Working Paper
of the
Graduate Institute of the Americas
College of International Studies
Tamkang University
US Studies Division
October 2008
Keynes vs. Fisher on the Real Rate of Interest
Professor David Kleykamp
Keynes vs. Fisher on the Real Rate of Interest
I. The Real “Real” Rate of Interest
Most economists identify the concept of the real rate of interest as having been
fully and correctly developed by Irving Fisher.
Of course, there is the
important issue of distinguishing between ex ante and ex post real rates, but the
general concept of the real rate as an inflation-adjusted nominal rate of interest
is as uncontroversial as supply or demand. No serious economist would ever
doubt the notion of the real rate existing or that its definition might be confusing.
Those same economists might nevertheless be surprised to find that J.M. Keynes
felt the Fisherian real rate was not at all clearly defined. Moreover, the idea
that nominal interest rates might not fully adjust to changes in expected inflation
(in a one to one fashion) lies at the very heart of his argument that stickiness of
nominal wages is not the main issue behind less than full employment
equilibrium.
Thus, this seemingly small and rather uncontroversial economic
relation – the Fisher Relation – is the lynchpin on which all modern arguments
about reduced nominal wages lowering unemployment rest.
Keynes’ view about Fisher’s real rate is not taken from some obscure and hidden
passage. It is prominently displayed in Book IV Chapter 11 and Section III of
the General Theory dealing with the marginal efficiency of capital.1
Keynes
Keynes writes – “It is difficult to make sense of this (Fisherian) theory as stated, because it is not
clear whether the change in the value of money is or is not assumed to be foreseen. There is no
escape from the dilemma that, if it is not foreseen, there will be no effect on current affairs; whilst, if it
is foreseen, the prices of existing goods will be forthwith so adjusted that the advantages of holding
money and of holding goods are again equalized, and it will be too late for holders of money to gain or
1
was using this part of the text to explain the effect of a change in prices or wages
on investment. The passages are crucial to his argument that labor is not
capable of lowering unemployment by collectively accepting a lower nominal
wage.
He was adamant that changes in prices affect the real economy directly
through the MEC and not indirectly through the interest rate, or some monetary
or wealth effect, for that matter.2
Economists today would find this odd since,
for them, real saving and investment should be a function of strictly real
variables. Yet, there is more to Keynes’ argument than meets the eye.
Economists typically derive the real rate in a two period model like
Pt Yt  Pt Ct  Pt S t
and
Pt S t (1  Rt 1 )  Pt e1Ct 1  0
with a putative real rate of interest  t 1 being defined such that
(1   t 1 )(1   te1 )  (1  Rt 1 )
where  te1  (
(1)
Pt e1
 1) represents an expected rate of inflation given the price
Pt
level Pt . It follows that the Fisher Relation in (1) is predicated on the
assumption that Pt e1 can change without a concomitant change in Pt .
to suffer a change in the rate of interest which will offset the prospective change during the period of
the loan in the value of the money lent.” (p.142 General Theory)
2
Again, Keynes writes – “The mistake lies in supposing that it is the rate of interest on which the
prospective changes in the value of money will directly react, instead of the marginal efficiency of a
given stock of capital.” (pp.142-143). And, slightly earlier, “This is the factor through which the
expectation of changes in the value of money influences the volume of current output. The expectation
of a fall in the value of money stimulates investment, and hence employment generally, because it
raises the schedule of the marginal efficiency of capital; i.e., the investment demand schedule…”
(pp.142-143 General Theory).
Keynes took strenuous objection to this.
He felt that a change in Pt e1 at time t
would most certainly generate an immediate change in Pt , so that the benefits
from the holding of goods and money would equalize.
As such, there would
then be no reason for Rt 1 to change. Obviously, static expectations,
with Pt e1  Pt , imply  te1  0 and therefore any change in the value of money (i.e.
a change in the price level) will have no effect on the interest rate since nothing is
foreseen – that is neither Pt e1 nor Pt change.
Suppose however, as Keynes
does, that the change in the value of money is perfectly foreseen. In this case,
Keynes claims again that there will be no effect on Rt 1 . To Keynes’ thinking,
perfect foresight means that any change in the expected price level will be
transmitted immediately to current prices.
This means that
Pt e1 Pt

Pt
Pt e1
(2)
which implies that  te1  0 . Again, there will be no effect on Rt 1 .
It is important to realize that (2) above is not saying that expected inflation is
equal to lagged actual inflation. Though these look like growth rates in
expected and actual price levels, they are not.
