Discussion Paper No.337
Keynesian and classical theories: static anddynamic perspectives

Hiroki Murakami
Faculty of Economics, Chuo University

October 2020

INSTITUTE OF ECONOMIC RESEARCH
Chuo University
Tokyo, Japan

Keynesian and classical theories: static and dynamic perspectives
Hiroki Murakami∗
Faculty of Economics, Chuo University†
October 29, 2020

Abstract
This paper reexamines the fundamental difference between the Keynesian and classical theories from both
static and dynamic perspectives. It is shown that the rigidity of wages plays a pivotal role in the distinction
between these theories in statics and that they can be differentiated in terms of long-run stability in dynamics.

Keywords: Classical theory; Keynesian theory; Long-run stability; Wage flexibility
JEL classification: C62; E12; E13; E24

1

Introduction

It is no exaggeration to say that the development of macroeconomics has been stimulated by controversies between
the Keynesian and classical theories. Generally speaking, the Keynesian theory allows for the existence of involuntary
unemployment, while the classical theory does not. There have been a lot of arguments on the similarities anddissimilarities between them. The main topic of these arguments is the fundamental cause of the difference inconclusion between them.
The purpose of this paper is to reconsider the fundamental difference between the Keynesian and classicaltheories from both static and dynamic perspectives. This paper is organized as follows. Section 2 summarizes thestatic properties of the Keynesian and classical theories from a static viewpoint. It confirms Modigliani’s (1944)
claim that the fundamental difference between them in statics is due to the rigidity of nominal wages. Section 3
presents a different view on the Keynesian and classical theories from a dynamic (and long-term) perspective. In
this section, two dynamic systems with Keynesian and classical features are examined to consider the differencein view on long-run stability between these theories. It is demonstrated that the same long-run equilibrium has
totally different characteristics in stability between the Keynesian and classical systems and that the fundamental
difference lies in the difference in stability properties of long-run equilibrium. Section 4 concludes this paper.
∗ E-mail: hmura@tamacc.chuo-u.ac.jp† 742-1, Higashi-Nakano, Hachioji, Tokyo 192-0393, Japan.

12Keynesian and classical theories in statics

This section examines the Keynesian and classical theories from static perspectives. We compare the Keynesianand classical theories in statics and look at the differences in assumption and conclusion between them.

2.1

Keynesian theory in statics

The static Keynesian theory can be illustrated by the well-established IS-LM (or AD-AS) system.1 The IS-LM
system examined in this section is the following one proposed by Modigliani (1944, 1963):2
M= L(r, Y ),P

(1)

I = I(r, Y ),(2)

S = S(r, Y ),(3)

S = I,(4)

Y = F (N ),(5)W= F 0 (N ),P(6)where the notations are given as follows: Y , aggregate real income or output; r, the rate of interest; I, aggregate
investment or the investment function; S, aggregate saving or the saving function; L, aggregate real demand formoney or the liquidity preference function; M , aggregate supply of money; N , aggregate employment of labor; F ,the short-run production or aggregate supply function; P , the price level; W , the nominal or money wage. In thissystem, aggregate supply of money M is treated as given, while the other variables, Y, r, I, S, N, P and W, areendogenous variables.
Eq. (1) represents the equilibrium condition for the money or assets market, whereby the rate of interest isdetermined, and illustrates the liquidity preference theory of interest (Keynes 1936, chap. 15).
Eqs. (2) and (3) reflect the Keynesian theory of consumption and investment (Keynes 1936, chaps. 8-12).3
Eq. (4) has a great importance in the Keynesian theory because it represents not only the equilibrium condition
1 The formal theory developed in Keynes’ General Theory is substantially static. Since the view is widespread that the IS-LM systemprovides a “distorted” interpretation of Keynes’ General Theory, it may be useful to argue against this view. Keynes did not oppose
but endorse Hicks (1937) as a summary of his General Theory. This fact can be confirmed by the following correspondence between
Keynes and Hicks:

At last long I have caught up with my reading and have been through the enclosed [Hicks (1937)]. I found it very
interesting and really have next to nothing to say by way of criticism. (the letter from Keynes to Hicks in Keynes (1973,p.80)).
2 The IS-LM system is taken as an expression of the static Keynesian system in this paper, but as recognized in Modigliani (1944), it
can be adapted to a dynamic context if time lags are appropriately introduced. For a recent study on the dynamic effects of time lags
concerning consumption and investment in the IS-LM system, see Murakami (2017).
3 Although it is argued in Keynes’ General Theory that aggregate investment is determined by the marginal efficiency of capital (orthe expected profit on capital) and the rate of interest, eq. (2) may be regarded as consistent with his theory of investment because
expected effective demand, which is a major determinant of the marginal efficiency of capital, is influenced largely by current effective
demand or aggregate output. Eq. (2) is also consistent with the profit principle of investment postulated in the Keynesian theories ofbusiness cycles of Kalecki (1935, 1939) and Kaldor (1940, 1951).

2

for the goods-services market but also the principle of effective demand (Keynes, chap. 3).4 It should, in theKeynesian theory, be interpreted to mean that aggregate saving, defined as the difference between aggregate income
and aggregate consumption, is determined by aggregate investment, the sum of which and aggregate consumption
is aggregate effective demand, not the other way round.
Eq. (5) is nothing but the aggregate supply function (Keynes 1936, chap. 3) and specifies the short-run
relationship between aggregate employment and aggregate output with aggregate stock of capital kept constant.
In the Keynesian theory, aggregate employment is adjusted by this equation to that level which is consistent withaggregate output determined by aggregate effective demand.
Finally, we would like to have a closer look at (6), because it may invoke some confusion; its meaning is different
between the Keynesian and classical theories. It represents the “first postulate” of the classical theory, adopted
as a realistic hypothesis in Keynes’ General Theory (Keynes 1936, chap. 2). In the classical theory, this postulate
has the implication that aggregate demand for labor or aggregate employment is adjusted to that level at which
the marginal productivity of labor is equalized to the given real wage. Thus, in the classical theory, eq. (6)describes the (aggregate) labor demand function, which relates aggregate employment to the given real wage. In
the Keynesian theory, on the other hand, aggregate employment or aggregate demand for labor is determined byaggregate effective demand for goods and services, as indicated by (4) and (5), and hence eq. (6) should not
be interpreted to represent the (aggregate) labor demand function. Indeed, it should be interpreted as the pricesetting function, which represents the price level as a function of aggregate employment for the given nominal wage;5
otherwise it is obviously inconsistent with the principle of effective demand.6 Thus, in the Keynesian theory. it
4 It seems that the reason why the investment and saving functions and the equilibrium condition for the goods-services market are
separately posited in Modigliani’s (1944, 1963) system is that he appreciated the difference between ex ante investment and saving. The
following statement is made in Modigliani (1944):

