DBJ Discussion Paper Series, No.1503

Ramsey’s Belief and the Plausibility of Keynesian Economics

Masayuki Otaki(Institute of Social Sciences, University of Tokyo)

November

2015

Discussion Papers are a series of preliminary materials in their draft form. No quotations,reproductions or circulations should be made without the written consent of the authors in orderto protect the tentative characters of these papers. Any opinions, findings, conclusions orrecommendations expressed in these papers are those of the authors and do not reflect the viewsof the Institute.

Ramsey’s Belief and the Plausibility of Keynesian Economics
Masayuki Otaki
The Institute of Social Sciences, University of Tokyo
ohtaki@iss.u-tokyo.ac.jp

Abstract
Ramsey’s theory of probability refers to the partial language associated with action. The
term “partial language’’ means that a language that does not describe the causality inquestion with perfect confidence. The degree of confidence, which is dubbed “belief ’’, orequivalently, probability, is estimated by a person’s action. Ramsey assumes that people
decide their action depending on the cost of knowing true causality. When such a cost is
not dear and they can easily access true causality, they act with deep confidence (i.e.，high probability). Using Ramsey’s formulation of probability theory, this study considers
the plausibility of alternative monetary theories --- Keynesian and monetarism ---,
which is defined by the people’s confidence in such theories. It is naturally assumed thatthe more complicated theory entails a dearer cost to understand. Keynesian equilibrium
is supported only through belief in the stable intrinsic value of money. Conversely,
monetarism requires that laypeople have special difficult knowledge so as to be
confident about its assertion; namely, the linear relationship between nominal money
supply and price; the actual value of Marshallian k and the volume of nominal money
supply. As such, the Keynesian belief is far more prevalent in the actual economy.

Keywords:Ramsey’sBelief,

KeynesianEquilibrium Selection
JEL Classifications; E12, E31, E42

1Economics,Monetarism,Plausibility,

1. Introduction
Not only the familiar probability theory, which is established by Kolmogrov relying on
the measure theory, but also Ramsey’s (1928) theory depend on the frequency theory ofprobability. The measure theoretic approach is ultimately meaningless without the lawof large numbers (i.e. ergodic property) because the exact shapes of the density function
and/or cumulative distribution function is unknowable.
Ramsey discards the law of large numbers because everyday samples are usually smallin size. Instead, he constructs his theory by relying only on the subjective mathematical
expectation, which, of course, does not relate to the law of large numbers at all. The
subjective probability, p , is defined the ratio of the subjectively expected number ofoccurrences, m , from the total number of incidences, n . That is,pm.
n(1)The most prominent feature of his theory is that he connects the subjective probability
with one’s action towards the proposition in question. He assumes that one can knowwhether the proposition is right if one pays some investigation cost, f  d  , where ddenotes the difficulty of knowing the true causality. In addition, he also assumes thatwhenever one acts in accordance with the belief that the proposition in question is true,
his expected subjective gain from such behavior must be equal to the gain obtained byknowing the truth. Unless such a relationship upholds, he surely ceases to act charted
by the belief (i.e. subjective probability), and instead, ascertains the true causality at
the cost f  d  when having decided on the action. It is natural to assume that f is anincrease function of d .
Let the benefit obtained when the proposition is true be denoted as a and as b when it
is false  a  b  . The subjectively expected gain from the n times trials,  , is  npa  n 1  p  b .

(2)Else, If one, otherwise, accesses the true causality in each trial, gain,  T , becomes T  na  nf  d  .