The familiar symbol ∆ in (2)
refers to a change taking place at a moment in time, not across time.
The best
way to describe (2) is that percentage jumps in both the expected and actual
(lagged) price level are equal.
The individual changes or jumps are not taking
place over time. They are taking place at the instant of time t.
An alternative way of reformulating the Fisher setup, so as to include Keynes’
important point about arbitrage, is as follows:
Pt ( Pt e1 )Yt  Pt ( Pt e1 )Ct  Pt ( Pt e1 ) S t
and
Pt ( Pt e1 ) S t (1  Rt 1 )  Pt e1C t 1  0
which implies
Yt  Ct 
Ct 1 Pt e1
1
0
(1  Rt 1 ) Pt ( Pt e1 )
(3)
This formulation does not contradict the clear and well accepted notion that the
consumer is a price taker.
It is standard to the theory of rational expectations.
This is because the consumer recognizes that his own projection if aggregated
will have a determinant effect on the current price level, about which he is only
vaguely aware.3
As was pointed out above, if the price change is totally
unexpected, then we have static expectations, Pt e1  Pt , whereas if the price
change is fully foreseen, then we have perfect foresight Pt e1  Pt 1 and there will
be attendant arbitrage among producers such that (assuming no transactions
costs) results in Pt  Pt 1 , implying that Pt e1  Pt .
In both cases, the interest
rate remains unaffected. However, in the later case, as Keynes states, the
3
Some economists maintain that, as we are constantly consuming, we must have full and precise
knowledge of the current price level, Pt. Thus, according to them, Pt in (3) above is known and
determinant at time t. But, if such were the case, it would be easy and efficient for the Department of
Commerce to estimate the price level through sampling by phone a few hundred consumers and taking
their views of what the price level is currently. Instead, the Department goes through an extensive and
very expensive sampling throughout the country and publishes these findings often with extensive
revisions months later. The argument that the public knows P t at time t is very tenuous at best.
Keynes, by contrast, does not make explicit the arbitrage relation establishing P t(Pet+1), since any such
attempt is doomed to failure by similar reasoning. This despite the fact that such an arbitrage no
doubt exists.
nominal interest rate Rt 1 cannot adjust in such a short time to compensate
savers or lenders for the change in purchasing power, as can be seen from (3).
The fact that Keynes saw Fisher’s “real” rate of interest as problematic no doubt
(among other things) led him to abandon the distinction between nominal and
real rates of interest – a distinction that Milton Friedman and others came to
believe was very important. But, contrary to what Friedman thought, this was
not due to a failure to distinguish real and nominal magnitudes on Keynes part,
though it may have been perceived as sloppiness on the part of Keynes’ later
disciples.4
He purposively avoided the distinction since he felt it was a false one.
For Keynes, there was only one rate of interest and it measured the strength of
people’s desire to part with liquidity.
How then is inflation reflected in the nominal rate of interest? Surely, a person
as astute as Keynes, who had advised Finance Ministers, had been the bursar of
Cambridge, and had written on inflation, hyperinflation and, the influence of
money on the economy, could not avoid the fact that nominal interest rates
tended to track inflation quite well.
For modest changes in the value of money,
his answer must clearly be that the interest rate is indirectly affected through the
MEC (see footnote 2 above).
For example, consider the case where current output prices increase steeply,
whereas current capital prices have yet to react.
4
Firms require a longer period
Friedman (1982, p.49-51) notes that Keynes avoided the distinction between real and nominal
magnitudes by couching his analysis in terms of wage units which he took as inflexible for a number of
seemingly ad hoc reasons. Friedman goes on to claim that Keynes’ disciples quite often failed to
make this important distinction between real and nominal magnitudes – even when their central results
depended on such distinctions.
of inflation (not simply a jump in the price level) before they accept that it will be
profitable to invest in new capital. The lag between the rise in output prices
and the rise in capital prices should be dependent on the conviction which
entrepreneurs hold these expectations. If firms believe that the output price
rise is merely a jump, then they will have little faith that the expectation will last
and therefore capital prices will not be affected very much.
This is similar to
Friedman’s distinction between permanent and transitory variables as applied to
output prices (or prospective profits from investments). However, provided the
rise in output prices is expected to last for a reasonable period, it will naturally
generate an increase in the MEC, thus raising investment demand.