In our case, the equilibrium of the “money market” is a condition of short-run equilibrium (that determines the rate ofinterest for each period) because it is the result of decisions that can be carried out into effect immediately. The conditionsaving = investment, on the other hand, is a condition of long-run equilibrium because the equality of ex ante saving andinvestment cannot be brought about instantaneously. This is a different way of stating the familiar proposition that themultiplier takes time to work out in its full effect. (p. 62).
5 Eq. (6) is equivalent toW.
F 0 (N )It can thus be taken as a behavioral equation for firms’ price setting. Also, if the curvature of the production function is gradual, thenthe price level is almost proportionate to the nominal wage and hence almost fixed for the given nominal wage.
6 According to Keynes’ interpretation, the first postulate just describes the correlation between aggregate employment and the real
wage, as the following statement indicates:P =It [the first postulate] means that, with a given organisation, equipment and technique, real wages and the volume of
output (and hence of employment) are uniquely correlated, so that, in general, an increase in employment can only occur
to the accompaniment of a decline in the rate of real wages. Thus I am not disputing this vital fact which the classicaleconomists have (rightly) asserted as indefeasible. In a given state of organisation, equipment and technique, the real
wage earned by a unit of labour has a unique (inverse) correlation with the volume of employment. Thus if employment
increases, in the short period, the reward per unit of labour in terms of wage-goods must, in general, decline and profits
increase. (Keynes 1936, p. 17)
His interpretation is more explicit in his debate with Dunlop and Tarshis on real wages:
In the passage quoted above [Keynes (1936, pp. 9-10)] I was dealing with the relation of real wages to changes in output,
and had in mind situations where changes in real and money wages are reflection of changes in the level of employment
caused by changes in effective demand. (Keynes 1939, p. 35)

3

describes the determination mechanism not of aggregate employment but of the price level.7
It is easily seen that the number of equations ((1)-(6)) is one less than that of endogenous variables Y, r, I, S, N, P
and W. Then, (at least) one of the endogenous variables cannot be solved in this system. In other words, there isindeterminacy in this system. As argued in Modigliani (1944, 1963), however, the indeterminacy problem in the
IS-LM system can be solved by fixing the nominal wage W to some predetermined level. With the nominal wage
fixed, the other variables can be determined by this system, (1)-(6). Aggregate employment so determined does
not necessarily correspond to full employment because aggregate supply of labor is not taken into account in thissystem. In other words, involuntary unemployment can arise in this system. Thus, the existence of involuntary
unemployment or underemployment equilibrium can be verified in the presence of rigid nominal wages. Due to this,
it is widely thought that the rigidity of nominal wages is the main cause of involuntary unemployment.8 On thesegrounds, the Keynesian theory, in statics, is viewed as a macroeconomic theory based on the assumption of rigid
nominal wages.9

2.2

Classical theory in statics

Now we proceed to explore the properties of the (neo-)classical theory from static perspectives. The static classicaltheory can substantially be delineated by the same IS-LM system, (1)-(6), with one missing equation. The one
missing equation is given by

N = NS

W 
P

,

(7)

where NS is the labor supply function.
To examine the differences between the (static) Keynesian and classical theories, we consider the meanings of(1)-(6), as well as of (7), in the classical theory. For this purpose, we first look at (6) and (7) and then turn to (5),7 It seems that Keynes should not have maintained (6) and replaced it with a naive but realistic mark-up pricing hypothesis toestablish the principle of effective demand as a foundation of his theory. In the Keynesian system based on the principle of effective
demand ((4) and (5)), eq. (6) may be better taken as a mark-up pricing rule than as a first order condition for optimality. Indeed, it
can be written as
N F (N ) W N
1 W
P =
=
,
F 0 (N ) Y
β Y /N
where β stands for the employment elasticity of production. If β is approximately constant (and less than unity), this equation isconsistent with the mark-up principle. In this sense, Kalecki’s (1939, 1971) theory, based on the mark-up principle, is more faithful tothe principle of effective demand. Indeed, in examining the Keynesian theory from dynamic perspectives (in the next section), we will
adopt a hypothesis on price settings based on Kalecki’s theory rather than Keynes’, because the former is more “Keynesian” than thelatter.
8 This view is asserted in Modigliani (1944) as follows:

It is usually considered as one of the most important achievements of the Keynesian theory that it explains theconsistency with the presence of involuntary unemployment. It is, however, not sufficiently recognized that, except in alimiting case to be considered later, this result it due entirely to the assumption of “rigid wages” and not to the Keynesianliquidity preference. (p. 65)
The liquidity-preference theory is not necessary to explain underemployment equilibrium; it is sufficiently only in alimiting case; the “Keynesian case.” In the general case it is neither necessary nor sufficient; it can explain this phenomenon
only with the additional assumption or rigid wages. (pp. 75-76).
9 As noted in the outset of Modigliani’s (1944) analysis, this view on the Keynesian theory is only valid in a static context.

Section 2.3.

