Discussion Paper No.234

Delay Kaldor-Kalecki Model Revisited
Akio Matsumoto
Chuo University

Ferenc Szidarovszky
University of Pécs

September 2014

INSTITUTE OF ECONOMIC RESEARCH
Chuo University
Tokyo, JapanDelay Kaldor-Kalecki Model Revisited
Akio Matsumotoy
Chuo University

Ferenc Szidarovszkyz
University of Pécs

Abstract
This paper studies dynamics of the Kaldor-Kalecki model of national
income and capital stock. The investment function is assumed to have
not only a Kaldorian characteristics, namely, a S-shaped form but also a
Kaleckian characteristics, that is, a gestation delay between "investment
decision" and "investment implementation." We divide the analysis into
two parts. In the …rst part, we assume that the time period under consideration is short enough so that the capital stock is not a¤ected by the ‡owof investment and then examine the delay e¤ect on dynamics of national
income. In the second part, taking the capital accumulation into account,
we draw attention to how the delay a¤ects cyclic dynamics observed in the
non-delay Kaldor-Kalecki model. It is demonstrated that the investment
delay quantitatively a¤ects the dynamic behavior but not qualitatively.

The authors highly appreciate the …nancial supports from the MEXT-Supported Program
for the Strategic Research Foundation at Private Universities 2013-2017 and the Japan Society
for the Promotion of Science (Grant-in-Aid for Scienti…c Research (C) 24530202, 25380238
and 26380316). The usual disclaimers apply.
y Professor, Department of Economics, Senior Researcher, Institute of Further Develpmentof Dynamic Economic Research, Chuo University, 742-1, Higashi-Nakano, Hachioji, Tokyo,
192-0393, Japan; akiom@tamacc.chuo-u.ac.jp
z Professor, Department of Applied Mathematics, University of Pécs, Ifjúság, u 6, H-7624,
Pécs, Hungary; szidarka@gmail.com

1

1

Introduction

In real economy, macroeconomic variables such as national income, capital accumulation, interest rate, etc. exhibit cyclic ‡uctuations. As a natural consequence, it has been the main interest in studying macro economic dynamics todetect endogenous sources of such cyclic behavior. Since investment is considered to be the key factor to cyclic dynamics for evolution of national economy,
a lot of e¤orts have been devoted to studying investment determinations. As
early as in 1930s, a few years before the publication of the Keynes’ General
Theory, Kalecki (1935) introduces an idea of the consumption function and themultiplier in his statistic analysis of macro economy. Furthermore, regarding
investment in the dynamic analysis, he assumes a lag between "investment order" and "investment installation" and call it a gestation lag of investment.1
Adopting a linear investment function, he constructs a macro dynamic model
as a delay di¤erential equation of retarded type. It had been already shownthat Kalecki’s model gives rise to a cycle due to the delay. Kaldor (1940), on
the other hand, studies the evolution of production and capital formation andbelieves that nonlinearities of behavioral equations could be a clincher in such
cyclic oscillations. He builts a 2D model of national income and captial withnonlinear investment and saving functions. Its basic movements consist of two
sorts. One is the movements along the nonlinear functions and the other is shiftsof these functions according to capital accumulations. So far, it has also been
con…rmed in various ways that the Kaldor model is capable of generating cyclic
behavior when nonlinearities become strong enough. Indeed, Ichimura (1955)
reduces the model to the Liénard equation, Chang and Smyth (1971) rigorously
show the existence of a limit cycle by applying the Poincaré-Bendixson theorem
and so does Lorenz (1993) by the Hopf bifurcation theorem. Furthermore, Grasman and Wentzel (1994) show multi-stability in the Kaldor model, that is, thecoexistence of stable and unstable cycles when the equilibrium is locally stable.
"Nonlinearity" and "delay" are now treated as two of the main ingredients for
endogenous cycles. More than a half century after Kaldor (1940), Krawiec and
Szydlowski (1999) combine the Kaldor model with the Kaleckian time delay ininvestment to built a delay business cycle model. Their model is often called(delay) Kaldor-Kalecki model. To emphasize the role of delay, they assume alinear investment function as in the Kalecki model and then show the occurrenceof a limit cycle with respect to time delay.
We will revisit the Kaldor-Kalecki model and concern the roles of nonlinearity and delay for the birth of limit cycles in order to complement Krawiec and
Szydlowski (1999). For this purpose, we …rst recapitulate the 2D Kaldor-Kalecki
model and specify the investment function. Then we proceed to the analysis ofdelay dynamics with two steps. At the …rst step, we investigate dynamics ofnational income, keeping the stock of capital …xed. As is well known, no cyclic
behavior arises in the Kaldor model without captial accumulation. We focus
attention on the e¤ect caused by delay on such stable dynamics. Since invest1 Since "lag" and "delay" do not have distinctive di¤erent meanings, we use these words
interchangeably. In particular, we mainly use "delay" in this study.