Of course,
the price of capital goods will naturally rise along with investment demand and
moderate the former increase in the MEC. Nevertheless, the rise in investment
demand will lift national income and will subsequently raise the demand for
money thus increasing the nominal interest rate through the expanded sale of
debt instruments (to banks), or what is commonly called the Keynes’ Effect.5
It
is through this mechanism that the nominal interest rate tracks inflation and not
through the Fisher effect, at least according to Keynes. Certainly there is no
reason to expect that the nominal rate of interest will move one for one in tandem
and contemporaneously with the movement in prices. Changes in prices must be
seen as exhibiting a permanent or long lasting increase in growth of the price
index in order to be effective at stimulating investment and thus becoming
embedded in nominal rates.
The scenario above also points up an important distinction between output
5
Leijonovud (1967) has strongly argued that this sequence is wrong. His point is that the change in
the MEC will either react directly on the loanable funds market thus raising the interest rate, or will
prices and capital prices.
All too often there is little distinction among
economists between output price inflation and capital price inflation.
It is
precisely this distinction that is being given close scrutiny today due to the
current housing debacle in the US.
Keynes theory asserts that output prices
rise first and subsequently cause capital prices to rise. In fact, it is not output
prices per se that rise and then cause capital prices to rise; rather it is prospective
profits which may be caused by higher output inflation or greater output growth.
This in turn leads to increased money demand and a rise in interest rates. It
was not an increase in output prices per se that caused the rise in interest rates
directly.
Note that Keynes’ argument does not rely upon savers being highly
sensitive to interest rates and thus to small changes in inflation (foreseen or
otherwise). Likewise, the general public does not need to be super sensitive to
the inflation cost of holding money balances in order for price expectations to
affect the economy. The effect is manifested through the MEC and is
transmitted to interest rates by firms and investors who ARE sensitive to small
changes in expected prices.
More importantly this example shows how that an
inflationary pressure can lead to asset bubbles in land, commercial structures,
and housing -- which then increase nominal interest rates unless accommodated
by the monetary authority.
How might one use data to falsify Keynes’ theory about the way that inflation
affects interest rates?
Given Keynes’ formulation of the MEC, a predictable rise in output prices at the
same rate as capital prices would clearly have no effect on the MEC and thus
ceteris paribus there would be no increase in investment demand (over and above
its current level). Rising nominal interest rates would tend to support Fisher’s
hypothesis whereas stable interest rates would tend to support Keynes’ theory.
This is only an idealized way of confirming one theory or the other. The actual
test would depend on numerous other factors.
Additional confirmation of Keynes’ theory would come from a situation where
lagged output prices were highly correlated with current capital prices, which
were then correlated with a lag with current interest rates. That is,
Highly Lagged Output Inflation → Lightly Lagged Capital Price Inflation
→Current Level of Nominal Interest Rates
Interestingly, data on US inflation, housing price inflation, and Moody’s AAA
bond rate show precisely this pattern. By contrast, the Fisher Effect requires
Current Output Inflation → Current Level of Nominal Interest Rates
There are at least two things interesting about Keynes’ argument.
First, Keynes is implicitly using an arbitrage relation which makes the current
price level at least in part dependent on the expected future price level.
He does
not make this arbitrage relation explicit, but he explains things in terms of a
balance between the desire to hold money and the desire to hold goods. Thus,
the typical modern assumption of describing Keynesian liquidity preference as
strictly a choice between money and bonds (broadly defined) is not true to
Keynes’ own writings in the General Theory.
Moreover, modern Keynesians
have emphasized over and over that it is the rigidity of prices, particularly
downwardly rigid prices and wages that account for real effects in the economy.
Nearly all economists today would agree that if prices are allowed to adjust
quickly, they will ensure a stable, full- employment equilibrium. This is the
conclusion we get from most modern Keynesian models and it has become
virtually axiomatic in current macro thinking. However, Keynes is arguing
here that mere changes in expectations of price changes (if generally held) will
lead to actual current prices adjusting.
adjusts very quickly indeed.
Keynes is arguing for a price level that
This is a far cry from the price rigidities typically
asserted by the New Keynesians.
Second, by making the current price level dependent on the future price level,
Keynes is giving a passing nod to rational expectations, although it is true that he
eschewed the explicit use of objective probability models as applied to business
and economic decision making.6
Nevertheless, one only needs an explicit
arbitrage model, mathematically relating Pt to Pt e1 , to complete the model
Keynes is asserting and one naturally is led to a rational expectations solution
Indeed, Muth’s original research in rational expectations is devoted to the
question of the speculative holding of inventory which is at the heart of Keynes’
argument above.
6
This was due to his belief that typically probabilities of economic events occurring could not be
compared with each other and that such probabilities were not sufficiently stable to allow a
mathematical formulation.