4See

(2)-(4) and (1) in order.
Eq. (6) truly expresses the first postulate of the classical theory, which implies that aggregate demand for labor
is determined by the real wage. In this sense, as we have argued above, it plays the role of the labor demand
function in the classical theory. Eq. (7), on the other hand, indicates that aggregate employment is equal toaggregate supply of labor corresponding to the given real wage. It reflects the “second postulate” of the classicaltheory, which states that the marginal disutility of labor, divided by the marginal utility of consumption, is equal
to the real wage (Keynes 1936, chap. 2). Thus, eqs. (6) and (7) imply that aggregate employment, which appears
as a symbol in both of them, is equal to aggregate demand for and supply of labor and hence that the labor market
is in equilibrium in the sense that there is no involuntary unemployment. In other words, aggregate employment
derived from (6) and (7) corresponds to full employment. The presence of (7) characterizes the classical theory
because it means full employment.10
Eq. (5) describes the short-run technique of production, but its role in the classical theory is different fromthat in the Keynesian theory. In the Keynesian theory, it derives aggregate employment from aggregate real
output corresponding to aggregate effective demand; in the classical theory, it derives aggregate real output fromaggregate employment corresponding to full employment. It implies different causal relationships between aggregate
employment and output in these theories.
Naturally, eqs. (2)-(4), in the classical theory, play a role different from that in the Keynesian theory. Since
aggregate real output is determined by (5), aggregate investment and saving can be reduced to functions of the rateof interest alone by (2) and (3). Then, what eq. (4) determines is not aggregate output but the rate of interest. It
reflects the classical (or loanable funds) theory of interest.
Eq. (1) is, of course, the equilibrium condition of the money market but does not describe the determination
mechanism of the rate of interest. Since aggregate output and the rate of interest are already determined by (5)and (4), respectively, the right-hand side of (1), or aggregate real demand for money, is constant. It then follows
from (1) that the price level is proportionate to aggregate supply of money, which implies that the quantity theoryof money holds. Eq. (1) gives the price level in response to the given quantity of money.
In the static classical theory, aggregate supply of labor is taken into consideration by (7). This is the maindifference from the static Keynesian theory. Because of it, the real wage is allowed to be adjusted to the level atwhich the labor market is in equilibrium in the sense that aggregate demand for and supply of labor equal eachother; the flexibility of (real) wages is implicitly assumed in the classical theory. As a result of this flexibility,
10 The equivalence of the second postulate and full employment is recognized in Keynes’ General Theory:

At different points in this chapter we have made the classical theory to depend in succession on the assumptions:(1) that the real wage is equal to the marginal disutility of the existing employment;(2) that there is no such thing as involuntary unemployment in the strict sense;(3) that supply creates its own demand in the sense that the aggregate demand price is equal to the aggregate supply
price for all levels of output and employment.
These three assumptions, however, all amount to the same thing in the sense that they all stand and fall together, any
one of them logically involving the other two. (Keynes 1936, pp. 21-22)

5

aggregate output is fixed at the level corresponding to full employment in the labor market and hence determined
independently from the quantity of money. In this respect, the classical dichotomy holds. This is reflected in the
conclusion that the price level is determined entirely by the quantity of money. Thus, the flexibility of wages, aswell as the labor supply function, plays a vital role for establishing full-employment equilibrium.

2.3

Keynesian and classical theories in statics: summary

We have so far discussed the Keynesian and classical theories from static viewpoints. We can now conclude thatwhether the nominal wage is flexible or not makes a great difference between the Keynesian and classical theories. It
then follows from this conclusion that the existence of underemployment equilibrium with involuntary unemployment
is only the outcome of the rigidity of nominal wages. According to this view, the Keynesian theory reduces to theclassical theory without the nominal-wage rigidity. This is the view shared among a lot of economists.11
The rigidity of nominal wages plays an essential role for verifying the Keynesian theory or the presence of
involuntary unemployment in statics; it is a sufficient (not necessary) condition for the existence of equilibriumwith involuntary unemployment. In statics, phenomena do not possess practical relevancy unless they are shownto occur in equilibrium. In this sense, the widespread view on the Keynesian theory is right in a static context.

3

Keynesian and classical theories in dynamics

In statics, as we have confirmed in the last section, the rigidity of nominal wages or prices is of great importancein explaining involuntary unemployment; it guarantees the existence of underemployment equilibrium. Involuntary
unemployment, however, need not be characterized as a phenomenon occurring at equilibrium. It can be described
as a disequilibrium phenomenon in a dynamic context. Indeed, if the economy stays away from the full-employment
equilibrium for a long while, it undergoes prolonged involuntary unemployment in disequilibrium or underemployment disequilibrium. From this point of view, we do not necessarily have to assume the rigidity of nominal wages
or prices to explain the phenomenon of underemployment equilibrium. We may thus argue that what mattersfor the practical relevancy of the Keynesian theory is not the existence of underemployment equilibrium but the(in-)stability of full-employment equilibrium.12
11 This view is made explicit in Ball et al. (1988) as follows:

According to the Keynesian view, fluctuations in output arise largely from fluctuations in nominal aggregate demand.
These changes in demand have real effects because nominal wages and prices are rigid. (p. 1)
12 This position is taken in Patinkin (1965) as the following statements indicate:Thus Keynesian economics is the economics of unemployment disequilibrium. It argues that as a result of interestinelasticity, on the one hand, and distribution and expectation effects, on the other, the dynamic process of Chapter
XIII:3—even when aided by monetary policy—is unlikely to converge either smoothly or rapidly to the full-employment
position. Indeed, if these influences are sufficiently strong, they may even render this process unstable. In such a case thereturn to full employment would have to await the fortunate advent of some exogenous force that would expand aggregate
demand sufficiently. (pp. 337-338)
While our interpretation takes off the analytical edge of Keynesian economics in one direction, it sharpens it in another,more vital one. It makes unmistakably clear—what should always have been clear—that the involuntary unemploymentof the General Theory need not have its origin in wage rigidities. Indeed, in this respect we are more Keynesian than6

This section discusses the Keynesian and classical theories in dynamics from the viewpoint of stability of fullemployment equilibrium. For this purpose, we examine two dynamic systems, one of which is with Keynesianfeatures and the other with classical features; they possess the same equilibrium corresponding to the naturalrate of unemployment, identified with full employment, but differ in dynamic adjustments in the price level andaggregate output. The issue in this section is the differences in stability properties of the same equilibrium between
the Keynesian and classical systems.
The subsections that follow explore Keynesian and classical systems. Section 3.1 sets up the basic frameworkcommon between our Keynesian and classical systems. Sections 3.2 and 3.3 formalize and examine dynamic systems
with Keynesian and classical properties, respectively.

