Goodwin Accelerator Model Revisited with Piecewise Linear Delay Investment

It is well-known that Goodwin’s nonlinear delay accelerator model can generate diverse oscillations (i.e., smooth and sawtooth oscillations). It is, however, less-known what conditions are needed for these dynamics to emerge. In this study, using a piecewise linear investment function, we solve the governing delay differential equation and obtain the explicit forms of the time trajectories. In doing so, we detect conditions for persistent oscillations and also conditions for the birth of such cyclic dynamics.


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.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.

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, .
Here k is the capital stock, y the national income, α the marginal propensity to consume, which is positive and less than unity, and the reciprocal of ε is a positive adjustment coefficient.Since the dot over variables means time differentiation, ( ) where 0 ν > and 0 n > .This function is piecewise linear and has three distinct regions.Accordingly, there are two threshold values of ( ) y t  denoted as 3n ν and n ν − and the investment is proportional to the change in the national income in the middle region, M I , but becomes perfectly inflexible (i.e., inelastic) in the upper region U I or the lower region L I .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, ( where the stationary point of the basic model is ( ) ( ) 0 y t y t = =  for all t.With Equation (2) it is reduced to either ( ) ( ) ( ) ( ) if ( ) y t  is in the middle region or ( ) ( ) ( ) where is in the upper region and d n = − if in the lower region.Equations ( 4) and ( 5) are linear and thus solvable.We see graphically and then analytically how dynamics proceeds.

Phase Plot
Solving (4) and ( 5) for ( ) y t presents an alternative expression of dynamic equation ( ) ( ) Once the initial value is given, the whole evolution of national income is de-  is described by a mirror-imaged N-shaped curve in the ( ) , y y  plane. 1 The stationary point is at the origin denoted by E. The locus of ( ) y f y =  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 y  , there is a unique corresponding y value determined to make a point ( ) , y y  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 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 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 ( ) , t t , 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.

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 ( ) ( ) and if ( ) 1 e and e for 2,3, 1 where i d n = − if i is even and ( ) presents an arrival time ( ) Since the time trajectory ( ) This rapid change is described by the vertical movement of the red curve along the vertical line at ( ) In the same way above, it is possible to show that It is also able to be shown that point 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 in- 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.
The length of one cycle given by 2 i i t t + − is about 5.218 years. 2 In the same way, the recession period from one peak to trough of the cycle is given by

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, 0 θ > , between deci- sions 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 With this modification, the dynamic Equation (3) turns to be 2 In [4], it is assumed that 1 θ = is one year delay.Under the specified values of the parameters, that we call the delay model. 3Equation ( 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 ( ) ( ) ( ) ( ) 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, ( ) ( ) for 0.
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 ( ) y t 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 ( ) y t 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.

Local Stability
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 ( ) 0 e t y t y λ = 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 ε and introducing the new variables  16) has at most finitely many eigenvalues with positive real parts.The eigenvalue is real and negative when 0 θ = .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 θ , 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 v ε ≤ , then the zero solution of the fixed delay model ( 13) is locally stable for all 0 θ > .
Proof. 1) It can be checked that 0 λ = is not a solution of (16) because subs- 16) must have a pair of pure conjugate imaginary roots with θ θ = .Thus to find the critical value of θ , we assume that i 1 a < is assumed, there is no ω that satisfies the last equation.In other words, there are no roots of (16) crossing the imaginary axis when θ increases.No stability switch occurs and thus the zero solution is locally asymptotically stable for any 0 θ > .

