Goodwin Accelerator Model Revisited with Piecewise Linear Delay Investment ()
1. Introduction
It has been well-known that Goodwin’s business cycle model with a delayed nonlinear accelerator [1] can generate multiple solutions. Depending on specified forms of the initial functions and specified parameter values, it gives rise to smooth cyclic oscillations or sawtooth (i.e., slow-rapid) oscillations. This paper aims to analytically and numerically investigate these cyclic properties of Goodwin’s model by solving the time delay equations and performing simulations.
[1] presents five different versions of the nonlinear accelerator-multiplier model with investment delay. The first version has the simplest form assuming a piecewise linear function with three levels of investment and aims to exhibit how non-linearities give rise to endogenous cycles without relying on structurally unstable parameters, exogenous shocks, etc. The second version replaces the piecewise linear investment function with a smooth nonlinear investment function. Although persistent cyclical oscillations are shown to exist, the second version includes unfavorable phenomena, that is, discontinuous investment jumps, which are not observed in the real economic world. “In order to come close to reality” ( [1] , p. 11), the third version introduces an investment delay. However, no analytical considerations are given to this version. The existence of an endogenous business cycle is confirmed in the fourth version, which is a linear approximation of the third version with respect to the investment delay. Finally, alternation of autonomous expenditure over time is taken into account in the fifth version, which becomes a forced oscillation system.
This paper reconstructs the third version having a piecewise linear investment function with fixed time delay. It is a complement to [2] in which the effects caused by investment delay as well as consumption delay are considered. It is also an extension of [3] in which the dynamics of Goodwin’s model is examined under continuously distributed time delays and the existence of the multiple limit cycles is analytically and numerically shown. Following the method of successive integration provided by [4] , we derive explicit forms of the solutions and obtain conditions under which the smooth or sawtooth oscillations emerge. With the same spirit, [5] examines Goodwin’s model. Their focus is mainly put on the relaxation (i.e., sawtooth) oscillations. We step forward and investigate periodic properties of the smooth oscillations, which will be called Goodwin oscillations henceforth. Our main concerns in this paper are on the role of the fixed delay for the birth of cyclic macro dynamics
The paper is organized as follows. In Section 2, the Goodwin model without delay is considered to see how nonlinearities of the model contribute emergence of cyclic dynamics. In Section 3, an investment delay is introduced to construct. Effects of investment delay on the smooth oscillations are considered in Section 4 and those on the sawtooth oscillations are done in Section 5. Section 6 contains some concluding remarks.
2. Basic Model
To find out how nonlinearity works to generate endogenous cycles, we review the second version of Goodwin’s model, which we call the basic model,
(1)
Here k is the capital stock, y the national income, a the marginal propensity to consume, which is positive and less than unity, and the reciprocal of e is a positive adjustment coefficient. Since the dot over variables means time differentiation,
and
are the rates of change in capital (i.e., investment) and national income. The first equation of (1) defines an adjustment process of the national income in which national income rises or falls according to whether investment is larger or smaller than savings. The second equation determines the induced investment based on the acceleration principle with which investment depends on the rate of changes in the national income in the following way,
(2)
where
and
. This function is piecewise linear and has three distinct regions. Accordingly, there are two threshold values of
denoted as
and
and the investment is proportional to the change in the national income in the middle region,
, but becomes perfectly inflexible (i.e., inelastic) in the upper region
or the lower region
. These values are thought to be “ceiling” and “floor” of investment where the ceiling is assumed to be three time higher than the floor as it was the case in Goodwin’s model.
Inserting the second equation of (1) into the first one and moving the terms on the right hand side to the left give an implicit form of the dynamic equation for national income y,
(3)
where the stationary point of the basic model is
for all t. With Equation (2) it is reduced to either
(4)
if
is in the middle region or
(5)
where
if
is in the upper region and
if in the lower region. Equations (4) and (5) are linear and thus solvable. We see graphically and then analytically how dynamics proceeds.