2ment and savings are short term, eliminating captial or …xing capital implies
short-run dynamic analysis. At the second step we consider delay dynamics
of national income and capital accumulation. As is seen above, the nonlinear
Kaldor model without delay can generate not only one limit cycle but also multiple limit cycles while the linear delay Kaldor-Kalecki model also give rise toa limit cycle. A natural question we rise is how the delay a¤ ects these cyclic
behavior in the long-run.
This paper is organized as follows. In Section 2, the basic elements of the
Kaldor-Kalecki model is rebuilt. In Section 3, short-run dynamics is examined
under the investment delay. In Section 4, after reviewing the Kaldor model
without delay quickly, we analytically and numerically detect the delay e¤ect
on cyclic dynamics of the Kaldor-Kalecki model from the long-run point of viewin the sense that capital accumulation is explicitly taken into account. Section
5 contains some concluding remarks.

2

Delay Dynamic Model

Kaldor (1940) relates investment to the level in income (i.e., pro…t principle)
and extends it by proposing a sigmoidal, instead of linear, investment function.
A brief description of his model is given by two equations,
Y_ (t) =[ (Y (t); K(t))

S(Y (t))]
(1)

_
K(t)
= (Y (t); K(t))K(t):

where Y (t) is national income at time t; K(t) denotes capital, (Y (t); K(t))
is an investment function, S(Y (t)) is a savings function and the parametersand denote the adjustment coe¢ cient and the rate of depreciation. The…rst equation describes the national income adjustment process and the second
describes capital accumulation process. As will be reviewed soon, the Kaldor
model is able to generate endogenous limit cycles. There are two ingredients forthe birth of the cycles, the nonlinearity of investment in Y and the dependency
of investment in K: These two prevent Y and K for global divergence when theequilibrium is locally unstable. Using our notation and following his spirit, aKaleckian investment function can be presented byI(t) = (Y (t); K(t))where denotes the gestation delay.2 Replacing the investment function in thesecond equation of system (2) with Kaleckian function yields the Kaldor-Kalecki
2 There are several extensions of this delay investment function. Kadder and Talibi Alaudi(2008) introduce time delay also in capital stock in capital accumulation equation (i.e.,(Y (t); K(t))). Zhou and Li (2009) assume that the investment function in the capital
accumulation depends on the income and the captial stock at di¤errent gestation periods (i.e.,(Y (t1 ); K(t2 )) with 1 6= 2 ).

3

model of the income and captial stock,
Y_ (t) =[ (Y (t); K(t))

S(Y (t))] ;
(2)

_
K(t)
= (Y (t); K(t))K(t):We plan to analyze dynamics generated by (2) with two steps. At …rst step, we
con…ne attention to short-run dynamics. To this end, we impose two assumptions:
Assumption 1. The time period under consideration is short enough so thatthe capital stock is not a¤ected by the ‡ow of investment.
Assumption 2. The investment function in the …rst equation of system (2)contains the delay.
As a consequence of these modi…cations, the second equation concerning theevolution of capital is eliminated and the variable K in the investment function
also disappears. Thus the modi…ed Kaldor-Kalecki model can be presented bya one dimensional nonlinear di¤erential equation with one time delay,
Y_ (t) =f'[Y (t)]

S[Y (t)]g(3)

After examining short-run dynamics, we proceed to the second step in which
the dynamic analysis of Y (t) and K(t) generated by system (2) is investigated.
It is considered to be long-run dynamics in the sense that the national income
as well as the capital stocks evolve over time. We will numerically con…rm theexisting analytical results and then consider the e¤ects of the delay on them.

