## Friday, August 03, 2018

### A Unique Natural Rate Of Interest?

1.0 Introduction

In explaining the policy implications of the Austrian Business Cycle Theory, Hayek argued that the central bank should try to keep the money rate of interest rate equal to the natural rate. Sraffa famously criticized Hayek by describing a model with multiple interest rates, not necessarily all equal. In reply, Hayek asserted that all the interest rates in Sraffa's example would be equilibrium rates. Sraffa had a rejoinder:

"The only meaning (if it be a meaning) I can attach to this is that his maxim of policy now requires that the money rate should be equal to all these divergent natural rates."

This interchange was part of the downfall of the Austrian theory of the business cycle. I thought I would try to shortly explain what is and is not compatible with a unique natural interest rate.

2.0 Multiple Interest Rates Compatible with a Unique Natural Interest Rate

When one talks about the interest rate or the rate of profits, one may be abstracting from all sorts of complications. And these complications may be consistent with multiple interest rates, in some sense. Yet these multiple interest rates were not in dispute between Hayek and Sraffa.

2.1 Interest Rates for Loans of Different Lengths

Suppose at the start of the year, one can obtain a one-year loan of money for an interest rate of 10%. At the same time, one can obtain a two-year loan for 21%. Implicit in these different rates is a prediction that a one-year loan will be available at the start of next year for an unchanged interest rate of 10%. This implication follows from some trivial arithmetic:

1 + 21/100 = (1 + 10/100)(1 + 10/100)

The yield curve generalizes these observations. A certain shape, with the interest rate increasing for longer loans is consistent with the interest rate being expected to be unchanged, for loans of a standard length, over time.

2.2 Interest Rates for Loans of Different Risks

One might also find interest rates being higher for loans deemed riskier, independently of the time period for which the loan is made. This variation is consistent with talk of the interest rate. Often, in finance, one sees something called the risk-free rate of interest defined and used for discounting income streams. In practice, the rate on a United States T-bill is taken as the risk-free rate.

2.3 Rate of Profits

One can also distinguish between finance and business income. One might refer to the interest rate for the former, and the rate of profits for the latter. Kaldor and others, in a dispute over a Cambridge non-marginal theory of the distribution of income, have described a steady state in which the interest rate is lower than the rate of profits. Households lend out finance to businesses and obtain the interest rate. Such a steady state is compatible with the existence of two classes of households. Capitalist households receive income only from their ownership of firms.

2.4 Rates of Profits Varying Among Industries

Steady states are also compatible with the rate of profits varying among industries, as long as relative profit rates are stable. Perhaps some industries require work in more uncomfortable circumstances. Or perhaps firms are able to maintain barriers to entry.

3.0 Interest Rates with Different Numeraires

I have shown above how money interest rates for loans of different lengths embody expectations of the future course of money interest rates. Interest rates need not be calculated in terms of money. They can be calculated for any numeraire. And the ratio of real interest rates embody expectations of how relative prices are expected to change.

As an example, suppose that at a given time t, both spot and forward markets exist for (specified grades of) wheat and steel. One pays out dollars immediately on both spot and forward markets. Consider the following prices:

• pW, t: The spot price of a bushel wheat for immediate delivery.
• pS, t: The spot price of a ton steel for immediate delivery.
• pW, t + 1: The spot price of a bushel wheat for delivery at the end of a year.
• pS, t + 1: The spot price of a ton steel for delivery at the end of a year.

The wheat-rate of interest is defined by:

(1 + rW) = pW, t/pW, t + 1

I always like to check such equations by looking at dimensions. The units of the numerator on the right-hand side are dollars per spot bushels. The denominator is in terms of dollars per bushel a year hence. Dollars cancel out in taking the quotient. The wheat interest rate is quoted in terms of bushels a year hence per immediate bushels.

Suppose all real interest rates are equal. So one can form an equation like:

pW, t/pW, t + 1 = pS, t/pS, t + 1

Or:

pW, t/pS, t = pW, t + 1/pS, t + 1

If spot prices a year hence were expected not to be in the ratio of current forward prices, one would expect to be able to make a pure economic profit by purchasing some goods now for future delivery. Hence, a no-arbitage condition allows one to calculated expected relative prices from quoted prices on complete spot and forward markets.