Keynes might have been thinking along the lines of commodity arbitrage
between spot and futures markets, although this is not explicit in his writings.
Instead, he chooses to couch his argument in the balance between producers who
decide when to bring goods to markets (thus seeking money) and consumers who
decide when to bring money to markets (thus seeking goods). The currently
price level adjusts to the changing expectation of future price movements to
bring these two to an equilibrium.
It is now possible to state the difference between the real rate of Keynes and that
of Fisher.
Fisher’s real rate is the textbook version where any percentage
change in the expected price level is fully reflected as a change in the expected
rate of inflation. It follows that Fisher’s ex ante real rate can be defined as
 t 1  (1  Rt 1 ) /(1   te1 )  1
The Keynesian ex ante real rate is perhaps best seen as
 t 1  (1  Rt 1 ) /(1  (1   ) t 1 )  1
where  is governed by the speed and magnitude by which the expected price
level affects the current price level.
A value of   1 corresponds to the case
where the current price level immediately and fully adjusts to changes in the
future expected price level so that the interest rate is unaffected and there is no
benefit from holding more goods or more money.
A value of   0
corresponds to the case where current prices are completely unaffected by future
expected prices and there is absolutely no arbitrage between goods now and
goods in the future.
Neither of these two extremes seems likely, but the Keynesian formulation
appears more general and thus there is no reason to expect that the familiar
Fisher Relation will hold in general.
should be a constant over time.
In addition, there is no reason why that 
Finally, the value of  is seen to depend on
the composition of goods in the economy. If most goods are nondurable, then
one would expect that  would be close to zero since arbitrage would not be
possible and the Fisher relation would tend to hold. However, if storage is
possible and durable goods constitute a great chunk of consumption, we would
expect  to be closer to 1 and thus the Fisher relation would not tend to hold.
II.
Jumps Versus Growth in the Price Level
As Section I has shown, Keynes felt that Fisher’s notion of the real rate of
interest was at best a vague concept since if changes in the value of money in the
future are perfectly foreseen, then the current value of money would adjust
immediately, and this would leave the nominal rate of interest unchanged. The
idea, that the current price level can adjust so quickly, leads to the issue of how
one might distinguish between jumps in the price level and secular changes (or
growth) in the price level.
The effect of a perfectly foreseen jump in the price
level on the nominal interest rate is profoundly different than one that is slow,
smooth, and predictable.
Jumps in the price level can be expected to occur for many reasons.
For
example, a change in a general sales tax or a jump in the price of oil can both
lead to once and for all jumps in the price level.
If the government was
expected to devalue its currency once and for all at some later date, and this
expectation was held with great conviction by a great many people, the current
price level might well experience a jump.
It is problematic if and how that this
jump might become incorporated into the nominal rate of interest. The typical
method of simply treating this as a foreseen increase in expected inflation, and
adding this in a Fisherian way to the existing nominal rate, does not make sense.
Yet, bond holders need and expect to be compensated in a fair and rationally
competitive way for the loss in purchasing power of their holdings.
Figure 1 below shows an upward jump in the price level Pt which is assumed to
occur at to and is perfectly foreseen by everyone. Before to inflation (both
expected and actual) is zero and the same remains true after to. It is only at to
that there is any change at all, and this is not inflation, but represents instead a
jump in the price level.
Clearly any bond maturing before to, say at time T’,
Figure 1
will be completely unaffected by the jump at to.
For simplicity, assume that the
nominal rate of interest before to is Rt = 0. Since both actual and expected
inflation is zero before to, the real rate is similarly equal to zero. By
contrast, any bond issued before to but maturing after time to, say at time T, must
incorporate in some way the loss of purchasing power of the face value and fixed
coupons. The exact way in which this occurs will depend on how often interest
is paid and how much time exists between issuance and the price level jump.
For bonds that are issued after to the nominal interest rate is again unaffected
since inflation is zero.
While the argument above seems perfectly reasonable, it nevertheless begs the
Figure 2
subtle question raised by Keynes. That is, if the jump is known or perfectly
foreseen by all, then why should we expect that the price level before to remains
constant? For example, if we know that devaluation of the currency will occur
at to, it is preposterous to expect that the price level before to will remain
unaffected. Some sort of rise is to be expected as well. This rise in the price
level before to will no doubt have an effect on the nominal interest rate Rt -- but,
by how much and through what mechanism?
This is the key to understanding
the difference between Keynes’ and Fisher’s notions of a real rate of interest.