3.1

Aggregate demand and aggregate supply

This subsection presents behavioral equations concerning aggregate demand and aggregate supply. In what follows,
the notations used in the last section are kept except when special mention is made.
We first consider aggregate demand. Aggregate saving is assumed to be represented by

S = sY,(8)

where s is a positive constant which represents the (marginal and average) propensity to save.
We postulate that aggregate (gross) investment is described in the following equation:
Y

I
=f
, r − π e = f (x, ρe ),
K
K

(9)

where K and π e stand for aggregate stock of capital and the expected rate of inflation, respectively; x and ρe
represent the output-capital ratio and the real rate of interest, respectively; f is the capital formation function
which is assumed to be twice continuously differentiable with

fx ≡

∂f
∂f
> 0, fρe ≡
< 0.
∂x
∂ρe

Keynes. For by unequivocally placing the center of emphasis on the inadequacy of aggregate demand in the commodity
market, and by recognizing the resulting involuntary unemployment to be phenomenon of economic dynamics, we havefreed ourselves from the necessity of static analysis to connect decreases in employment with increases in the real wage
rate. We have been able to explain the existence of involuntary unemployment without placing any restrictions on the
movement of the real wage rate. (pp. 340-341)
A similar argument is made in Tobin (1975):
Very likely Keynes chose the wrong battlefield. Equilibrium analysis and comparative statics were the tools to which
he naturally turned to express his ideas, but they were probably not the best tools for his purpose. (p. 195)
The real issue is not the existence of a long-run static equilibrium with unemployment, but the possibility of protracted
unemployment which the natural adjustments of a market economy remedy very slowly if at all. So what if, within therecherché rules of the contest, Keynes failed to establish an “underemployment equilibrium”? The phenomena he described
are better regarded as disequilibrium dynamics. Keynes’s comparative statics were an awkward analytical language unequal
to the shrewd observations and intuitions he was trying to embody. (pp. 195-196)

7

(10)

This equation reflects the Keynesian theory of investment.13
We postulate that the nominal rate of interest is represented by the following equation:

r=r

PY 
M

= r(υ)(11)

where υ represents the ratio between aggregate nominal income to aggregate nominal supply of money; r is interest
rate function which is assumed to be twice continuously differentiable withrυ ≡ r0 (υ) > 0,(12)This relation can be derived from the money-market equilibrium condition given byM= l(r)Y,P(13)

l0 (r) < 0.

(14)

where l is the liquidity preference function with

We turn to aggregate supply. Aggregate real output is determined by the following Cobb-Douglas technique:

Y = K 1−β (AN )β ,(15)

where A stands for the labor augmenting technology; β is a positive constant less than unity.
With the capital formation and interest rate functions (10) and (11), aggregate capital formation can be describedby14
 Y PY 

K̇
=f
,r
− π e − δ,
K
K
M

(16)

where δ is a positive constant which represents the rate of capital depreciation.15
The natural-rate level of aggregate employment, proportionate to aggregate supply of labor,16 and the labor
13 Eq. (9) is consistent with the marginal efficiency theory of investment (Keynes 1936, chap. 11) and also with the profit principle
of investment (cf. Kalecki 1935, 1939; Kaldor 1940, 1951). For an intertemporal microeconomic foundation on the Keynesian theory ofinvestment, see Murakami (2016a).
14 Throughout this paper, v̇ denotes the time derivative of v, i.e., v̇ = dv/dt.
15 Since aggregate ex ante investment and saving are distinguished, aggregate ex post capital formation may not be equal to aggregate
ex ante investment net of capital depreciation. For simplicity, however, the difference between ex ante and ex post investment is ignoredin (16). For a more general formalization of capital formation, see Stein (1969); for a further generalization, see Murakami (2014).
16 If the natural rate of unemployment is constant over time, the natural-rate level of aggregate employment is proportional to aggregate
supply of labor.

8

augmenting technology are assumed to change at constant rates as follows:
Ṅ ∗
= n,
N∗
Ȧ
= a,
A

(17)
(18)

where N ∗ stands for the natural-rate level of aggregate employment; n is a real constant which represents the rateof change in aggregate supply of labor; a is a nonnegative constant which represents the rate of technical changewith δ + n + a > 0.17
Given the production technique described by (15), the natural-rate level of aggregate real output is defined by
the natural-rate level of aggregate employment as follows:Y ∗ = K 1−β (AN ∗ )β ,(19)

where Y ∗ stands for the natural-rate level of aggregate real output. For this variable, the following three ratios canbe defined:Y,Y∗Kk = ∗,
Y
M
m=
,
PY ∗
y=

(20)
(21)
(22)

where y, k and m are called the rate of utilization, the capital coefficient and the money coefficient, which corresponds to the Marshallian k at the natural rate of employment, respectively.
It follows from (19) that the rate of change in the natural-rate level of aggregate output can be written as Ȧ Ṅ ∗ 
Ẏ ∗
K̇
=
(1
−
β)
+
β
+
,
Y∗
K
A N∗
which can, by (16)-(18) and (20)-(22), be reduced to
h y

i
Ẏ ∗
= (1 − β) f , r − π e − δ + β(n + a),
∗
Y
k

(23)

With (16)-(18) and (20)-(22), the rate of change in the capital coefficient can also be given byh y  yik̇= β f ,r− π e − (δ + n + a) .
k
k
m

(24)

17 Provided that this condition holds, the rate of change in aggregate supply of labor, which can be identified with the rate of change

in population, can be negative.