2) In case of ε ν
= in which 1 a = , the characteristic equation becomes ( ) It is clear that 0 λ = is not a solution of (17) since 0 b > .Thus we can assume that a root of (17) has non-negative real part, u iv λ = + with 0 u ≥ for some 0 θ > .From (17), we have where the last inequality is due to where the direction of inequality contradicts the assumption that 0 u ≥ and 0 b > .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 0 θ > .Lemmas 1 and 2 imply the following theorem concerning local stability of the delay model (13).
Theorem 3 For any 0 θ > , the zero solution of the delay model ( 13) is locally asymptotically stable if ν ε ≤ and unstable if ν ε > .
We call 0 y in Assumption 2 an initial value for convenience.Fixing the length of delay at 1 θ = , we illustrate a bifurcation diagram with respect to the initial value in Figure 3.For given value of 0 y , the dynamic system runs for 0 500 t T ≤ ≤ = .The solution for 450 t ≤ are discarded to eliminate the initial disturbances and the maximum and minimum values of the resultant solutions for 450 500 t ≤ ≤ are plotted against 0 y .The bifurcation parameter 0 y in- creases from −10 to 6 with increment of 0.01 and for each value of 0 y , 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 max

Time Trajectories
We omit consideration in interval [ ] where the last inequality implies discontinuity of the derivative at S t t = .
In the second interval [ ] 03 successive integration for dynamic equation ( ) ( ) where gives the following forms of the solution and its time derivative, ( ) termines the following evolution of ( ) y t and ( ) .
( )     ( ) The length of the period is about 10 years. 5Very roughly speaking, the recovery period could be approximately 4.7 years from I t to V t and then the recession period is 5.3 years.The same cycle repeats itself for

Phase Plot
Calculating the boundary values of each interval i I , we have the following set of points ( ) ( ) ( ) in the phase diagram of Figure 5.With 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  4) and (7) at which Equations (13) have at different forms of ( )  ( ) ( ) 2) At point (5) with 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 0 y of the initial function and the length of delay θ are selected such as

Sawtooth Oscillations
Under Assumptions 1 and 2 with 2 S o y = − , Figure 6 illustrates trajectories of ( ) y t (blue curve) and ( ) . The blue trajectory has kinks and the red trajectory jumps at i t nθ = .These are initial parts of the tra- jectories 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. 6Our main aim of this section is to analytically reproduce these numerical results to understand why a trajectory ( )  ( ) y' t (red) for 0 5 t ≤ ≤ .

Time Trajectories
The constant initial function 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 1 I are ( ) ( ) ( ) ( ) α is determined so as to satisfy ( ) ( ) , implying the continuity of the blue curve at 1 t .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 ( ) y t and ( ) y t satisfy the corresponding dynamic equations at where 1 0 t t θ = + and 0 0 t = implying that ( ) ( ) 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 The boundary values for the end points of interval where the blue curve is continuous and the red curve jumps at The boundary values for the end points of interval The boundary values for the end points of interval The boundary values for the end points of interval The boundary values for the end points of interval As seen above, the delay differential Equation ( 13) describes dynamic behavior of ( ) while the linear ordinary Equation ( 5) determines the form of ( )

Phase Plot
We now turn attention to the phase diagram in the 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 1 I at 0 t = and the delay Equation .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.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:

Concluding Remarks
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 ( ) y t kinked and ( ) for n and then a dynamic equation defined over interval n I 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.
The initial function ( ) explicit forms of the solution and its derivative ( ) ( ) and e with .
and as can be seen in Figure 4, the red curve is below the lower dotted line or ( ) Derivation of (G-I) where 2 2 t = .
(A-1) implies ( ) where solving ( ) ( ) and the following holds, ( ) Intervals 2 and 3: where In the same way as in interval

Derivation of (G-II)
On the other hand for 4 b t I ∈ , the investment is delayed and (12) with ( ) ( ) Multiplying both sides by the term e e e e .d Integrating both sides yields ( ) ( ) Thus the form of the solution is The derivative of ( ) It can be checked that ( ) As in interval 4 I , the threshold value 5.03 , , b II I t t = .For  ( ) In consequence, the form of the solution and its derivative in .
Therefore blue and red trajectories in Derivation of (G-VI) On the other hand, ( ) ( )    ( )