Figure 2 shows how that arbitrage can close the original gap. This process of
raising the price level before the jump and lowering the price level after the
jump is due to arbitrage and therefore should be understood to all parties.
What is not known is the extent to which this gap can be closed.
Thus, a
residual jump remains, which depends on factors outside the scope of the
participants.
If the original gap is only vaguely foreseen, then the narrowed gap
also becomes a random residual to the process of arbitrage.
In addition, there
is no reason why rates of growth in the price level should be the same before and
after the jump.
III.
Wage Rigidities and Full Employment
Keynes’ belief (that Fisher was wrong about the real rate of interest) is important
since it under girds his theory how that changes in the value of money affects the
economy by affecting the desire to investment.
In particular, Keynes was
adamant that a general fall in nominal wages would fail to bring about greater
employment if such reduction in wages was expected to continue into the future.7
The standard way in which the MEC is used to determine whether a project is
carried out, or not, is by the following
R1e
R2e
C

(1  m) (1  m) 2
where C is a nominal price of the asset and the R’s are the nominal prospective
yields or profits earned by using the asset. Most economists would reject the
use of nominal figures and would transform the above to its real form as the
following
Keynes writes – “If, on the other hand, the reduction leads to the expectation, or even to the serious
possibility, of a further wage-reduction in prospect, it will have precisely the opposite effect. For it
will lead to the postponement both of investment and of consumption”. (p. 263 General Theory)
7
C

p1e (1   e ) p 2e (1   e ) 2

(1  m)
(1  m) 2

p1e (1  mr )  p 2e (1  mr ) 2

p1e
p 2e

(1  mr ) (1  mr ) 2
where mr  m   e
The real MEC would then be compared to the Fisherian real rate to determine
whether or not to undertake the investment.
Furthermore, if the nominal MEC
and the nominal interest rate are merely assumed to be equal to real rates plus
an (independent) expected inflation rate, then changes in expected inflation
cannot affect investment. General movements in nominal wages that are
reflected in the price level cannot affect investment and hence aggregate demand.
The effect of a general fall in nominal wages will then increase aggregate supply
without any significant change in aggregate demand. Hence, a fall in nominal
wages will increase employment and output. This is quite contrary to Keynes’
belief that a general fall in money wages (if expected to continue into the future)
will reduce investment and thus aggregate demand.
IV. A Suggested Resolution
It would be wrong to suppose that the growth of the price level cannot be well
predicted, nor would it be right to say that jumps must all be unforeseen.
Reason dictates that part of the movement in the price level is predicted well
while another part is unforeseen.
Keynes has said as much by noting that an
expected jump in the future price level that is foreseen must lead (at least to some
extent) to a concomitant rise in the current price level, via some implicit
arbitrage. This represents a smoothing of the jump, making the rise less of a
clear break and more of a continuous expected growth. It follows that a price
level series may be decomposed into smooth expected growths and sudden
unforeseen jumps. Very large jumps in the price level must by their nature be
unforeseen, since if they were foreseen they would be properly smoothed by
arbitrage. The typical method of taking the nominal interest rate and
subtracting growth in the overall price level to arrive at an ex post real rate
misses the all important aspect of the problem; viz. distinguishing between
unexpected jumps and expected growths in the price level.
What is more, expectations of a future jump in prices may only lead to a modest
rise in the current price level if such expectations are not held with great
conviction by all of the public. To provoke an arbitrage, the conviction the
public has about the movement of future prices must be sufficient to overcome
obvious costs to carrying out the arbitrage, even with fully developed futures
markets. Expectations that are not held with great force will not be capable of
affecting the current price level much and thus will not affect current nominal
rates of interest. This is where rational expectations models fail to follow
through since they typically consider only the expectation of the public and not
the strength of the conviction in which the public holds such expectations.
One simple way of decomposing the unforeseen jump from foreseen growth
using a time series on the price level is as follows
Pt e1  (1  growtht 1 ) Pt
and
Pt 1  Pt e1  Jumpt 1
That is, given a reasonable and appropriate way of constructing a series on
foreseen growth, which is consistent with Keynes’ arbitrage above, the resulting
jump is then calculated as a residual.
The real rate of interest would then be
the nominal interest rate minus such foreseen growth in the price level.
If
growth in the price level is equated with the actual rate of inflation, then the
unforeseen jump becomes zero and one has perfect foresight (   0) .
On the
other hand, if growth is equal to zero, then the jump is the entire (actual) change
in the price level (   1) .
These two situations represent polar extremes for the
foreseen growth rate of the price level.