3

Short-run Dynamics

We …rst make the following two assumptions for the sake of analytical simplicity.
Assumption 3. The saving function is linear and has no autonomous savings,
S(Y ) = sY; 0 < s < 1:
Assumption 4. The investment function has the S-shaped form,'(Y ) = A 2

1(CY +D)2+ BY

K:Parameter K in '(Y ) is positive implying the …xed level of the capital stock.3
At a stationary state of equation (3), two conditions, Y_ (t) = 0 and Y (t) =Y (t) = Y e for all t0; are satis…ed. Economically, these conditions canbe restated as investment is equal to saving at the equilibrium. In Figure 1, we
3 This is a simpli…ed version of the investment function adopted in Lorenz (1987). It isreplaced with the full version in the latter half of this paper.

4superimpose the linear saving function, sY with s = 0:282, on the three sigmoid
investment functions with A = 35, B = 0:02; C = 0:01; D = 0:00001 and threedi¤erent values of K (i.e., K = 5; 10; 15). The equilibrium state is determinedby the intersection of these curves.

Figure 1. Determination of equilibriumwith di¤erent values of K
When the …xed value of K is small, we will have a high level of investment andthus a high short-run equilibrium level YHe of national income. As K increases,
the investment curve shifts downward. Due to the S-shaped form, there is acase where the saving curve crosses the investment curve three times. These
intersections are denoted by the black dotes and their x-coordinates are thecorresponding equilibrium levels of national income, Y1e < Y2e < Y3e . A further
increase of K shifts the investment curve downward enough resulting in only
one intersection denoted by the lower green dotted point and the corresponding
national income is YLe . Investment is small here and thus the equilibrium levelis also small.
We draw attention to stability of the three equilibrium points obtained underthe middle value of K. Let Y i = Y Yie for i = 1; 2; 3: The linear approximationof equation (3) is
Y_ i (t) = i Y i (t)
sY i (t)(4)
where i = '0 (Yie ) > 0 denotes the slope of the investment curve at the equilibrium income Yie . Substituting an exponential solution, e t u; yields the characteristic equation,
+ s
= 0:
(5)
ie
In the absence of time delay (i.e.,= 0), we simply have,

i =

( i
5

s):

(6)

If i < s or i < 0; then the equilibrium income Yie is stable. Since investment
is greater or less than savings in the left or right of the intersection, any nonequilibrium Y approaches its equilibrium level by the multiplier. Similarly it islocally unstable if i > s or i > 0: The sign of i s depends on the slopes of theinvestment and saving curves evaluated at the equilibrium point. Apparently,
among three equilibrium values, 1 s < 0 at e1 ; 2 s > 0 at e2 and 3 s < 0
at e3 . The middle one (i.e., Y2e ) is locally unstable and the remaining high andlow ones (i.e., Y1e and Y3e ) are locally stable. It is also clear that the uniqueequilibrium YLe or YHe is locally stable. As a benchmark, we summarize theseresults as follows.
Theorem 1 When equation (4) with = 0 has three equilibria, then the middle one is locally unstable while both of the larger and smaller ones are locallyasymptotically stable.
Kaldor (1940) focuses on the unstable equilibrium. The main key to it is thecaptial accumulation that shifts the short-term investment function over time
which gives rise to a cycle. On the other hand, we focus on the stable equilibrium
and consider whether the delay a¤ects its stability. Hence we return to equation(5) with > 0: It should be noticed that = 0 does not solve this equation
unless s = i . So if stability of Yie switches at = ; then equation (5) musthave a pair of pure imaginary roots there. Since roots of a real function always
come in conjugate pairs, we assume = i! with ! > 0. Substitution of thisroot divides equation (5) into the real and imaginary parts,
s

i cos

! = 0;!+i sin! = 0:(7)
Moving the …rst term in each equation to the right, squaring the resultant
equations and adding them together yield!2 =2( i + s)( i

s)