(3)

It is assumed that Equation (2) is equalized to (3); then, taking Equation (1) intoconsideration, we obtains the subjective probability (i.e. the belief on the validity of theproposition), p , as2

p  1f d .
a b(4)Equation (4) implies a serous meaning. Along with an increase in the difficulty ofunderstanding and checking the proposition in question, individuals become lessconfident to it. This is because the more complex the proposition becomes, the morecostly the examining its truthfulness becomes when individuals wish. Such kind oftendency is quite prevalent in everyday life.
In relation to economic theory, there is acute difference on the difficultness necessary
for understanding between Keynesian theory and monetarism. As Otaki (2007, 2015)shows, Keynesian theory upholds when people merely believe that the value of money,
which is defined by the inverse of the price index, is not affected by a change in nominal
money supply. In other words, as long as people rationally believe the intrinsic value ofmoney, money becomes non-neutral.
In contrast to such a simple structure of Keynesian economics, as Otaki (2015) shows,monetarism requires much information and complex processing to understand andbelieve it; every individual is incessantly concerned about the movement of nominal
money supply and all must investigate whether the price level is proportionately moves
with nominal money supply; and they all must exactly know the value of Marshallian k .
To summarize, the value, d , of monetarism can be deemed to be much larger than thatof the Keynesian thesis.
This implies that, ceteris paribus, Keynesian economics becomes more confident
among laypeople than monetarism. As such, if people behave in accordance with theirconfidence in a theory, the economy in reality becomes to possess the properties ofKeynesian economics.
The rest of this paper is organized as follows. Section 2 constructs a model based on
Otaki (2007, 2015) with Ramsey’s (1928) concept of belief and shows the plausibility ofKeynesian economics being superior to monetarism. Section 3 is the brief concluding
remarks.
2. Plausibility of Keynesian Economics
2.1 Two Alternative Models
A two-period overlapping-generations model with infinite time horizon is considered, asin Otaki (2007, 2015). There is no population growth and every individual has an
identical utility function:3

U t   c1t   c2t 

1

11  1 1
 1
  t   , 0    1, cit    cit    01(5)where cit    is the consumption level of good  of an individual, who is born at thebeginning of period t , during his i -th stage of life.  t is a definition function the value of
which takes unity when the individual is employed and zero when unemployed.  is thedisutility of labor supply.
For simplicity, it is assumed that the marginal productivity of labor is unity and that
there is no fixed cost. All firms are price takers, and thus, earned profits become zero inequilibrium. In addition, this model assumes that the labor market is in interior
equilibrium where some individuals are always unemployed, and the equilibrium
nominal wage stacks at the nominal reservation wage. These assumptions yield thefollowing equation:pt    Wt R ,(6)where pt   is the price of good  and Wt

Ris the nominal reservation wage during

period t . From the property of the indirect utility function, which corresponds toEquation (5), the nominal reservation wage becomes1

Wt   Pt P


R1
t 1

1
1
 1
 1
, Pt     pt    d  ,     1    .
0
(7)Combining Equations (6) and (7), the equilibrium inflation rate,  * , is determined as1

   1 . Furthermore, an elementary calculus gives us the following aggregateconsumption function of the young generation, Ct :Ct   

Wt R
Lt    yt ,Pt(8)

where yt is the real GDP in terms of the current goods. Consequently, one obtains thegoods market equilibrium condition asyt    yt Mt,
Pt

(9)

where M t is the nominal money supply at the beginning of period t . The second term ofthe right-hand side of Equation (9) is the real total expenditure of the old generation,

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M t 1
M  M t 1
, and the government expenditure financed by printing money, t. It isPt
Pt
assumed that the economy is located at the stationary state after period t , and hence
the monetary authority keeps the real money supply constant. That is, the growth rateof money is fixed at  * .
Equation (9) contains two undetermined endogenous variables; real GDP yt ; the initial
price level Pt . This means that the structure of an economy essentially depends on how
individuals evaluate the value of money. If they believe that the intrinsic value of money
is independent of nominal money supply, the initial price level remains intact even after
a change in the nominal money supply under rational expectations equilibrium, andmoney becomes non-neutral (the Keynesian proposition).
On the other hand, when people believe that the value of money is not absolute but
relative, this assertion is summarized by the following equilibrium price function (themonetarism proposition):Pt   1  M t(10)One must note both resulting equilibria are self-fulfilling. From Equations (9) and (10),
it is clear that the equilibrium GDP earned under Keynesian proposition, y k , is larger
than that earned under monetarism, y m , as long as the initial price level is sufficiently
low and the value of money is high enough. Henceforth, let be assumed that y k  y m . Inaddition, the corresponding Marshallian k s are denoted as  , km, respectively(    )1.
km