9

This is a dynamic equation common to our Keynesian and classical systems explored in the next subsections.
Account is also taken of the labor market. It is seen from (15), (19) and (20) that the ratio between actual andnatural rates of employment can be related to the rate of utilization as follows:N= y 1/β .
N∗(25)

The rate of change in the nominal wage is supposed to be determined by the actual and natural rates of
employment and the expected rate of inflation in the following fashion:N − N∗ Ẇ+ π e + a,=γ
WN∗
where γ is a positive parameter that measures the speed of adjustment in the labor market and also the degree ofwage flexibility. This equation is a natural-rate version of the Phillips curve augmented with the rate of change inlabor augmenting technique. It then follows from (25) that this Phillips curve can be written asẆ= γ(y 1/β − 1) + π e + a.
W(26)

We suppose that the monetary authority changes aggregate supply of money at a constant rate in the following
way:
Ṁ
= µ,
M

(27)

where µ is a real constant.
Finally, it is assumed that the expected rate of inflation is revised adaptively in the following fashion:π̇ e =  ṖP− πe ,(28)

where  is a positive constant which represents the speed of revisions of inflation expectations.18

3.2

Keynesian theory in dynamics

In dynamics, the Keynesian and classical theories can be distinguished in terms of adjustments in the price level
and aggregate output. In the Keynesian theory, as the principle of effective demand implies, aggregate output ischanged in response to fluctuations in aggregate effective demand; in the classical theory, the gap between aggregate
demand and output is filled by adjustments in the price level.19 Following this view, this and the next subsections
18 As verified by Muth (1961), the “adaptive expectations” rule is “rational” (or efficient) if the actual rate of inflation is composedof both permanent and transitory disturbances.
19 This is a common view as observed in Leijonhufvud (1968):10

consider the Keynesian and classical theories in a long-run and dynamic context to discuss the difference in viewon long-run stability between them.
The principle of effective demand is no doubt the core of the Keynesian theory. In statics, it maintains that
aggregate real output is equalized to aggregate effective demand. In dynamics, it is interpreted to mean that
aggregate output is adjusted to meet aggregate effective demand in the short run. In other words, it can be
regarded as the hypothesis that it is quantity, rather than prices, that is adjusted in response to the gap between(aggregate) demand and supply. Given that a “long term” is a consequence of a “short term,” the “quantity
adjustment” should also characterize the long-term Keynesian theory.
Reflecting this view, we may postulate for dynamic analysis that aggregate output is adjusted to aggregate
demand. In the long term, however, the level of potential output, identified with the natural-rate level of output,
increases through changes in the supply side (e.g., capital formation, population changes and technical progress).
In this respect, it is reasonable to suppose that aggregate output is determined relative to its potential level in
response to changes in the demand side. In other words, the rate of utilization, the ratio between the two, should
be taken as the adjusting variable in a long-term version of Keynesian quantity adjustment.
Thus, we assume that the rate of utilization is changed in response to the discrepancy between aggregate effective
demand and supply in the following fashion:ẏ = αE − Y Y∗=αI − S 
Y∗

,

(29)

where E stands for aggregate effective demand equal to the sum of aggregate consumption and investment;20 α is apositive constant which represents the speed of adjustment. This equation says that the rate of utilization is varied
in response to the ratio between aggregate excess demand and the natural-rate level of aggregate output. It can beseen as a long-term version of quantity adjustment.
Substituting (8), (9), (11) and (20)-(22) in (29), we obtain the following:h y  y i− π e k − sy .
ẏ = α f , r
k
m

(30)

This equation, as well as (24), constitutes our Keynesian system.
According to the principle of effective demand, the gap between aggregate demand and supply is filled by quantity
In general equilibrium flow models, prices are the only endogenous variables which enter as arguments into the demandand supply functions of individual households. Tastes and initial resource endowments are parametric. In “Keynesian”
flow models the corresponding arguments are real income and the interest rate. Of these, real income is a measure ofquantity, not of prices. On a highly abstract level, the fundamental distinction between general equilibrium and Keynesianmodels lies in the appearance of this quantity variable in the excess demand relation to the latter. The difference is dueto the assumptions made about the adjustment behavior of the two systems. In the short run, the “Classical” system
adjusts to changes in money expenditures by means of price-level movements; the Keynesian adjusts primarily by way ofreal income movements. (p. 51)
20 Government expenditure or net exports can be included in aggregate effective demand as exogenous factors, provided that they areproportionate to the natural-rate level of aggregate output.

11

adjustment rather than price adjustment. In the Keynesian theory, the price level should then be determined bya mechanism different from price adjustment. In Keynes’ General Theory, as we have observed, the “marginal”
principle plays a role in price determination; “[t]he general price-level depends partly on the rate of renumeration of
the factors of production which enter into marginal cost and partly on the scale of output as a whole” (Keynes 1936,
p. 294). As we have argued, however, the mark-up principle, proposed by Kalecki (1939, 1971), is more suited to
the Keynesian theory. In Kalecki’s (1939, 1971) theory, the mark-up ratio is influenced by the degree of monopoly,
and aggregate share of labor is determined by the (average) mark-up ratio. As argued by him, aggregate share oflabor so determined is subject to fluctuations in the short run but is fairly constant in the long run.21
Reflecting the Kaleckian theory, we may postulate that the price level is determined so that aggregate share oflabor would be constant at the natural rate of unemployment:

P = (1 + θ)

WN∗
,
Y∗

(31)

where θ is a positive constant which represents the mark-up ratio. This is consistent with the short-run fluctuations
and long-run stability of aggregate share of labor.
It follows from (31) that the rate of inflation is given by
Ṗ
Ẇ
Ṅ ∗
Ẏ ∗
=
+ ∗ − ∗,P
W
N
Y
which can, with (17), (23) and (26), be reduced toh y  y iṖ= γ(y 1/β − 1) − (1 − β) f , r− π e − (δ + n + a) + π e .
P
k
m

(32)

The rate of change in the money coefficient (defined by (22)) can then be calculated as
ṁ
Ṁ
Ṗ
Ẏ ∗
=
− − ∗,m
M
P
Y
which can, by (23) and (27), be reduced toṁ= µ − n − a − γ(y 1/β − 1) − π e .
m(33)

It is seen from (28) and (32) that the expected rate of inflation is revised as follows:nh y  y ioπ̇ e =  γ(y 1/β − 1) − (1 − β) f , r− π e − (δ + n + a) .
k
m

(34)

21 The long-run stability of aggregate share of labor has been confirmed by Jones (2016). According to him, the U.S. share of capital
was almost constant (about 34.2 percent) until around 2000, though it has recently been rising (to 38.7 percent by 2012).