(8)

where the …rst two factors on the right hand side are positive. If i s > 0;then there is a positive ! and stability switch can occur. The condition, i >s; could be possible only at the middle equilibrium point Y2e : However since it
is already shown to be locally unstable for = 0, Y2e still remains unstable forany > 0: On the other hand, i < s holds at the other equilibrium points. If
s 0; then there is no ! > 0 implying that stability switch does not occur.
i
Summarizing these results yields the following:
Theorem 2 For any equilibria of the delay equation (3), no stability switch
occurs for any positive value of the delay.
Theorem 2 implies that the delay does not a¤ect asymptotic dynamics of thedelay model (3). In spite of this result, we show that it really matters in transient
dynamics. In particular, following Beddington and May (1975), we show that6the delay increases the real parts of the eigenvalues for a stable equilibrium pointand decreases the magnitude for an unstable equilibrium point. We proceed toillustrate these e¤ects of the investment delay in the three equilibria case. We
assume that = x + iy with y
0 and substitute it into equation (5), Afterarranging the terms, we obtainx + iy =

s+

ie

x

cos y + i

x

ie

sin y

:

Comparing both sides …nds that the real and imaginary parts are
ie

x

x=

s+

cos y ;y=x
sin y ;
ie(9)from which we derive the form of the real part depending on y and ;x=sy cot y :(10)

Moving the …rst term in the right hand side of the …rst equation in (9) to the left
hand side, squaring both sides of the resultant equation and adding the squareof the second equations to it yield the imaginary part depending on x and ,q22(x + s) :(11)
y = ( i ) e 2x

Solving equation (9) with y = 0 for x yields the real part functional for . The
equation isx + s = ie x
(12)which clearly has a unique real solution for x that is denoted by x( ): For
xi (0) = ( i

= 0;

s)

which is identical with equation (6). By implicitly di¤erentiating this equation(12), we have that
dx( )
=
d

ie

x( )

dx( )
d

x( )

implying that
dx( )
=
d

x( )

ie

1+

i

x( )

e

x( )

:

So if x( ) > 0; then dx( )=d < 0 implying that x( ) decreases, and if x( ) < 0;then dx( )=d > 0 implying that x( ) increases. In both cases jx( )j decreases.
Hence we have Theorem 3.
Theorem 3 Larger delays result in the smaller absolute value of the real parts
and thus slow down convergence speed to stable equilibrium.

74

Long-run Dynamics

4.1

Kaldor Model

We review the original Kaldor model (1). So far, in this model, it is demonstrated that the nonlinearity of investment functions leads to the two remarkable
results. One is the existence of a stable limit cycle shown by Chang and Smyth(1971) with applying the Poincáre-Bendixson theorem when the equilibrium islocally unstable equilibrium and the other is the coexistence of a stable limit
cycle and a unstable limit cycle by Grasman and Wentzel (1994) with the useof a Hopf bifurcation theorem when the equilibrium is locally stable. Figure
2 graphically con…rms these results with the following form of the separable
investment function,4(Y; K) = 25 2

1
(0:015Y +0:00001)2+ 0:05Y + 5

320
K

3

:

and
= 3. The positive sloping dashed curve is the K_ = 0 locus and theconvex-concave dashed curve is the Y_ = 0 locus. The intersection of these curves
determines the stationary equilibrium point denoted by (Y e ; K e ): In Figure 2(A)where we take s = 0:3; the stationary point (Y e ; K e ) ' (54:05; 324:32), is locallyunstable as we will see shortly and two trajectories starting in a neighborhoodof the stationary point explosively oscillate and approach the limit cycle. In
Figure 2(B), s is decreased to 0:282 and any other parameters are kept to be…xed. The equilibrium point (Y e ; K e ) ' (67:72; 381:94; 54:05) becomes stable,
which is enclosed by an unstable inner limit cycle which is, in turn, enclosed byan outer stable limit cycle. A green trajectory starting inside of the inner limit
cycle converges to the equilibrium point while both the red trajectory starting
outside of the inner limit cycle and the blue trajectory starting outside of theouter limit cycle approaches the outer cycle. The black dots denote the initial
4 Lorenz (1987) uses this form of the function to show the occurrence of chaotic motion in a
multisector Kaldorian business cycle model. Grasman and Wentzel (1994) also use this form.