2.2. Ramsey’s belief and Keynesian EconomicsThis subsection applies the Ramsey’s formula (4) to examine which of the two above
mentioned macroeconomic theses is plausible. Before undertaking the analysis, onemust clarify what kind of knowledge is needed for a layperson in economics tounderstand each theory.
(i) For the Keynesian theory, the layperson must be able to confirm that the current
price level is unchanged despite a change in the nominal money supply.
(ii) For monetarism (a) the layperson must be able to confirm that the current priceNote that knowing the resulting benefits empirically does not necessarily mean that
all individuals are informed of the whole structure of the economy.

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level changes proportionately with the nominal supply, (b) he must be conversant withthe linear function to understand the relationship in (a), and (c) he must know the valueof the proportional constant  (Marshallian k ) and the exact quantity of nominal money
supply.
As such, it is evident that the monetarist thesis is far more difficult and laborious tounderstand than the Keynesian theory. Accordingly the value of d in Equation (4) isthought to be much larger in the case of monetarism.
In addition, whenever the true proposition is the Keynesian thesis, it is assumed thatthe subjective distribution when the proposition is false is uniform within the interval k     m . Thus, the difference in gains between the action based on the correct
proposition and that based on the wrong one, which corresponds to a  b in Equation (4),isyk  ym. Similarly, whenever the true proposition is monetarism, the probability
2

distribution of such proposition being false is uniform within      . Accordingly,
kmyk  ym
 m 1 km such a difference also becomes y   y   y  y   ..
22mHence, the denominators in Equation (4) are the same value independent of whether
the true proposition is Keynesian or monetarism2. This implies, therefore, that thebelief in Keynesian thesis, p , is much higher than that in monetarism, p . If peoplekm

behave in accordance with proposition more confident about, such behavior becomes
self-enforcing, and the economy moves as predicted by Keynesian economics. This studydefines this property as the plausibility of Keynesian economics.
3. Concluding Remarks
This study develops the concept of plausibility in alternative monetary theories withthe key concept being Ramsey’s (1928) belief. This theory assumes that belief (subjectiveprobability; confidence) can be measured by human’s action with regard to alternative
proposition depending on the cost incurred to know the truth. When such a cost is notdear, people are more confident about the true proposition because they can approach
the truth more easily if they wish.
While Keynesian economic is upheld only by the belief in the stable value of money interms of goods, monetarism is not sustained until laypeople in economics understand
not only the exact linear relationship between the nominal money supply and the priceNote that there is no economic surplus in this simplified setting. However, as Otaki(2007, 2015) shows, if the equilibrium concept of goods markets is interchanged withmonopolistic competition instead of perfect competition, there emerges an economic
surplus that is proportional to the real GDP.

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index but also macroeconomic statistics with which they are almost unconcerned ineveryday life. Consequently, the cost for knowing the truth is much cheaper in theKeynesian thesis. This implies that Keynesian economics is more plausible thanmonetarism although both theories possess the self-fulfilling property. In this sense,
Ramsey’s theory can be regarded as an effective equilibrium selection procedure.

References
Otaki M (2007). The dynamically extended Keynesian cross and the welfare-improving
fiscal policy, Economics Letters 96, 23-29.
Otaki M (2015). Keynesian economics and price theory: re-orientation of a theory ofmonetary economy, Springer-Japan, Tokyo.
Ramsey, FP (1928). Truth and probability, in Mellor DH (ed.) Philosophical papers,1990,Cambridge University Press, Cambridge.

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