2.1. Phase Plot
Solving (4) and (5) for
presents an alternative expression of dynamic equation
where
(6)
Once the initial value is given, the whole evolution of national income is determined. The phase diagram with
is shown in Figure 1 in which
is described by a mirror-imaged N-shaped curve in the
plane.1 The stationary point is at the origin denoted by E. The locus of
is the positive sloping line in the middle region while it is the negative sloping upper or lower line in the upper or lower region. For each value of
, there is a unique corresponding y value determined to make a point
satisfy Equation (6) and it is also determined whether y is increasing or decreasing at that point. So the direction of the trajectory is given in all points of the phase diagram. The directions are shown by arrows. Let A denote the local maximum point of the curve with positive y and
always, and let C be the local minimum point with negative y and
. Point B and D have the same y values as at points A and C, respectively. Notice that the direction of the dynamic evolution goes from
to C, from the origin to C, from the origin to A and also from
to A. Selecting the initial point denoted as S on the positive-sloping line, the evolution starts at this point and moves upward until point A as indicated by arrow. Then it cannot continue on the continuous curve after point A since the direction of evolution changes. Therefore it jumps to point B and continues along the same direction until point C, where the same problem occurs, so another jump occurs to point D and evolution continues until point A, at which the next round repeats itself. Thus the differential Equation (3) with the piecewise linear investment function (2) can give rise to a closed orbit ABCD constituting a self-sustaining slow-rapid oscillation. The stationary point is unstable if
, however, the oscillation is stable in a sense that the locus
sooner or later converges to the same oscillation regardless of a
1Mathematica version 11 is used to perform simulations and illustrate this and the following figures. The color versions of the figures are found in DP278 at http://www.chuo-u.ac.jp/research/institutes/economic/publication/disccusion.
Figure 1. Slow-rapid oscillation along
with
.
selection of the initial point. This is a simple exhibition of emerging a stable endogenous cycle of national income.
The vertex of the closed oscillation in Figure 1 are
where the maximum and minimum values of
along the cycle are
while the maximum and minimum values of y along the cycle are
It should be noticed that the instability and the nonlinearity is crucial sources for the birth of persistent oscillations since the instability of the stationary point prevents trajectories from converging and the nonlinearities such as the ceiling and floor prevent trajectories from diverging.
Jumping behavior leads to the kinked time trajectory of
and the discontinuous time trajectory of
that are shown as the blue and red curves in Figure 2.
holds at the upper kinked point of the blue curve and
at the lower kinked point. The red trajectory from 0 to
describes the movement of
from point S to A. At time
when the left-most red curve defined on interval
arrives at the upper horizontal dotted line, the red curve jumps straightly down to the starting point of the lower red curve defined on interval
, the point of which correspond to point B. At time
, the red curve crosses the lower dotted line from below and the intersection corresponds to point C at which the red curve jumps straightly
Figure 2. Time trajectories of
(the kinked blue curve) and
(the discontinuous red curves).
up to the downward-sloping red curve defined on interval
, that then crosses the upper horizontal dotted line and the cross-point correspond to point A. Figure 2 illustrates the same dynamics of Figure 1 from a different view point.
2.2. Explicit Solutions
Selecting an initial point, we can determine an explicit form of the corresponding time trajectory and its rate of change. In particular, we take an initial point on the positive sloping part of the
curve such as
where
. If
is in the middle region, Equation (4) yields explicit forms of the solution
and its time derivative,
(7)
and if
enters the upper or lower region, Equation (5) yields the following forms of the solution
and its time derivative,
(8)
where
if i is even and
if i is odd.
Since
and
is increasing in t, solving
for t presents an arrival time
,
Substituting
into
and
yields
implying that point
corresponds to point A in Figure 1. Solutions in (7) describe the movement from point S to point A. At
, the dynamic system (4) is switched to Equation (5) with
. Equations in (8) with
presents explicit forms of the solution and its time derivative,
(9)
Since the time trajectory
is continuous in t, solving
gives the value of
,
With this
, we have
Thus point
corresponds to point B to which point A jumps. This rapid change is described by the vertical movement of the red curve along the vertical line at
in Figure 2. Since
and
increases in t, solving
gives a necessary time to arrive at
,
In the same way above, it is possible to show that
which is point C. The movement from point B to C is described by solutions in (9). At time
, the dynamic system with
is switched to the dynamic system with
and (8) with
presents explicit forms of the solutions
(10)
Solving
presents the value of
,
It is also able to be shown that point
is identical with point D, implying a jump to point D from point C. Solving
presents a time when
arrives at
,
At
, we can confine that the trajectory comes back to point A at which the following holds,
A new round starts as time goes further and the same procedure is applied to obtain explicit forms of the solutions for
. Time segments of
and
that constitute one cycle of national income are now given by (9) and (10). Since the length of one cycle is measured by the time period between one upper (or lower) kinked point and next upper (or lower) kinked point, it is given by
Further the length of the recession period along segment BC in which national income is decreasing is
while the length of the recovery period along segment DA in which national income is increasing is
In what follows, we will perform numerical simulations with the set of the parameter values given below which are the same parameter values used in [1] and [2] . Needless to say, these particular values of the parameters are selected only for analytical simplicity and do not affect qualitative aspects of the results to be obtained.