8

points in simulations.

(A) One limit cycle(B) Two limit cycles

Figure 2. Limit cycles in the Kaldor model
Kaldorian nonlinear dynamics is often examined with respect to the value ofthe adjustment coe¢ cient .5 Figure 2 indicates that Kaldorian dynamics has
also strong sensitivity to the value of s: Two di¤erent dynamics illustrated in
Figure 2 imply that there is a threshold value of s and the changing of the parameter through this value causes a qualitative change in the nature of dynamics.
A bifurcation diagram gives good insights into what is happening to evolutionof the equilibrium point as the value of the parameter is changed. In Figure
3(A), the value of s is increased from 0:25 to 0:38 with an increment 1=10000.
For each value of s, the dynamic system (1) runs for t 2 [0; 500] and the local
maximum and minimum values of the speci…ed trajectory for 400 t 500 are
plotted. If the diagram shows one point against s; it implies that the equilibrium is stable and the trajectory converges to it. If it shows two points, then
the trajectory has one maximum and one minimum point, implying an emergence of a limit cycle. Figure 3(A) roughly indicates that the equilibrium point
is locally stable for smaller values of s; loses its stability bifurcating to a limit
cycle for medium values and then gains stability for larger values. Figure 3(B)is an enlargement of Figure 3(A) around the two threshold values, s and s : If
s < s ; the Kaldor system generates a stable equilibrium. As s passes s ; thetrajectories of the system converges to a stable limit cycle with the radius equal
to the distance between the upper and lower red branches. On the other hand,for s 2 [s ; s ] ' [0:281; 0; 283],6 cyclic dynamics can emerge. Notice that thedotted vertical lime at s = 0:3 crosses the bifurcation diagram twice in Figure3(B) and the corresponding limit cycle is depicted in Figure 2(A). Similarly, the5 See, for example, Lorenz (1993)
6 The value of s is numerically obtained and the value of s is analytically determined asawill be seen.

9

dotted vertical line at s = 0:282 crosses the blue curve twice and the red curve
twice in Figure 3(B) and the subcritical Hopf bifurcation leads to multi-stability,that is, the coexistence of a stable equilibrium, an unstable limit cycle and astable limit cycle as illustrated in Figure 2(B). A distance between the upperand lower blue branches corresponds to the radius of the unstable limit cycle.

(A) 0:25s

0:38(B) 0:28s

0:284

Figure 3. Bifurcation diagrams with respect to s

4.2

Kaldor-Kalecki model

We now turn attention to the Kaldor-Kalecki model (2) that has the same
stationary equilibrium point, (Y e ; K e ); as the Kaldor model (1). As is alreadymentioned, Krawiec and Szydlowski (1999) show the existence of limit cycle in
the Kaldor-Kalecki model. In this section we revisit this property and compare
the results in the non-delay model with the results in the delay model to …nd
how the delay a¤ect dynamics.
Let Y = Y Y e and K = K K e : By linearizing (2) at the equilibrium
point, we have
Y_ (t) = [(s)Y (t)K (t)] ;(13)
K_ (t) = Y (t) ( + )K (t);
where

1

=

@
0:52 2 (0:015Y e +0:00001)2= 0:05 +3@Y(0:015Y e + 0:00001)

and
=

@
=
@K

10

15

(320)3:(K e )4

Notice that the values ofand depend on the the point where they areevaluated, although their dependency is not explicitly expressed in the following. Suppose exponential solutions, Y (t) = e t u and K (t) = e t v: Then thecharacteristic equation is written as2+ a + b + ce

=0

(14)

where
a=( + )
b=

(

(

s);

s)( + )

andc=
We …rst examine the non-delay case (i.e.,characteristic equation is reduced to2:= 0) in which the corresponding+ a + b + c = 0:

