TOBIN VS. KEYNES ON LIQUIDITY PREFERENCE R. L. Crouch *

I The Respective Positions T HE demand to hold money as an asset

(when interest bearing bonds are also available) has been rationalized in different ways by John Maynard Keynes and James Tobin. Keynes based his explanation on in- elastic bond price expectations. This makes bond prices (the interest rate) "sticky." Asset- holders have in mind some "safe" level of in- terest rates and if the current rate falls below the safe rate they flood the market with bonds as they scramble for cash, confident that they will be able to buy back their bonds at a lower price in the future when the interest rate con- firms their expectations and regresses towards its safe level. As asset-holders unload bonds this has the effect of driving the interest rate back up so fulfilling their expectations. To quote Leijonhufvud [2, p. 361], "in Keynes' theory, short-run variations in the interest rate are thus constrained by the prevailing market opinion of the ('safe') level of long-rate . . . I

Tobin [3], on the other hand, bases his ex- planation on uncertainty about the future course of bond prices. The asset-holder is "as- sumed . . . to be uncertain about [capital gains (or losses) and] to base his actions on his estimate of its probability distribution. This probability distribution . . . has an expected value of zero and is independent of the level of r, the current rate on consols. Thus the inves- tor considers a doubling of the rate just as like- ly when the rate is 5 per cent as when it is 2 per cent, and a halving of the rate just as likely when it is 1 per cent as when it is 6 per cent,"

[3, p. 72].2 It is apparent that in Tobin's model, asset-holders have unit-elastic bond price expectations. They expect the current bond price, or any future bond price estab- lished, to prevail.

Both Keynes' inelastic bond price expecta- tions and Tobin's uncertainty vis-A-vis bond prices are logically acceptable alternative building blocks on which to establish an in- terest responsive liquidity preference function. Their hypotheses must, therefore, be dis- criminated between on empirical grounds. Some simple tests are proposed and applied in the next section.

II Some Tests

Statistically, the difference between Keynes' and Tobin's hypotheses can be stated in terms of a first-order autoregressive structure. Name- ly,

>}'t -- at-1 Aort - + Et(l

where r is the interest rate and the usual as- sumptions are made about E.' According to Tobin, a is equal to zero for all t with the im- plication that changes in the interest rate are normally distributed with mean zero and that every change Et is uncorrelated with every other change E, - for all k # 0. In other words, changes in the interest rate are "Brownian" and follow a "random walk." Uncovering such be- havior in interest rate changes would not re- fute Tobin's hypothesis. It would, however, refute Keynes' hypothesis. According to Keynes, successive changes in the interest rate should be negatively autocorrelated. (This is implied by "stickiness.") In terms of equation (1), Keynes assumes a is between zero and minus unity. This implies Keynes' asserted

* I would like to thank Professors T. Hatanaka, M. B. Johnson, H. G. Johnson, D. Laidler, A. Leijonhufvud, and L. Phillips together with an anonymous referee for their helpful comments on an earlier draft of this paper with- out implicating them in any errors of fact or judgment that may still remain.

' In Leijonhufvud [21 consider also: ....speculative activity . . . stabilizes yields ." (p. 29); "To Keynes, the problem . . . lies . . . in the inflexibility of the long- term rate of interest." (p. 42) and ". . the level of long rate (is) quite stable in the short run . ." (p. 199, original emphasis). All these quotations from the most recent ex- egesis of Keynes' thought on the interest rate indicate "stickiness" in the interest rate.

[ 368 1

2 Compare this to Keynes [1, p. 202] where 'a long- term rate of interest of (say) 2 per cent leaves more to fear than to hope . . ."

'From here on, the discussion is presented in terms of the interest rate. Both Tobin's and Keynes' models were developed in terms of Consols and it is well-known that there is a unique relation between the price of Consols and their market yield.

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LIQUIDITY PREFERENCE 369

negative autocorrelation and that successive changes in the interest rate are not Brownian and do not follow a random walk.

Thus, to discriminate between Tobin's and Keynes' hypotheses one can test to see whether successive changes in the interest rate are inde- pendent, identically distributed random vari- ables. Tobin's hypothesis implies that they are whereas Keynes' hypothesis implies that inter- temporal dependence exists. Two standard techniques for testing for independence in a series are autocorrelation analysis and an analysis of runs. These techniques were applied to the interest rate.