Assumption 1
and
In particular, Figure 1 and Figure 2 are illustrated under Assumption 1 and the initial values of
and
, where the pair
corresponds to point S in Figure 2. The critical times at which the system switching occurs are given
by solving
2in [4] , it is assumed that
is one year delay. Under the specified values of the parameters,
, implying the length is considered to be 5.218 years.
The length of one cycle given by
is about 5.218 years.2 In the same way, the recession period from one peak to trough of the cycle is given by
years for i being even while the recovery period from one trough to peak by
years for i being odd. The constant
solves
with
and the numerical results are as follows,
3. Delay Model
We now investigate how the investment delay affects time paths of national income. Observing the fact that, in real economy, plans and their realizations need time to take effects, [1] introduces the investment delay,
, between decisions to invest and the corresponding outlays in order, first, to come closer to reality and second, to eliminate unrealistic discontinuous jumps. Consequently the investment function (6) is modified as follows
(11)
With this modification, the dynamic Equation (3) turns to be
(12)
that we call the delay model.3 Equation (12) is reduced to a linear delay differential equation of neutral type if the delayed rate of change in national income stays in the middle region
(13)
and it remains to be a linear ordinary differential Equation (5) if the delayed rate is in the upper or lower region. To solve the delay equation, we need an initial function that determines behavior of y prior to time zero,
Although [1] does not analyze delay dynamics generated by the third version, [4] , in addition to numerical analysis, derive the explicit forms of the piecewise continuous solutions of
under the piecewise linear investment function (11). We follow their method of successive integration to solve the delay equation and derive the explicit forms of time trajectories of
and
. Since a cyclic oscillation has been shown to exist in the basic model, our main concern here is to see how the presence of the investment delay and the selection of the initial function affect characteristics of such a sawtooth oscillation obtained in the basic model.
It has been examined by [4] that the birth of oscillations in the Goodwin model are caused by a selected form of the initial function and the length of delay. For the sake of analytical simplicity, we assume the constant initial function in the following numerical simulations.
Assumption 2
(14)
3.1. Local Stability
3Goodwin assumes a smooth form of the nonlinear investment function in his third version.
It is well known that if the characteristic polynomial of a linear neutral equation has roots only with negative real parts, then the stationary point is locally asymptotically stable. The normal procedure for solving this equation is to try an exponential form of the solution. Substituting
into (13) and rearranging terms, we obtain the corresponding characteristic equation:
To check stability, we determine conditions under which all roots of this characteristic equation lie in the left or right half of the complex plane. Dividing both sides of the characteristic equation by e and introducing the new variables
(15)
we rewrite the characteristic equation as
(16)
[6] derive explicit conditions for stability/instability of the n-th order linear scalar neutral delay differential equation with a single delay. Since (13) is a special case of the n-th order equation, applying their result (i.e., Theorem 2.1) leads to the following: the real parts of the solutions of Equation (16) are positive for all
if
. The first result on the fixed delay model is summarized as follows:
Lemma 1 If
, then the zero solution of the fixed delay model (13) is locally unstable for all
.
On the other hand, if
or
, characteristic Equation (16) has at most finitely many eigenvalues with positive real parts. The eigenvalue is real and negative when
. The roots of the characteristic equation are functions of the delay. Although it is expected that all roots have negative real parts for small values of
, the real parts of some roots may change their signs to positive from negative as the lengths of the delay increases. The stability of the zero solution may change. Such phenomena are often referred to as stability switches. We will next prove that stability switching, however, cannot take place in the delayed model.
Lemma 2 If
, then the zero solution of the fixed delay model (13) is locally stable for all
.
Proof. 1) It can be checked that
is not a solution of (16) because substituting
yields
that contradicts. If the stability switches at
, then (16) must have a pair of pure conjugate imaginary roots with
. Thus to find the critical value of
, we assume that
, with
, is a root of (16) for
. Substituting
into (16), we have
and
Moving b and w to the right hand sides and adding the squares of the resultant equations, we obtain
Since
and
as
is assumed, there is no w that satisfies the last equation. In other words, there are no roots of (16) crossing the imaginary axis when q increases. No stability switch occurs and thus the zero solution is locally asymptotically stable for any
.
