International Journal of Economic Behavior and Organization
Volume 3, Issue 2-1, April 2015, Pages: 77-85

Dynamic Economic Systems with Two Time Delays

Akio Matsumoto1, Ferenc Szidarovszky2

1Department of Economics, Chuo University, Higashi-Nakano, Hachioji, Tokyo, Japan

2Department of Applied Mathematics, University of Pécs, Ifjúság, u. 6, Pécs, Hungary

Emal address:

(A. Matsumoto), (F. Szidarovszky)

To cite this article:

Akio Matsumoto, Ferenc Szidarovszky. Dynamic Economic Systems with Two Time Delays. International Journal of Economic Behavior and Organization. Special Issue: Recent Developments of Economic Theory and Its Applications. Vol. 3, No. 2-1, 2015, pp. 77-85. doi: 10.11648/j.ijebo.s.2015030201.22


Abstract: An elementary analysis is developed to determine the stability region of certain classes of ordinary differential equations with two delays. Our analysis is based on determining stability switches first where an eigenvalue is pure complex, and then checking the conditions for stability loss or stability gain. In the cases of both stability losses and stability gains Hopf bifurcation occurs giving the possibility of the birth of limit cycles.

Keywords: Multiple Delays, Monopoly Model, Multiplier-Accelerator Model, Double-Edged Effect on Stability


1. Introduction

Delay differential equations have many applications in quantitative sciences including economics, biology, engineering among others. The single delay case is well established in the literature (Hayes, 1950, Bellman and Cooke 1963, Matsumoto and Szidarovszky, 2013a), however the presence of a second delay makes the models much more complicated. The works of Hale (1979) and Hale and Huang (1993) can be considered as major contributions. Matsumoto and Szidarovszky (2012) developed a simple analytic method, which is limited to examine only some special model variants. Gu et al. (2005) developed a geometric approach applicable for analyzing a more general class of models.

In this paper two particular models are examined and the two major approaches illustrated. A brief simulation study illustrates the theoretical findings.

2. Model 1

We first extend a text-book model of monopoly in microeconomics, following Matsumoto and Szidarovszky (2013b). A single product monopoly sells its product to a homogeneous market. Let q denote the output of the firm,  the price function and  the cost function. Since  and , we call a the maximum price and b the marginal price. It is assumed that the firm knows the marginal price but does not know the maximum price. In consequence it has only an estimate of it at each time period, which is denoted by . So the firm believes that its profit is

(1)

and its best response is

(2)

Further, the firm expects the market price to be

(3)

However, the actual market price is determined by the real price function

(4)

Using these price data, the firm updates its estimate. If the firm revises the estimate in such a way that the growth rate of the estimate is proportional to the difference between the expected and actual prices, the adjustment or learning process can be modeled by the differential equation

(5)

where  is the speed of adjustment. Substituting (3) and (4) reduces this to a differential equation with respect to  as

or multiplying both sides by  generates the logistic model

(6)

which is a nonlinear differential equation.

If the firm uses two past price information, then the differential equation turns to be the delay differential equation

(7)

where  and  are positive constants while δ and η denote the delays in the price information. It is clear that unique stationary state of this equation is a. By introducing the new variable,  equation (7) is written as

(8)

where = and = are positive constants. We will first examine the asymptotical stability of the delay differential equation

The characteristic equation can be obtained by looking for the solution in the exponential form . By substitution,

or

(9)

Introduce the new variables

to reduce equation (9) to the following:

(10)

Because of symmetry we can assume that  In order to find the stability region in the  plane we will first characterize the cases when an eigenvalue is pure complex, that is, when . We can assume that  since if  is an eigenvalue, its complex conjugate is also an eigenvalue. Substituting  into equation (10) we have

If there is no delay, then  and equation (10) becomes

with a negative eigenvalue  so the system is asymptotically stable.