1) A utocorrelation The autocorrelation coefficient pa is a mea-

sure of the relationship between the value of a variable at time t and its value 6 periods earlier. Defining it to be the percentage change in the interest rate between t- 1 and t, the autocorrelation coefficient for lag 6 is:

pa co (2) var(*t)

Some autocorrelation results are given instable 1. In this case the basic interest rate series em-

TABLE 1. - AUTOCORRELATION RESULTS

Lag, Autocorrelation weeks Coefficient

1 +.0123 2 +.0516 3 +.1022 4 +.1066 5 -.0092 6 +.0561 7 -.0346 8 +.0449 9 -.0144

10 -.0398 11 -.0403 12 -.0615

Note: The interest rate used was the yield on U.K. 21/2 per cent Consols quoted on Fridays (or the nearest working day) between 12/21/62 and 11/25/66 (N = 206). Source: Bank of England Quarterly Bulletin.

ployed was the weekly yield on United King- dom 2 1/2 per cent Consols. (As mentioned above, both Keynes and Tobin presented their models in the context of Consols and the United Kingdom (U.K.) provides the most accessible source of such a security.)

The significance points when N = 206 for

the autocorrelation coefficient at the 5 per cent level are + 0.1096 and -0. 1193. Thus, none of the autocorrelation results reported in table 1 are significant. Changes in the weekly yield on U.K. 212-) per cent Consols do, then, conform to a random walk process. Such evidence is contrary to Keynes' hypothesis but not to Tobin's. Although the evidence reported in table 1 is only limited, many other autocorre- lation analyses were performed. For example, one group used changes in the daily yield on Consols with lags from one to thirty days with the autocorrelation coefficients being calcu- lated for the whole sample and various sub- samples. Another group used changes in the monthly yield on Consols with lags from one to twelve months with the autocorrelation co- efficients also being calculated for the whole sample and various sub-samples. The results thus obtained were all consistent with the ran- dom walk hypothesis, i.e., tending to refute Keynes' and not refute Tobin.'

The objection could be made that the first- order autoregressive process is too unsophisti- cated to capture Keynes' hypothesis and, thus, that the insignificant first-order autocorrelation coefficients that have been observed do not pro- vide a conclusive refutation of his hypothesis. It might be argued, for example, that the inter- temporal dependence of the interest rate posited by Keynes must be represented by a general autoregressive process:

rt + airti + Id + aqrt-q = Et. (3) Such is the case if "investors are regarded as taking a considerable time-span of past ex- perience into account in forming their views of what is currently a 'safe' [interest rate] . . ." Leijonhufvud [2, p. 199]. It can be shown, however, that the evidence already ad- duced is sufficient to reject this hypothesis, too.

'Of course, every autocorrelation coefficient was not in- significant. This was only to be expected when such a large number of samples were taken. The crucial point is, how- ever, that the number of significant autocorrelation coeffi- cients observed was not inconsistent with a statistical chance process and, moreover, there was no observable pattern among those significant autocorrelation coefficients that were observed from series to series and from sample to sample. In addition, of those significant autocorrelation co- efficients that were observed, as many were positive as were negative. Thus, there was no support for Keynes' asserted dominance of inelastic bond price expectations. These more extensive results are available on request.

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370 THE REVIEW OF ECONOMICS AND STATISTICS

It has been demonstrated that: E f (*t - Eit) (*t - t - f) E _ yO) = 0

for all 0 except 0 0; yo > 0 when 0 = 0. If (3) is multiplied by rt-, rtq, rt-q_-, ... successively and expected values of both sides are taken then we obtain:

(1I + a2) '/l + a,3/2 + . . * + a~'q - 1=-al -1o

. . . . . . . . . . . . . . . . . . .

aq-1 71 + aq-2 72 + * * * + 7q-1 = q -l 70

. . . . . . . . . . . . . . . . . . .

aqll . ..... + al7q+Zk-1 + alq+/ 0

k 1. But it is known that yo > O and y1 Y2 0 .., = y(,+j = 0 from which it is immediately obvious that a, = a,, a, = 0. Thus, the insignificant first-order auto- correlation coefficients already observed are sufficient to refute (3) as well.