2) In case of
in which
, the characteristic equation becomes
(17)
It is clear that
is not a solution of (17) since
. Thus we can assume that a root of (17) has non-negative real part,
with
for some
. From (17), we have
where the last inequality is due to
for
and
. Hence
where the direction of inequality contradicts the assumption that
and
. Hence it is impossible for the characteristic equation to have roots with nonnegative real parts. Accordingly, all roots of (17) must have negative real parts for all
.
Lemmas 1 and 2 imply the following theorem concerning local stability of the delay model (13).
Theorem 3 For any
, the zero solution of the delay model (13) is locally asymptotically stable if
and unstable if
.
We call
in Assumption 2 an initial value for convenience. Fixing the length of delay at
, we illustrate a bifurcation diagram with respect to the initial value in Figure 3. For given value of
, the dynamic system runs for
. The solution for
are discarded to eliminate the initial disturbances and the maximum and minimum values of the resultant solutions for
are plotted against
. The bifurcation parameter
increases from −10 to 6 with increment of 0.01 and for each value of
, the same calculation procedure is repeated. As is seen in Figure 3 and already pointed out by [5] , the delay dynamic system with the constant initial function has the two threshold initial values
and
such that the sawtooth oscillations arise for
and so do the Goodwinian oscillations otherwise. These values depend on the length of the delay and are numerically determined as
and
under
. In the following, we first set
and consider Goodwinian oscillations in Section 4 and then examine sawtooth oscillations with
in Section 5.
4. Goodwinian Oscillations
Given Assumptions 1 and 2 with
, the time trajectories of
and
are illustrated by the blue and red curves, respectively, in Figure 4 in
Figure 3. Bifurcation diagram with respect to
.
Figure 4. Time trajectories of
and
.
which we can see that delay time trajectories show sharp differences from non-delay time trajectories depicted in Figure 2. The interval including the whole parts of one cycle where
starts at point S and ends at point E is divided into eight subintervals, each of which is distinguished by heavy or light gray color. Solving non-delay dynamic Equation (5) or delay dynamic Equation (13), we will derive the explicit forms of these trajectories in each subinterval where the detailed derivations are presented in Appendix I.
4.1. Time Trajectories
We omit consideration in interval
with
as behavior there strongly depends on a choice of initial point. In the first interval
where
and
solves the equation
, the blue and red trajectories are controlled by Equation (5) with
,
4The explicit forms are given in Appendix I.
where
. At, the red trajectory
crosses the lower horizontal dotted line at
and the crossing point is denoted by the left most green dot in Figure 4. The boundary values of this interval are
As seen in Figure 4, the blue trajectory is kinked and the red curve jumps downward at
. This discontinuity is shown as follows. Let
and
be a solution and its derivative in interval
.4 Then, constant
is determined so as to satisfy
, that is, the end point of
is coincided with the starting point of
. Hence it first implies the continuity of
at
. Secondly, the solutions of
and
should satisfy the following dynamic equations respectively,
Subtracting the second equation from the first presents
where the last inequality implies discontinuity of the derivative at
.
In the second interval
with
, applying successive integration for dynamic equation
gives explicit forms of the solutions,
where
The integral constant
is obtained by solving
. The boundary values for the end points of interval
are
It can be numerically as well as graphically checked that
and
. In consequence
and
are smoothly connected (i.e., continuous and differentiable). On the other hand,
for
induces system change at
leading to that
and
are connected with kink (i.e., continuous and non-differentiable).
For t in the third interval
with
, we have
. Hence, successive integration implies that Equation (13) with
yields the trajectories described by
where
Since
and
hold, the blue and red trajectories are continuous at
. The boundary values for the right endpoint of interval
are
As is seen in Figure 4, the red curve
crosses the upper horizontal dotted line from below at
and
for
. This crossing point is also denoted by the green dot.
In the fourth interval
with
, Equation (13) with
determines the trajectories,
where
Since interval
is very narrow, the right end point of
is labelled at the upper part of Figure 4 to avoid the notational congestion. Due to the continuity of the blue and red trajectories at
and
hold. The boundary values for the right end point of interval
are calculated as
Since
holds in the fifth interval
where
and
solves
, Equation (13) with
gives the following forms of the solution and its time derivative,
where
. The crossing point of the red curve with the upper horizontal dotted line at
is denoted by the green dot. The boundary values for the right end point of this interval are
Since
holds for t of the sixth interval
with
dynamic Equation (13) with
determines the following evolution of
and
:
where
The boundary values for the right end point of interval
are
It is seen that the red curve crosses the lower horizontal dotted line from above at
.