Derivation of (G-VIII)
On the other hand ( ) ( )

Appendix II
Our main aim of this appendix is to analytically reproduce these numerical results of sawtooth oscillations to understand why a trajectory ( )  and differentiation gives its derivative form ( ) Both of which form (S-I) defined in Section 5.1.Since ( ) ( ) Derivative of (S-II) 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 ( ) Since the trajectory of ( )   Due to the value of a t , the interval 3 I is divided into two subintervals, different dynamic systems are defined on different subintervals.So we derive the solution of the differential equation in each subinterval.
In this subinterval, Equation (8) with the first equation of (A-11) yields the solution ( ) and its derivative ( ) and integrating both sides give the following form of a solution, ( ) ( ) where solving ( ) ( )     ) ( )        To avoid notational congestion in Figure 6, f t and g t are not labelled.Interval lV-3:

[ 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 un-A.Matsumoto et al.DOI: 10.4236/apm.2018.82010179 Advances in Pure Mathematics stable parameters, exogenous shocks, etc.The second version replaces the piecewise linear investment function with a smooth nonlinear investment function.

A.
Matsumoto et al.DOI: 10.4236/apm.2018.82010181 Advances in Pure Mathematics termined.The phase diagram with ε ν < is shown in Figure 1 in which ( ) y f y =

1 0 2 ,Figure 2 .
Figure 2. Time trajectories of ( ) y t (the kinked blue curve) and ( ) y t  (the disconti- to point S in Figure2.The critical times at which the system switching occurs are given , implying the length is considered to be 5.218 years.
Moving b and ω to the right hand sides and adding the squares of the resultant equations, we obtain the Goodwinian oscillations otherwise.These values depend on the length of the delay and are numerically determined

Figure 3 .
Figure 3. Bifurcation diagram with respect to 0 y .

Figure 4 .
Figure 4. Time trajectories of ( ) y t and ( ) y t  .

5
at which Equation(5) with d n = − is changed to Equation (13); Point (5) at which Equation (13) to Equation (5 above two dynamic equations are identical and thus two solutions of these dynamic equations take the same values at delay model can generate smooth oscillations of national income. t has kinks and its derivative ( ) y t  makes jumps.To this end, we start to divide the interval [ ] 0,5 I = into five subintervals with respect to the length of delay θ , Detailed derivations of the forms of ( ) y t and ( ) y t  in each interval are presented in Appendix II.

Figure 6 .
Figure 6.Time trajectories of ( ) y t (blue) and the same types of the solutions are obtained and the size of b k I shrinks, implying that as k increases, the resultant shape of the solution form of ( ) y t approaches the sawtooth shape at whose vertices ( ) y t  jumps.
solution and its derivative that are the same as the ones obtained in 4 b I , with (A-13) implies that the solution of the differential equation to Equation (2) with (A-17) yields the following forms of the solutions, seen in Figure6, the red curve crosses the horizontal dotted lines at 3n k and n k − with (A-18) presents the following forms of the solutions, 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, t  and ( ) y t  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 A. Matsumoto et al.DOI: 10.4236/apm.2018.82010180 Advances in Pure Mathematics investment is larger The movement from point B to C is described by solutions in .Matsumoto et al.  S t t= .This discontinuity is shown as follows.Let ( ) 0 y t and ( ) 0 y t  be a solution and its derivative in interval [ ] 0, S t .4Then,constant  can verify the following.
IIy t should satisfy the dynamic equations at As shown in the Appendix II, a t is the value at which the red curve crosses the upper horizontal dotted line once from above and divides the interval in continuous time scales.Assuming a piecewise linear investment function and specifying the values of the model's parameters, explicit This paper presented Goodwin's nonlinear accelerator model augmented with A. Matsumoto et al.DOI: 10.4236/apm.2018.82010202 Advances in Pure Mathematics investment delay Repeating the same procedure done just above, we can derive the explicit forms of a solution of the delay dynamic equation.