Having refuted (3), we can now make an- other interesting refutation.) An intuitively ap- pealing specification of Keynes' "safe" interest rate rs would be a weighted average of passed interest rates.6 Thus, let:

q

rts i rt-i (4) i=1

where the A sum to unity and eventually de- cline to zero at q + 1. Now, according to Keynes, this period's change in the interest rate will be a function of the divergence between the "safe" interest rate and the observed inter- est rate in the previous period. That is,

q

it -Y [4,rt-i- rt l) (5) j=j

However, lag (5) one period and subtract the result from (5) to obtain, after re-arrange- ment:

rt = (yi - y + 1) rt-i + ?yA2rt-2 + . . * + 7iq/t-q. (6)

But (6) is completely analogous to (3) and we have already proved that the a1 in that equa- tion are zero. It follows immediately that the coefficients in (6) must also be zero. Exami- nation of the coefficient on r,_1 reveals that this coefficient can only be zero if y = 1 and /x =

0. Now if y = 1, all the other ,ui must also be zero. We conclude, therefore, that no set of weights exists from which Keynes' "safe" in- terest rate can be calculated from past interest rates. And note that the only restrictions placed on the At, were that they decline to zero eventually and sum to unity. This conclusion, very economically arrived at, corroborates Starleaf and Reimer's [4] findings for their weights (which were arbitrarily selected to correspond to the weights used by Friedman to calculate permanent income).

2) Analysis of Runs For the interest rate, there are three different

possible types of change and, therefore, three different possible types of run. Positive changes give rise to plus runs, negative changes to minus runs, and zero changes to no change runs. A plus run, for example, is a sequence of consecu- tive positive price changes preceded and fol- lowed by either a negative change or a zero change. For a stationary series in which any given change is independent of all previous changes the total expected number of runs of all signs is given by:

m = [N(N+1) - EnI2]/N (7)

which has the standard error: ( Eni [2[n,2+N (N+ 1) ]-2N:n,3-N3 ) ?

(Til = X lN2(N-1) ( (8)

where N is the total number of price changes, and the ni are the number of price changes of each sign.7

According to Keynes' hypothesis the number of runs observed should be greater than the number expected in a stationary independent series since the interest rate is alleged to be "sticky" due to the fact that a fall (rise) in the current rate generates expectations of a rise (fall) in the future rate. Thus, if this is true, there will be a larger number of reversals in sign than a stationary independent series would generate. According to Tobin's hypoth- esis, successive changes are independent and, thus, the expected number of runs should corre- spond to that computed from (7).

An analysis of runs was made on the same 'The following piece of analysis was originally suggested to me by the comments of the anonymous referee.

6 Such an approach has been adopted by Starleaf and Reimer [4, p. 73]. The specific weights they employ are those used by Friedman to calculate permanent income.

'See Wallis and Roberts [5, p. 569]. The summations in every case are over i = 1 . . . 3.

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LIQUIDITY PREFERENCE 371

body of data as that used in the autocorrelation analysis discussed in the previous section. That is to say, daily, weekly, and monthly changes in the yield on U.K. 21/2 per cent Consols for various periods and sub-periods thereof. Every such analysis refuted Keynes' hypothesis and corroborated Tobin's hypothesis. In no single test was the observed number of runs statistical- ly larger than the expected number of runs. By way of illustration (using the same data as that employed to calculate the autocorrela- tion coefficients presented in table 1) the ex- pected number of runs is 1 15 while the ob- served number was 112. The ninety-per cent confidence intervals in this test were 103 and 125. Clearly, the observed number of runs is consistent with that to be expected from a sta- tionary independent series and inconsistent with that to be expected from a series containing negative dependence.

By way of summary, then, all the evidence marshalled in preparation of this note (of which that reported is the merest fraction since

the repetition of essentially duplicate results would seem to serve no useful purpose) refutes Keynes' rationalization of the asset demand for money based on inelastic bond price expecta- tions and does not refute Tobin's rationalization based on uncertainty about future bond prices.

REFERENCES

[1] Keynes, J. M., The General Theory of Employment, Interest and Money (New York: Macmillan Com- pany 1936).

[2] Leijonhufvud, A., On Keynesian Economics and the Economics of Keynes (London: Oxford Uni- versity Press, 1968).

[3] Tobin, J., "Liquidity Preference as Behavior To- wards Risk," XXV, Review of Economic Studies, (Feb. 1958).

[4] Starleaf, D. R., and R. Reimer, "The Keynesian Demand Function for Money; Some Statistical Tests," Journal of Finance, XXII, no. 1, (Mar. 1967).

[5] Wallis, W. A., and H. V. Roberts, Statistics; A New Approach (Glencoe, Illinois: Free Press, 1956).

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