. He expects that over a long enough period, the federal authority's monetary policy will average out, thereby achieving this normal rate. The individual expects the price of T-notes to eventually rise when the current long-term interest rate is above that normal long-term rate and to fall when the current long-term rate is below that normal rate. Here, too, the possibility for speculative gains influences the individual in his allocation of his savings between T-notes and T-bills.

3.0 Markets
Consider a range of the price of T-notes. For a high enough price, those who are bears on this market (who expect the long term interest rate to rise) would dominate the bulls (who expect the long term interest rate to fall). More would be selling than buying, and the price would fall. The opposite is true for a low enough price. The equilibrium price at an instant of time balances bulls and bears:

"In the Treatise [Keynes] pictures the Bulls and Bears of the gilt-edged market going into and out of bonds as they individually come to think that the next price movement will be up or down. In this speculative market the price of bonds and thus their yield, the interest rate, can only settle if opinion is divided, so that those who wish to sell for fear of a fall find their offers matched by the bids of those who wish to buy in hope of a rise. It is thus, as Keynes says, a variety of opinion in the gilt-edged market which gives stability to the interest rate and some control over it to the monetary authorities." -- G. L. S. Shackle, "Simplicity in Keynes's Theory of Money and Employment", The South African Journal of Economics, v. 51, n. 3 (1983): 357-367

Elsewhere Shackle talks about equilibrium in such a speculative market as inherently restless.

4.0 Conclusions and a Policy Implication
I suppose one could express the above model in mathematics, if one were so inclined. One might start with some distribution of agents' beliefs about the conventional long term interest rate, and allow each agent to slowly update their view, maybe with the addition of random noise. (One might draw on Shackle's "The Bounds of Unknowledge" (in Beyond Positive Economics (ed. by J. Wiseman) Macmillan, 1983) in specifying this updating.) And the agents would decide on the distribution of their savings based on their views. Maybe the model should have more types of assets. One would want a model in which a diversity of opinion is maintained among agents, and in which time series for stock equilibria exhibit hysteresis and non-ergodicity. It wouldn't surprise me if somebody has already published such a model.

Keynes had something to say about policy based on this sort of analysis:

"Thus a monetary policy which strikes public opinion as being experimental in character or easily liable to change may fail in its objective of greatly reducing the long-term rate of interest... The same policy, on the other hand, may prove easily successful if it appeals to public opinion as being reasonable and practicable and in the public interest, rooted in strong conviction, and promoted by an authority unlikely to be superseded." -- J. M. Keynes, The General Theory of Employment, Interest and Money (1936): p. 203