12

Thus, we can obtain the following system of equations:ih y  y − π e k − sy ,ẏ = α f , rkmh y  y ik̇− π e − (δ + n + a) ,= β f ,rkkmṁ= µ − n − a − γ(y 1/β − 1) − π e .
mionh y  y − π e − (δ + n + a) .
π̇ e =  γ(y 1/β − 1) − (1 − β) f , r
k
m

(30)
(24)
(33)
(34)

We call the system of these equations “System (K)” (to mean Keynesian) and examine it to look at the dynamic
characteristics of the Keynesian theory.22
The long-run equilibrium values of y, k, m and π e , denoted by y ∗ , k ∗ , m∗ and π ∗ , respectively, are solutionsof the following simultaneous equations:23
y  y 
0 = f ,r− π e k − sy,kmy  y 
0 = f ,r− π e − (δ + n + a),km
0 = µ − n − a − γ(y 1/β − 1) − π e ,ih y  y − π e − (δ + n + a) .
0 = γ(y 1/β − 1) − (1 − β) f , rkm
It follows that y ∗ , k ∗ and π ∗ can be calculated asy ∗ = 1,

(35)

s
,
δ+n+a

(36)

π ∗ = µ − n − a.

(37)

k∗ =

It is also seen that m∗ is given byfδ + n + a
s

,r

 1 
m∗


− µ + n + a = δ + n + a.

(38)

It can be found from (35) that the natural-rate level of aggregate output is attained at the long-run equilibrium andthe long-run expected (and actual) rate of inflation is determined by the rate of change in the quantity of money.
22 System (K) has a lot in common with the Kaldor-Tobin models (synthesizing Kaldor (1940) and Tobin (1975)) of Asada (1991),
Chiarella and Flaschel (2000, chap. 6), Chiarella et al. (2013, chap. 13) and Murakami and Asada (2018) but differs from them in thefollowing points: (i) the rate of utilization is not identified with the output-capital ratio; (ii) the price level is determined based on thenatural-rate (long-run) levels (not actual levels) of employment and output (cf. (31)). It may also be viewed as a long-term extension of
the medium-term Keynesian models of Murakami (2014, 2016b). Note that the purpose of our analysis is not to present a generalized
Keynesian model integrating the related ones but to consider the difference in view on stability between the Keynesian and classicaltheories making use of fairly standard models.
23 k = 0 or m = 0 is ruled out as an equilibrium value of k or m.

13

To guarantee the existence (and hence uniqueness) of long-run equilibrium, it is assumed that24

lim f

m→+0

δ + n + as,rδ + n + a  1 − µ + n + a < δ + n + a < lim f−µ+n+a .
,r
m→∞
m
s
m

1

(39)

Due to (10) and (12), this ensures the existence and uniqueness of m∗ > 0 that satisfies (38).
The long-run stability of System (K) can be discussed by examining the Jacobian matrix evaluated at the∗long-run equilibrium, denoted by JK
:






∗
JK = α(fx∗ − s + k ∗ fρ∗e rυ∗ /m∗ )α(s − fx∗ )/k ∗−αk ∗ fρ∗e rυ∗ /(m∗ )2−αk ∗ fρ∗eβ(fx∗ + k ∗ fρ∗e rυ∗ /m∗ )−βfx∗ /k ∗−βk ∗ fρ∗e rυ∗ /(m∗ )2−βk ∗ fρ∗e−γm∗ /β00−m∗

[γ/β − (1 − β)(fx∗ /k ∗ + fρ∗e rυ∗ /m∗ )](1 − β)fx∗ /(k ∗ )2

(1 − β)fρ∗e rυ∗ /(m∗ )2(1 − β)fρ∗e






,




where ∗ signifies the value evaluated at the long-run equilibrium. The characteristic equation associated with the∗is given by
Jacobian matrix JK

λ4 + a1 λ3 + a2 λ2 + a3 λ + a4 = 0,

where


k∗
f∗
a1 = α s − fx∗ − ∗ fρ∗e rυ∗ + β x∗ − (1 − β)fρ∗e ,mkn h k∗
1 k∗ ∗rυ∗ irυ∗ o ∗
a2 = α
γ −
r
γ
−
(1
−
β)s
−
βs
 fρe ,+(1−β)ββ m∗ υ
m∗m∗h
1 k∗ ∗ rυ∗rυ∗ i ∗
a3 = α s −
r
γ
−
s
γ
+
(1
−
β)s
 fρe ,
υ
β m∗
m∗
m∗
r∗
a4 = −αγsfρ∗e υ∗ > 0,
m

(40)
(41)
(42)
(43)

The inequality holds under (10) and (12).
According to the Routh-Hurwitz criterion, the following is the necessary and sufficient condition for the local
asymptotic stability of the long-run equilibrium of System (K):a1 > 0, a3 > 0, a4 > 0,(a1 a2 − a3 )a3 − a21 a4 > 0.
24 A similar assumption is made for the same purpose in Asada (1991).

14

It then follows from (36), (41) and (42) that necessary conditions for the stability is given by

a2 > 0, a3 > 0,

or
h
αs

r∗ i
r∗
1
r∗
1
γ
−
(1
−
β)
−
β
+
(1
−
β)
 < 0,γ −β(δ + n + a)β(δ + n + a) ηr∗
ηr∗
ηr∗
h
1
r∗
r∗
r∗ i
1−
γ − ∗ γ − (1 − β) ∗  < 0,∗β(δ + n + a) ηr
ηr
ηr

(44)
(45)

where

r∗ = r

 1 

m∗
∗ ∗
m
r
ηr∗ =
.
∗
rυ

,

(46)
(47)