In the special case of  the equation becomes

The real and imaginary parts imply that

We can assume first  so from the first equation

so no stability switch is possible. If  then

implying that  and so  showing that there is no pure complex root. Hence for  the system is asymptotically stable with all

Assume now that  . The real and imaginary parts give two equations:

(11)

and

(12)

We consider the case of  first and the symmetric case of  will be discussed later. Introduce the variables

then (11) implies that

or

(13)

From (12),

implying that

(14)

Combining (13) and (14) yields

from which we can conclude that

(15)

and then from (14),

(16)

Equations (15) and (16) provide a parameterized curve in the () plane:

(17)

In order to guarantee feasibility we have to satisfy

(18)

and

(19)

Simple calculation shows that with  these relations hold if and only if

From (17) we have four cases for  and since

or

and similarly

or

However from (11) we can see that  and  must have different signs, so we have only two possibilities:

(20)

and

(21)

For each  these equations determine the values of  and  At the initial point  we have

and if  then

Therefore the starting point and end point of  are given as

and

Similarly, the starting and end points of  are as follows:

and

With fixed value of   and  have the same end point, however the starting point of  is the same as that of  Therefore the segments  and  with fixed  form a continuous curve with . They are shown in Figure 1 for . The curves  are shown in red color and curves are given in blue.

Figure 1. Partition curve in the () plane with fixing

Consider first the segment  Since  is strictly increasing in   is strictly decreasing in  By differentiation

(22)

Consider next segment  similarly to (22) we can show that

which is the same as in , since from (21),  Similarly

(23)

where we used again equation (11).

We will next examine the directions of the stability switches on the different segments of the curves  and . We fix the value of  and select  as the bifurcation parameter, so the eigenvalues are functions of  By differentiating the characteristic equation (10) implicitly with respect to we have

implying that

(24)

From the characteristic equation we have

so

If  then

and the real part of this expression has the same sign as

Hence

Consider first the case of crossing any segment  from the left. Here , so both  and  are positive. Hence stability is lost everywhere on any segment of  Consider the case when crossing the segments of  from the left. Stability is lost when  increases in and stability is gained when  decreases in At all intersections with  and  Hopf bifurcation occurs giving the possibility of the birth of limit cycles.

We can also show that at any intersection with  or  the pure complex root is single. Otherwise  would satisfy both equations

and

from which we have

By substituting  and comparing the real and imaginary parts yield

Therefore this intersection is at an extremum in  of a segment  and also at an extremum of a segment  which is impossible.

Assume next that  Then equations (11) and (12) imply that

and the curves  and  are simplified as follows:

(25)

and

(26)

The stability switching curves are shown in Figure 2 in which the stability region is the gray area.

Figure 2. Partition curve in the () plane with  

3. Model 2

In this section we consider a simple dynamic system in macroeconomics, following Matsumoto and Szidarovszky (2015). Based on Philips (1954), we construct the following dynamics model,

(t-

Here C, I and Y denote consumption, investment and national income, η>0 and δ>0 are the consumption delay and investment delay. The first equation is the consumption function, the second equation is the induced investment where the acceleration principle is assumed with >0  and . E(τ)=C(τ)+I(τ) is the total expenditure and  the last equation indicates that national income lags behind the expenditure and this delay is of exponential form. Differentiating the last equation with respect to t and substituting delayed consumption and investment into the resultant expression presents a differential equation with two fixed delays,

(27)

This is the dynamic model we will analyze. Notice that  with  is the unique stationary equilibrium. To consider its local stability, equation (27) is linearly approximated,

where . With the notation

it becomes

(28)

Here  and  Similarly to the previous model it is easy to prove that the system is stable without delays and also with a single delay, when either ,  or The corresponding characteristic equation is obtained by substituting an exponential solution, ,

(29)

Dividing its both sides by and introducing the new functions,

simplify equation (29),

(30)

The terms of this function are shown in Figure 3.

Figure 3. Triangle formed by 1,  and

Suppose that  with  then

(31)

and

(32)

Their absolute values are

and their arguments are

The triangle can be above the real line and also under the real line. In the two cases the following relations hold for angles  and :

(33)

and

(34)

In a triangle consisting of three line segments, the length of the sum of any two adjacent line segments is not shorter than the length of the remaining line segment,

and

Substituting the absolute values renders these three conditions to the following two conditions,

and

Both  and  have the same discriminant,

In the following we draw attention to the case of , otherwise  for all  implying no stability switch. Solving  gives the solutions

and so does solving ,

Since both  and are negative and both  and are positive, the two conditions,  and , are satisfied when is in interval

The internal angles,  and of the triangle in Figure 3 can be calculated by the law of cosine as

(35)

and

(36)

Solving equations (33) and (34) for  and yields

and

so we have again two stability switching curves with fixed values of  and

and

They are shown in Figure 4 for the case of . They have the same initial point  and arrive at the same end point  as  increases from  to . With fixed by increasing the value of , stability is lost at point  and regained at point . These curves are shifted to the right by increasing the value of  and up by increasing the value of .