The necessary and su¢ cient conditions for the roots of the quadratic equation
to be negative if real and to have negative real parts if complex are a > 0 andb + c > 0: Under the speci…ed values of the parameters, b + c > 0 always. In
Figure 4, the negative sloping black curve is the locus of (s; Y e ) and the closed
red curve is the locus of a = 0: The black locus crosses the red curve fourtimes at s = si for i = 1; 2; 3; 4:7 It is veri…ed that a < 0 inside and a > 0
outside.8 Hence, the stationary equilibrium obtained along the black curve islocally stable for s < s2 and s > s3 and locally unstable for s2 < s < s3 wheres1 ' 0:265; s2 ' 0:283; s3 ' 0:346 and s4 ' 0:416:
7 Notice that s

2 is identical with s : See footnote 6.
8 We use the green curve later when the delay model is examined.

11

Figure 4. Stability and instability
regions
We now return to equation (14) and examine the delay case (i.e., > 0).
Suppose that = i! with ! > 0 is a solution of the equation for some > 0:
Substituting it into the equation and separating the real and imaginary parts
present! 2 + b + c cos ! = 0;(15)
!a c sin ! = 0:Thus
c2 = (! 2b)2 + (!a)2 :

Hence
!4

(2b

2b

a2

a2 )! 2 + b2

c2 = 0and its roots are
2

! =

p
(2b

a2 )2

4(b2

c2 )

2

where
2b

a2 =

( + )2 +2(s)2 < 0:If b2 c2 < 0; then ! + > 0 and there is only one imaginary solution, = i! + :On the other hand, if b2 c2
0; then both roots are negative or complex
so no imaginary solution exist. In Figure 4, the green curve is the locus ofb2 c2 = 0: It is veri…ed that b2 c2 > 0 in the left side of the green curveand b2 c2 < 0 in the right side. Thus stability of Y e along the black curveis a¤ected by the value of s: Along the black curve, stability of Y e (as well as
stability of K e ) can be switched to instability for s1 < s < s2 and s3 < s < s412

for some value of while Y e is locally unstable for s2 < s < s3 regardless of thevalue of : In case of stability switch, solving the …rst equation of equation (15)for yields the partition curve9
=

1
cos 1
!+

! 2+ bc(16)that divides the parameter region into two subregions, stability is reserved in
one subregion and stability is lost in the other subregion. Figure 5 presents thedivision of the (s; ) plane. For = 0 (i.e., no-delay case), as it is alreadycon…rmed, stability is lost for s2 < s < s3 . As increases and becomes positive,
we have two results. One is that as far as s 2 [s2 ; s3 ]; the equilibrium is locallyunstable regardless of the values of (i.e., in the white region of Figure 5). The
other is that the instability interval of s becomes larger. Stability is preservedin the yellow region and lost in the blue region. The boundary of these regions
is the partition curve described by equation (16) that is downward-sloping fors 2 [s1 ; s2 ] and upward-sloping for s 2 [s3 ; s4 ]. The blue regions are the enlarged
instability regions due to the positive delay.

Figure 5. Partition of the (s; ) plane
Figure 5 gives the bifurcation diagrams with respect to s and reveals thee¤ects caused by increasing on dynamics with respect to s from a di¤erent
view point. The red curve is for = 0 and is identical with the one given in
Figure 3(A) although multi-stability phenomenon occurred around s2 is omitted
for the sake of graphical simplicity. The blue curve is for = 5 where s isincreased along the dotted line at = 5 in Figure 5 where the dotted line
crosses the partition curves at points a and b: Let us denote the s-values of the
9 Solving the second equation yields the same partition curve in a di¤erent form.