Anyways, a steady state requires constant ratios of spot prices and, thus, real interest rates to be independent of the numeraire. This is the condition Hayek imposed in his exposition of Austrian business cycle theory in Prices and Production. And this is the condition that he dropped in his argument with Sraffa, leaving his macroeconomics a confused mess.

I might as well note that a steady state is consistent with constant inflation. If all prices go up at, say, ten percent, relative spot prices do not vary. On the other hand, relative spot prices differ with the interest rate in comparisons across steady states.

4.0 Temporary Equilibrium with Consistent Plans and Expectations

Perhaps Hayek was willing to get himself into a muddle about the natural rate because he had already investigated another equilibrium concept in previous work.

Suppose above that real interest rates vary among commodities. Then forward prices show expected movements in spot prices. One might go further and assume a complete set of forward markets do not exist. Markets clear when each agent believes they can carry out their plans, consistent with their expectations, including of future spot prices. Should one call such market-clearing an equilbrium, even if agents plans and expectations are not mutually consistent?

Concepts of temporary, intertemporal, and sequential equilibrium were to become more important in mainstream economics after Hayek quit economics, more or less. John Hicks was a major developer of these ideas, under Hayek's influence at the London School of Economics. He eventually came to accept that the mainstream notions could not be set in historical time and were, at best, of limited help in understanding actual economies.

5.0 Conclusion

The above has outlined multiple ways in which multiple interest rates and multiple rates of profits are compatible with steady states. Nevertheless, such circumstances are often described by models in which one might talk about the rate of interest.

I have also described an equilibrium in which one cannot talk about the interest rate, whether natural or not. Advocates of Austrian business cycle theory have never clarified how it can be set in a temporary equilibrium. One can sometimes find Austrian fanboys asserting that critics do not appreciate distinctions between:

• Sources of exogenous shocks in central banks and supposed determinants (inter temporal preferences, technology) of the natural rate
• Money rates of interest and real rates
• Subjectivism and objectivism
• Interest rates and relative prices.

But assertions do not constitute an argument. One would have to do some work to show that these distinctions can serve to rehabilitate Austrian business cycle theory. No matter how much you send somebody chasing through the literature by Kirzner, Lachmann, Jesus Huerta de Solo, and Garrison, they will find the work has yet to be done. (Robert Murphy probably knows this.)

References
• Hahn, Frank. 1982. The neo-Ricardians. Cambridge Journal of Economics 6: 353-374.
• Hayek, F. A. 1932. Money and Capital: A Reply. Economic Journal 42: 237-249.
• Kaldor, Nicholas. 1966. Marginal Productivity and the Macro-Economic Theories of Distribution: Comment on Samuelson and Modigliani. Review of Economic Studies 33(4): 309-319.
• Sraffa, Piero. 1932. Dr. Hayek on Money and Capital. Economic Journal 42: 42-53.
• Sraffa, Piero. 1932. A Rejoinder. Economic Journal 42: 249-251.

Blissex said...

From my point of view, my understanding of "neoclassical" Economics, that is not quite right.

The assumption is that there is a single rate of riks-free interest which coincides with the risk-adjusted rate of profit. The argument are

* If the risk adjusted rate of profit were different from the risk-free rate of interest then investors would arbitrage it away by switching into financial or real investment whichever gives the better returns.
* If there were different risk-free interest rates then both lenders and borrowers would arbitrage it away too, borrowing at the lowest rate or lending at the highest rate.

Both arguments depend critically on perfect intertemporal "tatonnement", which is the opposite of the hayekian observation that information is diffuse and markets are information discovery mechanisms.

Equivalent to perfect intertemporal "tatonnement" is the assumption of "rational expectations", that is that on average predictions about the future come true.

Also it is difficult to speak meaningfully of "the" interest rates (even in the plural) without taking into account the spread between active and passive rates.

Things become clearer considering two points of JM Keynes and one of H Minsky:

* That "liquidity" is not the same as "money", and
active interest rates are the price of "liquidity",
not of "money".
* That much of the future is radically uncertain and
therefore any calculation of discounted valued of
investments is largely arbitrary.
* That "money" is institutionally defined, and there
therefore there are many types of "money" in a
political economy.

An interesting quote from H Minky:

«A fundamental insight of Keynes is that an economic theory that is relevant to a capitalist economy must explicitly deal with these two sets of prices. Economic theory must be based upon a perception that there are two sets of prices to be determined, and they are determined in different markets and react to quite different phenomena. Thus, the relation of these prices - say, the ratio - varies, and the variations affect system behavior.
When economic theory followed Sir John Hicks and phrased the liquidity preference function as a relation between the money supply and the interest rate, the deep significance of Keynesian theory as a theory of behavior of a capitalist economy was lost.»