In the seventh interval
with
, applying successive integration to dynamic Equation (13) with
yields the forms of the solutions,
where
The boundary values for the right end point of interval
are
Finally, in the eighth interval
with
,
for
implies dynamic Equation (5) with
controls the trajectories,
where
. Notice that
and
hold. The length of the period is about 10 years.5 Very roughly speaking, the recovery period could be approximately 4.7 years from
to
and then the recession period is 5.3 years. The same cycle repeats itself for
.
4.2. Phase Plot
Calculating the boundary values of each interval
, we have the following set of points
in the phase diagram of Figure 5.
Point denoted by (S) and (E) is the starting point and the ending point of the cyclic oscillation, both of which are identical. Equation (5) with
governs decreasing movements from (S) to (2) and from (8) to (E) along the lower red line whereas Equation (5) with
controls upward movements
Figure 5. Phase diagram of Goodwin Equation (12).
5With the same parameter values but different initial values, [1] analytically obtained a 9 years cycle and [4] numerically got an 8.12 year cycle.
from (5) to (6) along the upper red line. On the other hand, movements from (2) to (5) and from (6) to (8) along the dotted curves between these two lines are described by Equation (13). The switching of dynamic equations occurs at the following points:
Point (2) at which Equation (5) with
is changed to Equation (13);
Point (5) at which Equation (13) to Equation (5) with
;
Point (6) at which Equation (5) with
to Equation (13);
Point (8) at which Equation (13) to Equation (5) with
;
Points (3), (4) and (7) at which Equations (13) have at different forms of
We can verify the following.
Theorem 4 The cyclic time trajectories of
and
are continuous at these switching points.
Proof. 1) At point (2) with
,
is connected to
and so is
to
. Integral constant
of
is determined so as to solve
. Further
and
should satisfy the dynamic equations at
,
holds at
and
by definition of
, both of which lead to
. Therefore the above two dynamic equations are identical and thus two solutions of these dynamic equations take the same values at
, namely,
and
.
2) At point (5) with
,
is connected to
and so is
to
. Integral constant
of
is determined so as to solve
. Further,
and
should satisfy the dynamic equations at
,
holds at
and
by definition of
, both of which lead to
. Therefore the above two dynamic equations are identical and thus two solutions of these dynamic equations take the same values at
, namely,
and
. The same procedure applies for points (6) and (8).
3) At point (3) with
,
and
satisfy the dynamic equations
and
as
. From (i), we already have
. Further the integral constant
of
solves
. Then substituting the second equation from the first equation presents
. The same procedure applies for Points (4) and (7).
This theorem confirms no jumps of the derivatives at the switching points of the dynamic system, implying the smooth time trajectory of national income just like observed business cycle. This is what [1] aims to obtain. So we summarize this results as follows:
Theorem 5 If the initial value
of the initial function and the length of delay
are selected such as
or
, then the delay model can generate smooth oscillations of national income.
5. Sawtooth Oscillations
Under Assumptions 1 and 2 with
, Figure 6 illustrates trajectories of
(blue curve) and
(red curve) for
. The blue trajectory has kinks and the red trajectory jumps at
. These are initial parts of the trajectories that eventually converge to sawtooth oscillations. The shapes of these trajectories are different from those in Figure 2 and Figure 4.
It has been pointed out by [4] that the delay model also gives rise to sawtooth-like oscillations.6 Our main aim of this section is to analytically reproduce these numerical results to understand why a trajectory
has kinks and its derivative
makes jumps. To this end, we start to divide the interval
into five subintervals with respect to the length of delay
,
where
and
with
. Detailed derivations of the forms of
and
in each interval are presented in Appendix II.
Figure 6. Time trajectories of
(blue) and
(red) for
.
6More precisely, [4] found at least twenty five other limit cycles were also solution to the delay model with the same parameter values. Further it was indicated that there were an infinite number of additional solutions.
