**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:

*p*_{W, t}: The spot price of a bushel wheat for immediate delivery.
*p*_{S, t}: The spot price of a ton steel for immediate delivery.
*p*_{W, t + 1}: The spot price of a bushel wheat for delivery at the end of a year.
*p*_{S, 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 + *r*_{W}) = *p*_{W, t}/*p*_{W, 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:

*p*_{W, t}/*p*_{W, t + 1} = *p*_{S, t}/*p*_{S, t + 1}

Or:

*p*_{W, t}/*p*_{S, t} = *p*_{W, t + 1}/*p*_{S, 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.