Bonus JM Keynes quote:

«But, having given the reason why the money-rate of interest unlike most commodity rates of interest cannot be negative, he altogether overlooks the need of an explanation why the money-rate of interest is positive, and he fails to explain why the money-rate of interest is not governed (as the classical school maintains) by the standard set by the yield on productive capital. This is because the notion of liquidity-preference had escaped him. He has constructed only half a theory of the rate of interest.»

Blissex said...

«perfect intertemporal "tatonnement", which is the opposite of the hayekian observation that information is diffuse and markets are information discovery mechanisms»

I'll try to complement the above to make explicit a very important "detail" the importance of which may be underestimated a bit by our idealistic blogger. What I am going to say was implicit in many of my previous comments, but now perhaps the time time make it more explicit.

When Economics was being formalized the dominant paradigm was that of laplacian rigid body mechanics, and Walras and Marshall ended up fantasizing that the political economy was a mechanical system, an orrery.

The problem is that it describing the political economy and in particular figuring out the effects of policy means solving an optimization problem, a "variational" problem.

That creates the question of whether there is a single maximum or many local maxima, and whether the variational problem can be solved with global information or only with local information.

Unfortunately the central truthiness of Economics is that income, absent government distortion of markets, is exactly and uniquely earned by productivity. That requires that either:

#1 there be no local maxima, but a single global maximum;
#2 if there are local maxima, that the state of the political economy always and rapidly converges to the globally highest local maximum, which requires global information.

Then #1 implies assuming extreme restrictions on Economics models to ensure their "convexity", such as a single agent and a single capital good ("leets"), and "rational expectations" for the intertemporal case.

Then #2 implies the assumption of global information, that is an omniscient auctioneer ensuring that "tatonnement" always converges rapidly to the highest local maximum, which can also be achieved by assuming "rational expectations" and a single agent, in which case the single agent is both the only market participant and the omniscient auctioneer.

The real meaning of Sraffa's little book and the reason why he subtitled it "Prelude to a critique of economic theory" is that he showed that even in the simplest models, even without marginalism, multiple local maxima ("reswitching") arise naturally, and therefore all models predicated on some form or another of "convexity" are bullshit.

Note: that was actually well known before him, but he just demonstrated it with particularly simple and unobjectionable examples. The truly original and huge achievement of his book is that he defined a way to create a composite commodity, a "numeraire" whose value does not depend on the distribution of income, which was what classical political economists had been seeking for centuries.

The problem with F Hayek's late re-discovery of markets as discovery mechanisms (which was well understood before him, but not as well expressed) is that it results in achievable global maximum, just imperfect local maximum seeking and thus the impossibility of the central truthiness of Economics.

Another note: I try to be careful about using the terms "Economics" (a despicable form of propaganda) and "political economy" (an interesting topic of investigation and research). Also I am afraid that most of my points above will be utterly incomprehensible to people indoctrinated by courses in Economics. Much has been lost...

Blissex said...

«is that it results in achievable global maximum, just imperfect local maximum seeking»

That should have been "in NO achievable global maximum".

BTW the work JM Keynes in general and as quotes above was came about when he realized, because of obvious events and his reflections on "money", that a single global maximum is unachievable. In his theory the two major reason why are:

* Demand for investment is subject to "uncertainty", and that makes it impossible to find an intertemporal maximum; that is as uncertainty can change unpredictably, the optimization landscape changes with time, creating path dependency.

* Since demand for "money" and demand for "liquidity" are different things in different markets, and "liquidity" (by construction) is unproductive, when businessmen switch "money" from spending on investment to hoarding "liquidity" they create a completely different local maximum in which the political economy can get trapped.

Robert Vienneau said...

Certainly, most academic mainstream economics is propaganda.

I think Sraffa is important for more than an internal critique of neoclassical economics. You can read him as rediscovering the logical structure of the theory of value and distribution in classical political economy.

The role and importance of his specially constructed numerate is open to debate. It varies with time and the technique in use. You can read it as demonstrating that no solution of Ricardo's problem is possible, rather than solving the problem. And then there is its connection with Marx's labor theory of value. If aggregates are in standard proportions, a lot of volume I of Capital makes sense.