5.1. Time Trajectories
The constant initial function
for
is selected. The dynamic Equation (13) with
yields the following forms of the solution and its derivative
The boundary values for the end points of interval
are
In the second interval
, solving (13) with
by successive integration yields the following forms
where
Integral constant
is determined so as to satisfy
, implying the continuity of the blue curve at
. The discontinuity of the red curve at that point can be shown in the same way as in the case of Goodwin cycle. The solutions
and
satisfy the corresponding dynamic equations at
,
where
and
implying that
. Subtracting the first equation from the second equation presents
The last inequality confirms the discontinuity of the red curve at
. The boundary values for the end points of interval
are
It is then numerically confirmed that
As shown in the Appendix II,
is the value at which the red curve crosses the upper horizontal dotted line once from above and divides the interval
into two subintervals,
and
where
. So we derive a solution of the differential equation in each interval. In interval
Equation (5) with
presents the forms of
and
:
where solving
gives constant value
. The boundary values for the end points of interval
are
where the continuity of the blue curve and the discontinuity of the red curve at
are also numerically confirmed,
On the other hand, in interval
, Equation (13) with
yields the solution and its derivative,
where
The boundary values for the end points of interval
are
where the blue and red curves are confirmed to be continuous at
,
Due to the values of
and
obtained in the Appendix II, interval
is divided into three subintervals
,
and
by
and
. Since
for
, Equation (5) with
yields the solution of the differential equation and its derivative
The boundary values for the end points of interval
are
where the blue curve is continuous and the red curve jumps at
,
In
, Equation (13) with
gives the solution and its derivative
where
The boundary values for the end points of interval
are
where the blue and red curves are continuous at
,
Since
for
Equation (5) with
yields the solution and its derivative
The boundary values for the end points of interval
are
where the blue and red curves are continuous at
Due to the crossing values
and
obtained in the Appendix II, interval
is divided into three subintervals
and
and
by
and
. Since
for
, Equation (5) with
implies the solution and its derivative,
The boundary values for the end points of interval
are
where the blue curve of
is continuous but kinked at
and accordingly, the red curve of
jumps at
,
In
Equation (13) with
yields the solution and its derivative
where
The boundary values for the end points of interval
are
Since
for
, Equation (5) with
yields the solution and its derivative
The boundary values for the end points of interval
are
where the blue and red curves are continuous at
,
Notice that the red curves in intervals,
and
intersect the upper and lower horizontal dotted curves and a difference between t-values is getting smaller,
As seen above, the delay differential Equation (13) describes dynamic behavior of
for
for
while the linear ordinary Equation (5) determines the form of
for
or
. For
, the same types of the solutions are obtained and the size of
shrinks, implying that as k increases, the resultant shape of the solution form of
approaches the sawtooth shape at whose vertices
jumps.
5.2. Phase Plot
We now turn attention to the phase diagram in the
plane. The boundary values of each trajectory that have been obtained are summarized in the following table and plotted in Figure 7. The red curves are the locus of
and the green parallelogram is a sawtooth limit cycle. Black dotted curve connects the boundary values. The following points are shown in Figure 7:
Point (1) is the starting point of interval
at
and the delay Equation (13) transports it to point (2) at
. At the beginning of the second interval
,
jumps, which makes the horizontal move of point (2) to point (3) that arrives at point (4) at the end of
. At the beginning of the third interval
,
jumps again and point (4) horizontally shifts to point (5) at which the dynamic system is changed to the linear Equation (5) with
making a move along the upper downward line to point (6) as t proceeds from
to
Figure 7. Phase diagram of sawtooth oscillation.
. The dynamic system is changed back to the delay equation at
and then the dotted trajectory leaves the upper red curve heading to point (7). This is because for
, the delay equation with
controls dynamic behavior. On the way, the
curve crosses the upper and lower horizontal dotted curves as seen in Figure 6. The move reaches point (7) at the end of
and jumps to point (8) at the beginning of interval
in which we see signs of sawtooth oscillations. Two intersections obtained in interval
causes two changes of the dynamic system; the linear Equation (5) with
governs the movement from point (8) to point (9) and the delay Equation (13) controls the movement from point (9) to point (10) and then the system is changed to the linear Equation (5) with
managing the movement along the lower downward line from point (10) to point (11). At the beginning of interval
, a jump from point (11) to point (12) occurs and the further movement along the upper red line to point (13) is controlled by the linear equation with
, point (13) to point (14) by the delay equation and point (14) to point (15) by the linear equation with
. Point (15) jumps to a point on the upper red line and the dynamic system change occurs as well in
for integer
as in
. By doing so, the trajectory gradually approaches to the green sawtooth limit cycle as time goes on. It is noticed that a jump occurs at the local maximum or minimum point in the non-delay model whereas even at the middle of these boundary values in the delay model.
6. Concluding Remarks
This paper presented Goodwin’s nonlinear accelerator model augmented with investment delay in continuous time scales. Assuming a piecewise linear investment function and specifying the values of the model’s parameters, explicit forms of sawtooth oscillations were derived when the initial value of the constant initial function was selected in the neighborhood of the steady state. Otherwise the same was done for Goodwin oscillation. With these numerical results, the paper exhibited valuable insights into the macro dynamics of market economies: the delay nonlinear accelerator-multiplier mechanism can be a source of various types of business cycles; economies starting in the neighborhood of the steady state could achieve regular ups and downs while economies starting away from the steady state presented persistent and irregular cycles.