Figure 4. Partition curve with  and

4. Simulations

In the first case, Figure 5(A) shows the six cigar-shaped domains obtained for  and  and their lower parts are colored in yellow. We fix  and increase  from  to  along the dotted horizontal line. The system is stable until  when stability is lost. It is regained at  and system remains stable until  where stability is lost, and regained again at  and so on. So stability is lost at points  and  and stability is regained at points   and  The bifurcation diagram shown in Figure 5(B) well demonstrates these observations.

Figure 5. Stability switches with

Figure 6. Stability switches with

In the second simulation we illustrate the curves  and  for  and  in Figure 6(A). The yellow domains are surrounded by  and , which are the same as in Figure 5(A). The green regions are surrounded by  and , and the orange and blue regions by  and  and by  and  respectively. The value of  is now selected. The dotted horizontal line crosses the stability switching curves many times, but not all intersections are stability switches. For example, between  and  the system is unstable regardless of several intersections between them. At  stability is regained, and lost again at . The bifurcation diagram shown in Figure 6(B) well illustrates these findings.

Let  be any point in the positive quadrant and not on the stability switching curves and consider the line segment connecting points  and  Let  be the number of intersections of this segment with the stability switching curves with stability loss and  the number of intersections with stability gain. The system is stable for  if  otherwise unstable.

5. Conclusions

Two particular economic models were examined. Both are first order ordinary differential equations with two delays. The stability switching curves were first determined where an eigenvalue is pure complex, and then the stability and instability regions were demonstrated. In the first case an elementary analytic approach was used, and in the second case a geometric approach was shown. This approach could be also used for solving the first model as well, however the more simple analytic approach cannot be used for the second model without major changes.

Acknowledgements

The first author highly appreciates the financial supports from the MEXT-Supported Program for the Strategic Research Foundation at Private Universities 2013-207, the Japan Society for the Promotion of Science (Grant-in-Aid for Scientific Research (C) 24530202, 25380238 and 26380316). The second author appreciates the hospitality of the Department of Economics of Chuo University, Tokyo while the research leading to this paper was conducted. The usual disclaimers apply. This paper is dedicated to Professor Toshikazu Ito on the occasion of his retirement at Ryukoku University.


References

  1. Bellman, R. and Cooke, K. (1963), Differential-Difference Equations. Academic Press,  New York.
  2. Gu, K., Niculescu, S. and Chen, J. (2005), On Stability Crossing Curves for General Systems with Two Delays. Journal of Mathematical Analysis and Applications, 311, 231-252.
  3. Hayes, N. D. (1950), Roots of the Transcendental Equation Associated with a Certain Difference-Differential Equation. Journal of the London Mathematical Society, 25, 226-232.
  4. Hale, J. (1979), Nonlinear Oscillations in Equations with Delays. In Nonlinear Oscillations in Biology (F. C. Hoppenstadt, ed.). Lectures in Applied Mathematics, 17, American Mathematical Society, 157-185.
  5. Hale, J. and Huang, W. (1993), Global Geometry of the Stable Regions for Two Delay Differential Equations. Journal of Mathematical Analysis and Applications, 178, 344-362.
  6. Matsumoto, A. and Szidarovszky, F. (2015), Nonlinear Multiplier-Acceelerator Model with Investment and Consumption Delay. Structural Change and Economic Dynamics, 33, 1-9.
  7. Matsumoto, A. and Szidarovszky, F. (2013a), An Elementary Study of a Class of Dynamic Systems with Single Delay. CUBO A Mathematical Journal, 15, 1-7.
  8. Matsumoto, A. and Szidarovszky, F. (2013b), Learning in Monopolies with Delayed Price Information.IERCH DP. No.203, Institute of Economic Research, Chuo University.
  9. Matsumoto, A. and Szidarovszky, F. (2012), An Elementary Study of a Class of Dynamic Systems with Two Time Delays. CUBO A Mathematical Journal, 14, 103-113.
  10. Phillips, A. (1954), Stabilization Policy in a Closed Economy.Economic Journal, 64, 290-323.

Article Tools
  Abstract
  PDF(410K)
Follow on us
ADDRESS
Science Publishing Group
548 FASHION AVENUE
NEW YORK, NY 10018
U.S.A.
Tel: (001)347-688-8931