It can be seen from (13), (14), (35) and (46) that ηr∗ , defined by (47), is equal to the interest elasticity of liquidity
preference evaluated at the long-run equilibrium.25
Conditions (44) and (45) are examined to discuss the long-run stability in our Keynesian system. As we haveobserved, the principle of effective demand implies that quantity adjustment is carried out fast enough. In thisrespect, the speed of quantity adjustment α should be large enough in the Keynesian theory.26 If both the degree ofwage flexibility γ and the speed of revisions of expectations  are large enough, condition (44) is violated and hence
the long-run equilibrium is unstable, provided that α is large enough. Also, if the interest elasticity of liquidity
preference ηr∗ (evaluated at the long-run equilibrium) is sufficiently large or if the long-run equilibrium is stuck at
the “liquidity trap,” both conditions (44) and (45) are violated and hence the long-run stability is lost, as long as
both γ and  are large enough.27 Thus, in the Keynesian theory, characterized by fast quantity adjustment or by
the “liquidity trap,” the long-run equilibrium is more likely to be unstable, as the nominal wage is more flexible
and expectations on inflation are more frequently revised. In particular, a high degree of wage flexibility is notconducive to stability in the Keynesian theory.28 This conclusion is contrasted with the classical conclusion on theeffect of wage flexibility on stability.
25 The interest elasticity of liquidity preference is given byηr = −

rl0 (r)
m r
=
> 0.
l(r)
y rυ

26 The principle of effective demand in statics, represented by (4), can be taken as the limiting case of α → ∞ in the quantity
adjustment described by (30).
27 This is consistent with the conclusion in Yoshikawa’s (1981) dynamic Keynesian model abstracting from capital formation.
28 This conclusion supports the following view on the flexibility of wages, presented in Keynes’ General Theory:

To suppose that a flexible wage policy is a right and proper adjunct of a system which on the whole is one of laissez-faire,is the opposite of truth. (Keynes 1936, p. 269)

15

It is possible to explain the mechanism of instability of System (K) in a plain way. Assume that the state of
the economy was originally at the long-run equilibrium but the rate of utilization y suddenly falls. If the degreeof wage flexibility γ and the speed of revisions of expectation  are large enough, the response to this fall can be
described by the following diagram:

y↓


(33)
(30)

 −
−→ m ↑ −−→ y ↑
(34)
(30)

 −
−→ π e ↓↓ −−→ y ↓↓.

where ↑, ↓ and ↓↓ indicate a rise, decline and sharp decline in the variable on the left side, respectively. This
diagram illustrates the destabilizing effects of wage flexibility and of expectations on deflation in our Keynesiansystem.29

3.3Classical theory in dynamics

In the classical theory, as we have argued, it is not aggregate output but the price level that is changed in responseto the difference between aggregate demand and output. In the presence of expectations on inflation or deflation,
it is also reasonable to assume that they are passed on to price changes.
Thus, we postulate that the rate of inflation is determined as follows:E − Y Ṗ=α+ πe ,
P
Y∗
which can be written as follows:h y  y 
iṖ= α f ,r− π e k − sy + π e ,
P
k
m

(48)

where α is a positive constant which is here interpreted as the speed of price adjustment. This is a formulation of
the “classical” version of price dynamics.30
In the classical theory, aggregate employment is determined by the first postulate of the classical theory. Given
the Cobb-Douglas production function (15), the first postulate implies that aggregate employment and output are
related in the following fashion:
WN
= β.
PY
It then follows that
Ẇ
Ṅ
Ṗ
Ẏ
+
− −
= 0.
W
N
P
Y
29 For discussions on the existence (and uniqueness) of persistent business cycles in related Keynesian models, see Murakami (2014,2018, 2019, 2020) and Murakami and Asada (2018).
30 Eq. (48) is a modified version of the law of price dynamics in the “Keynes-Wicksell” model (cf. Stein 1969; Fischer 1972).

16

which can, by (20) and (25), reduced toẆ  1 ẏ Ṅ ∗  Ṗ  ẏ Ẏ ∗ 
− −
= 0.
+
+
+
W
β y N∗Py Y∗
It is seen from (17), (23), (26) and (48) thatih y  y ioẏβ n h y  y 
α f ,r− π e k − sy + (1 − β) f , r− π e − (δ + n + a) − γ(y 1/β − 1) .
=
y
1−β
k
m
k
m

(49)

This describes quantity dynamics in the classical theory.
Given the definition of the money coefficient (38), its rate of change can be derived as follows:
ṁ
Ṁ
Ṗ
Ẏ ∗
=
− − ∗.
m
M
P
Y
It then follows from (23), (27) and (48) thath y  y ih y  y iṁ= µ − n − a − α f ,r
− π e k − sy − (1 − β) f , r− π e − (δ + n + a) − π e .
m
k
m
k
m

(50)

It is seen from (28) and (48) that revisions of the expected rate of inflation can be described byih y  y − π e k − sy .
π̇ e = α f , r
k
m

(51)

Thus, we obtain the following system for the classical theory in dynamics:
ih y  y ioẏβ n h y  y =
α f ,r− π e k − sy + (1 − β) f , r− π e − (δ + n + a) − γ(y 1/β − 1) ,
y
1−β
k
m
k
m
h




i
y
y
k̇
= β f ,r− π e − (δ + n + a) ,kkmh y  y ih y  y iṁe= µ − n − a − α f ,r
− π k − sy − (1 − β) f , r− π e − (δ + n + a) − π e ,
m
k
m
k
m
h y  y iee− π k − sy .
π̇ = α f , rkm
The system of these equations is denoted by “System (C)” (to mean “Classical”).31
31 System (C) may be regarded as a long-term extension of Tobin’s (1975) M (Marshall) model.

17

(49)
(24)
(50)
(51)

The long-run equilibrium values of y, k, m and π e are given byih y  y ih y  y − π e k − sy + (1 − β) f , r− π e − (δ + n + a) − γ(y 1/β − 1),
0 = α f ,rkmkm
y  y e
0 = f ,r− π − (δ + n + a),kmh y  y ih y  y i
0 = µ − n − a − α f ,r
− π e k − sy − (1 − β) f , r− π e − (δ + n + a) − π e ,kmkmy  y 
0 = f ,r− π e k − sy.
km
It is seen that these equilibrium values in System (C) are identical with those in System (K), characterized by(35)-(38). The existence and uniqueness of long-run equilibrium can then be guaranteed by (39) in System (C).
The stability of System (C) can be examined in the same way as that of System (K). The Jacobian matrix of
System (C) evaluated at the long-run equilibrium, denoted by JC∗ , can be given by





JC∗ = βj21 + (βj41 − γ)/(1 − β)β[j22 + j42 /(1 − β)]β[j23 + j43 /(1 − β)]∗βk j21∗