Appendix I
In this Appendix, we provide mathematical underpinnings for Goodwin oscillations. Since the investment delay could make
kinked and
discontinuous at
for integer n, the time interval for
is reconstructed as the union of unit intervals
for n and then a dynamic equation defined over interval
is solved to obtain explicit forms of time trajectory and its derivative. Dynamic equation is solved with successive integration in which an initial point or function is the solution of dynamic equation defined in the proceeding subinterval.
Interval 0:
where
and
.
The initial function
determines dynamics for
. Since
by Assumption 2, solving
presents explicit forms of the solution and its derivative
and as can be seen in Figure 4, the red curve is below the lower dotted line or
(A-1)
Derivation of (G-I)
Interval 1:
where
.
(A-1) implies
for
and then (8) with
are
where solving
gives
and the following holds,
(A-2)
Intervals 2 and 3:
where
for
.
In the same way as in interval
, (8) with
leads to the identical forms of
and
for
as the ones defined in
,
Notice that the red curve crosses the lower dotted horizontal line from below in interval
. Solving
for t gives
with which the following inequalities hold
(A-3)
Interval 4:
where
and
.
Due to (A-3),
divides interval
into two subintervals,
and
. First, (2) with
for
presents the same forms
So far we have seen that in
with
, a time trajectory of
is described by
where
and
. A form of
is obtained by time-differentiating
and is identical with
.
and
construct system (G-I) defined in Section 4.1.
Derivation of (G-II)
On the other hand for
, the investment is delayed and (12) with
is written as
Multiplying both sides by the term
and arranging the terms present
Integrating both sides yields
Thus the form of the solution is
where
The integral constant
is obtained by solving
,
The derivative of
is
where
It can be checked that
(A-4)
where
Interval 5:
where
.
As in interval
, the threshold value
divides interval
into two subintervals,
and
. For
, Equation (12) with
leads to the solution and its derivative that are the same as the ones obtained in
,
Therefore time trajectories for
are described by
both of which form (G-II) defined in Section 4.1 where
and
for
are written as
and
.
Derivation of (G-III)
Dynamic Equation (12) with
for
has a solution
where
Solving
presents
Differentiating
gives
where
Since the
curve crosses the upper horizontal line from below at
we then have
(A-5)
Interval 6:
where
.
Due to the two values,
and
, we define two threshold values
and
, both of which then divide interval
into three subintervals,
and
. Accordingly conditions in (A-5) determine the induced investment as
In consequence, the form of the solution and its derivative in
are
Therefore blue and red trajectories for
are described by
both of which form (G-III) defined in Section 4.1 where
and
for
are written as
and
.
Derivation of (G-IV)
For
, successive integral leads to the solution
where
Solving
yields
Differentiating
with respect to t is, after arranging the terms, it can be written as
where
Therefore blue and red trajectories in
are described by
both of which form (G-IV) defined in Section 4.1 where
and
for
are written as
and
.
Derivation of (G-V)
For
, Equation (9) implies a form of the solution,
where solving
gives
Interval 7:
where
.
Since Equation (2) implies
for
and
, Equation (9) implies that
Notice that the
curve intersects the horizontal dotted line at
from above at the following point,
Thereby,
(A-7)
Interval 8:
where
.
The threshold value in interval
defines a new threshold value
in interval
that divides interval
into two subintervals,
and
. Since
for
, (9) implies
Therefore trajectories in
are described by
and
construct (G-V) defined in Section 4.1 where
is replaced with
.
Derivation of (G-VI)
On the other hand,
for
, successive integration implies that the solution of
has the form,
where
Time differentiation of
is
where
It can be checked that
(A-8)
Interval 9:
where
.
The threshold value
divides interval
into two subintervals,
and
. Since the first equation of (A-8) implies
for
, the solution of (12) is
Therefore trajectories in
are described by
both of which form (G-VI) defined in Section 4.1where
and
for
, are written as
and
.
Derivation of (G-VII)
On the other hand,
for
implies the following form of the solution,
where
and
where
It is to be noticed that the
curve intersects the horizontal line at
from above at
with which
and
Interval 10:
where
.
The threshold value
divides interval
into two subintervals,
and
. Since
for
, we have
Therefore time trajectories in
are described by
both of which form (G-VII) defined in Section 4.1 where
and
for
are written as
and
.