13intersections by sa ' 0:277 and sb ' 0:386. Stability is lost for s = sa andregained for s = sb : The similar bifurcation cascade is obtained for = 10 anddescribed by the green curve. The dotted line at = 10 crosses the partition
curves at points A and B whose s-values are sA ' 0:274 and sB ' 0:399.10
Stability is lost for s = sA and regained for s = sB : Since qualitatively di¤erent
dynamics arises according to s < s2 or s > s2 ; we …rst consider dynamics
for s > s2 : In Figure 6(A) where the bifurcation diagrams are expanding as
increases, we observe the following,(i) the equilibrium point bifurcates to a limit cycle when s crosses the downwardsloping partition curve;(ii) the amplitude (or ups and downs) of the cycle becomes larger as delay
becomes larger as illustrated by the expansion of the bifurcation diagrams;(iii) the stability-regain value of s increases as becomes larger, implying thatthe larger delay has the stronger destabilizing e¤ect by expanding the
instability region more.
We turn attention to dynamics for s < s2 to …nd that it is harder to generate
multi-stability as becomes larger. In Figure 6(B) the red curve describes thebifurcation diagrams for s < s2 ; which is the same as Figure 3(B) without theblue curves. We increase the value of from 1 to 7 and denote the corresponding
threshold values of s determined by equation (16) as k for k = 1; 2; :::; 7: Noticethat stability is lost for s = k for = k since the real parts of the eigenvalues
are zero. The black curves ending at the red dotted line at s = k imply thatmulti-stability occurs for s between the starting point of the black curves andthe ending point for = k: It is observed that the lengths of the black curves
become shorter with larger length of delay. Furthermore, for = 7; the black
curves almost disappear. Therefore it becomes harder to have multi-stability as
increases.
1 0 In Figure 5, s

A is not labelled to avoid confusion.

14

(A) Shift due to delay(B) Multi-stability

Figure 6. Bifurcation diagrams with respect to s
We perform two more numerical simulations to see how the stable equilibrium is destabilized via increasing value of (i.e., the delay e¤ect). In the above
simulations we change the value of s with …xed value of : In these simulations,
we increase the value of ; taking the value of s …xed. In the …rst example,
we take = 5 and s ' 0:277 2 [s1 ; s2 ] that is obtained via equation (16). For= 0; the equilibrium is locally asymptotically stable. When arrives at the
downward-sloping partition curve at = 5, stability is lost and further increased
induces the equilibrium to bifurcate to a limit cycle. Figure 7(A) shows a bifurcation diagram with respect to : It is seen …rst that the red curve jumps toa limit cycle at 1 = 5 via a subcritical Hopf bifurcation. It is further seen thatthe blue curve appears for 0 ' 3:45; implying the occurrence of multi-stability
for 2 [ 0 ; 1 ]. It is also veri…ed that the occurrence of multi-stability becomes
harder as the value of increases. In the second example, we change the valueof s to s ' 0:386 2 [s3 ; s4 ]. The increasing arrives at the upward-sloping
partition curve for 1 = 5. Figure 7(B) indicates that the stable equilibrium
loses stability at = 1 and bifurcates to a limit cycle via supercritical Hopf
bifurcation. Summarizing the delay e¤ects obtained above gives the following
results:(i)The equilibrium point bifurcates to a limit cycle via supercritical Hopf bifurcation when increasing crosses the upward-sloping partition curveand via subcritical Hopf bifurcation when it crosses the downward-sloping
partition curve;

15

(ii) Multi-stability can happen with respect to delay.

(A) s ' 0:277(B) s ' 0:386

Figure 7. Bifurcation diagrams with respect to5Concluding Remark

We investigate the Kaldor-Kalecki model in which the investment function has a
S-shaped form and a gestation lag of investment between "investment decision"and "investment installation." The main analysis can be divided into two parts.
In the …rst part, with a constant level of the capital stock, short-run dynamics
of national income is examined and two results are obtained. First, the delay
does not a¤ect asymptotical dynamics in the sense that no stability switch occurs for any values of the delay. Second, the convergence speed gets faster as
the delay becomes larger. In the second part, evolution of national income andthe capital accumulation are simultaneously examined. Two nonlinear phenomenon, the birth of a limit cycle and coexistence of stable and unstable limit
cycles around the stabile equilibrium point, both of which can emerge without
delay, are preserved even if the delay is introduced. However it is numericallycon…rmed that the amplitude of trajectory’s ‡uctuations becomes larger as thedelay becomes larger. It can be concluded that the delay a¤ects the long-run
as well as short-run dynamics quantitatively but not qualitatively.

16

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