βk j22∗

βk j23−[(1 − β)j21 + j41 ]m∗−[(1 − β)j22 + j42 ]m∗−[(1 − β)j23 + j43 ]m∗

j41

j42

j43

β[j24 + j44 /(1 − β)]






βk j24,−[(1 − β)j24 + j44 ]m∗ 

j44
∗

where
1 ∗
1 ∗ ∗
1
1
f +
f e r , j22 = − ∗ 2 fx∗ , j23 = − ∗ 2 fρ∗e rυ∗ , j24 = −fρ∗e ,k ∗ x m∗ ρ υ
(k )
(m )


k∗
1
k∗
j41 = α fx∗ − s + ∗ fρ∗e rυ∗ , j42 = α ∗ (s − fx∗ ), j43 = −α ∗ 2 fρ∗e rυ∗ , j44 = −αk ∗ fρ∗e .
m
k
(m )
j21 =

The characteristic equation associated with JC∗ can be calculated as follows:
λ4 + b1 λ3 + b2 λ2 + b3 λ + b4 = 0,

where
h β

∗
γ
β rυ∗ i∗ rυ−α(s − fx∗ ) + k ∗ fρ∗e  −
−
f
,
e
ρ
1−β
1−β
1 − β m∗
m∗
h α


r∗ 
β fx∗
r∗ i
r∗  βb2 =k ∗ fρ∗e  − υ∗ +− fρ∗e υ∗ γ − αfρ∗e υ∗
s+ ,
∗
1−β
m
1−β k
m
m 1−β
∗ 
∗ i
αγ ∗ h rr
b3 =f e  βs − k ∗ υ∗ − βs υ∗ ,
1−β ρ
m
m
∗
αβ
r
b4 = −
γsfρ∗e υ∗ > 0.
1−β
m
b1 =

The last inequality holds under (10) and (12).

18

(52)
(53)
(54)
(55)

The necessary and sufficient condition for the local asymptotic stability in System (C) can be obtained in thesame way as in System (K):b1 > 0, b3 > 0, b4 > 0,(b1 b2 − b3 )b3 − b21 b4 > 0.
With (46), (47) and (52)-(55), they can be reduced to32

α

h β

γ
β r∗ ir∗+(s − fx∗ ) + k ∗ fρ∗e  −− fρ∗e ∗ > 0,∗1−β
1 − β ηr
1−β
ηr

∗
∗
r
r
 βs − k ∗ ∗ − βs ∗ < 0,ηr
ηrg(γ)(= (1 − β)3 [(b1 b2 − b3 )b3 − b21 b4 ]/(αγfρ∗e ))nh α
∗ ih 
∗
r∗ 
β fx∗
r∗ i
r∗ o 2∗ r∗r=
k ∗ fρ∗e  − ∗ +
−
f

βs
−
k
−
βs
+
s
γ + g1 γ + g2 < 0
e
ρ
1−β
ηr
1 − β k∗
ηr∗
ηr∗
ηr∗
ηr∗

(56)
(57)

(58)

To discuss the stability in our classical system, we make the “classical” assumption that the interest elasticity
of liquidity preference (evaluated at the long-run equilibrium), ηr∗ , is nearly equal to zero (as in the purest form of
the quantity theory of money). Under this classical assumption, conditions (56) and (57) are satisfied. Moreover,condition (58) is also fulfilled if the degree of wage flexibility γ is large enough, as long as the speed of revisions of
expectations  is not so large. This confirms the stabilizing effect of wage flexibility in the classical theory. As wehave observed, it is in the effect of wage flexibility on stability that the classical theory can be distinguished mostfrom the Keynesian theory.
The mechanism of stability of System (C) can be explained with a schematic diagram. Assume as usual thatthe state of the economy was originally at the long-run equilibrium but the rate of utilization y suddenly falls. If
the degree of wage flexibility γ is large enough, the response to this fall can be described by the following diagram:

y↓


(49)

 −
−→ y ↑↑
(51)
(49)

 −
−→ π e ↓ −−→ y ↓.

This illustrates the stabilizing effect of wage flexibility in our classical system.33

3.4

Keynesian and classical theories in dynamics: summary

We have explored the dynamic properties of two dynamic systems with Keynesian features and classical features.
Our Keynesian and classical systems possess the same long-run equilibrium with full employment (or the natural rateof unemployment) but are characterized by different dynamic adjustments. Because of the difference in dynamics,
32 The exact values of g

1 and g2 in (58) are not necessary for our analysis.
33 For the possibility of persistent cyclical fluctuations in a classical model (with public capital), see Murakami and Sasaki (2020).

19

the same long-run equilibrium can have totally different stability properties between the Keynesian and classicalsystems. Indeed, the flexibility of wages is conducive to stability in our classical system while it is not in our
Keynesian system.
As we have argued, it is not the existence of underemployment equilibrium but the (in-)stability of fullemployment equilibrium that makes the fundamental difference between the Keynesian and classical systems. Our
analysis has confirmed this view in a long-run context.

4

Conclusion

This paper has reexamined the fundamental difference between the Keynesian and classical theories from staticand dynamic perspectives. As we have seen, the rigidity of nominal wages plays an essential role in establishing
involuntary unemployment as an equilibrium phenomenon. For this reason, the Keynesian theory is widely considered a theory built on the rigidity of nominal wages. If involuntary unemployment is taken as a disequilibrium
phenomenon in contrast to full-employment equilibrium, however, wage rigidity is no longer a prerequisite for it;
unless the long-run equilibrium has stability, full employment does not possess practical relevancy and involuntary
unemployment prevails for a long while. Indeed, the (same) long-run equilibrium is more likely to be stable in ourclassical system while it is more likely to be unstable in our Keynesian system, as the nominal wage is more flexible.
Our analysis shows that the distinction between the Keynesian and classical theories in terms of wage rigidity may
not be appropriate even in a long-term context.

Acknowledgments
This work was supported by Chuo University Personal Research Grant and JSPS KAKENHI Grant Number
18K12748.

Conflict of interest
The author declares that he has no conflict of interest.

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