Derivation of (G-VIII)
On the other hand
for
implies that
and
The end point
is obtained by solving
for t. Therefore time trajectories in
are given by
and
form (G-VIII) defined in Section 4.1 where
is denoted by
. A cycle starts at
and finishes at
with
. The length of this cycle is equal to
that is about 10.06.
Appendix II
Our main aim of this appendix is to analytically reproduce these numerical results of sawtooth oscillations to understand why a trajectory
has kinks (alternatively, its derivative
makes jumps). To this end, we start to divide the whole interval
into five subintervals with respect to the length of delay
,
with
and
.
Derivative of (S-I)
Interval I:
where
and
.
Equation (13) with the constant initial function
yields the solution of the form
and differentiation gives its derivative form
Both of which form (S-I) defined in Section 5.1. Since
,
and
for
,
stays in the middle region,
(A-9)
Derivative of (S-II)
Interval II:
where
.
Due to (2) and (A-9),
which is substituted into (13) to obtain,
Since this equation can be written as
Integrating both sides and arranging the terms present the solution of
, denoted as
,
Since the trajectory of
is piecewise continuous, solving
presents
The form of
is rewritten as
with
A time derivative of
is
with
These
and
form (S-II) defined in Section 5.1. Under Assumption 1, we calculate the boundary values of interval
,
The red curve crosses the upper horizontal dotted line once from above at point
with which the following inequalities hold, as is seen in
in Figure 2,
(A-10)
Derivations of (G-IIIa), (G-IIIb) and (G-IIIc)
Interval III:
where
.
Due to the value of
, the interval
is divided into two subintervals,
and
where
. Delay investment is differently determined according to conditions in (A-10),
(18)
different dynamic systems are defined on different subintervals. So we derive the solution of the differential equation in each subinterval.
Interval III-1:
.
In this subinterval, Equation (8) with the first equation of (A-11) yields the solution
and its derivative
where solving
gives
These two functions form (S-IIIa) where
and
are denoted by
and
. Boundary values of this interval are
Interval III-2:
.
In this interval, we have Equation (13) with the second equation of (A-11) that is rewritten as
where
Rewriting the dynamic equation as
and integrating both sides give the following form of a solution,
where solving
gives
Then the form of
is rewritten as
where
and
A derivative of
is
where
and
Boundary values are
and
These
and
form (S-IIIb) in which
and
are denoted as
and
. As is seen in Figure 6, the red curve crosses the horizontal dotted lines at
and
once at points
It is apparent from Figure 6 that
(A-12)
Derivation of (S-IVa), (S-IVb) and (S-IVc)
Interval IV:
where
.
Due to the properties described in (A-12), interval
is divided into three subintervals by
and
in which
(A-13)
(A-14)
and
(A-15)
Interval IV-1:
Equation (2) with (A-13) implies that the solution of the differential equation
is given by
where
solves
A derivative of
is
These
and
form (S-IVa) in which
and
are denoted as
and
We then have the boundary values of
of interval
,
Interval IV-2:
.
Rewriting the delay differential Equation (2) with (A-14) as
where
Successive integration yields the solution,
or
with
where
solves
A derivative of
is
with
These
and
form (S-IVb) in which
and
are denoted as
and
The boundary values of the interval are
Further the downward-sloping curve of
intersects each of the two dotted horizontal lines at
and
at the points
and
Interval IV-3:
Equation (A-15) implies that the dynamic Equation (2) has the following forms for the solution
and its derivative
an
where
solves
These
and
form (S-IVc) in which
and
are denoted as
and
. The boundary values of the interval are
It is seen that
Derivation of (S-V)
Interval V:
Due to the threshold values,
and
, interval
is divided into three subintervals by
and
in which
(A-16)
(A-17)
and
(A-18)
Repeating the same procedure done just above, we can derive the explicit forms of a solution of the delay dynamic equation.
Interval V-1:
.
Equation (8) with (A-16) presents the following forms of the solutions,
and
where
solves
. These
and
form (S-Va) in which
and
are replaced with
and
.
Interval V-2:
.
Applying successive integration to Equation (2) with (A-17) yields the following forms of the solutions,
with
where
solves
. A derivative of
is
with
These
and
form (S-Vb) in which
and
are replaced with
and
. As is seen in Figure 6, the red curve crosses the horizontal dotted lines at
and
once at points
To avoid notational congestion in Figure 6,
and
are not labelled.
Interval lV-3:
Equation (8) with (A-18) presents the following forms of the solutions,
and
where
solves
. These
and
form (S-Vc) in which
and
are replaced with
and
.
It is seen that