Nonlinear Problems: Mathematical Modeling, Analyzing, and Computing for Finance 2016View this Special Issue
Research Article | Open Access
Analysis of the Stability and Hopf Bifurcation of Currency Supply Delay in an Opened Kaldorian Business Cycle Model
First of all, we establish a three-dimension open Kaldorian business cycle model under the condition of the fixed exchange rate. Secondly, with regard to the model, we discuss the existence of equilibrium point and the stability of the system near it with a time delay in currency supply as the bifurcating parameters of the system. Thirdly, we discuss the existence of Hopf bifurcation and investigate the stability of periodic solution generated by the Hopf bifurcation; then the direction of the Hopf bifurcation is given. Finally, a numerical simulation is given to confirm the theoretical results. This paper plays an important role in theoretical researching on the model of business cycle, and it is crucial for decision-maker to formulate the macroeconomic policies with the conclusions of this paper.
Economy is a pivotal factor in the national development, and it not only closely links to the daily life of singular people but also plays an important role in maintaining the stability of society. Because of the complexity and the instability of the economic system operation, any tiny external condition changes may give rise to instability of the whole system. In recent years, more and more economists put their eyes on the field of nonlinear dynamic system and use it to build relevant economic models and do some research on the operation law of the economic system [1–3]. As is well known, a time delay always exists in most of economic systems. Therefore, as a branch of nonlinear dynamics, the delayed nonlinear dynamics have been used widely by economists. Kaldor used ordinary differential equations to research the business cycle model, and he proposed that nonlinear investment function and saving function were an important factor to generate periodical fluctuation in economic system . His research had given a new direction in business cycle research. Kalecki  proposed a new viewpoint in researching the business cycle: the capital for operation is related to the time. In other words, original part of the investment capital is the profit saving in a period of time previously; the investment decision will influence the capital stock , and a period or a time delay always exists in capital investment for production in economic operational process. After summing up the idea of Kaldor and Kalecki, Krawiec and Szydlowski  bring about a time delay in Kaldor model and get the Kaldor-Kalecki model which shows thatReference , in allusion to the model above, did some research on the influences of the time delay on the stability of the system and analyzed the generation condition of the Hopf bifurcation. Kaddar and Talibi Alaoui  thought that investment decision would exert an influence on both the gross value of product and capital stock. They modified the second equation in (1) and transfer the investment function into ; that is,
Reference  puts the time delay in Kaldor-Kalecki model and did some research on its dynamic properties. Wang and Wu found that the time delay would destroy the stability of the system and concluded that the variation of capital stock caused by investment decision was a factor of leading the economic system to an instable status.
Most of researches on Kaldor-Kalecki model, from the viewpoint of the functional differential equation or the viewpoint of the ordinary differential equation, concentrate on the enclosed type of economic system. Therefore, we will focus on an open Kaldorian business model following the research method and conclusion of the Kaldor-Kalecki model [11, 12]. Kaldorian business cycle theory is based on the initial thoughts of Kaldor and was enriched by Lorenz , Gandolfo , and Agliari et al. . According to , the Kaldorian model can be written in the following form:where is actual gross domestic product, is actual consumption expenditure, is physical capital stock, is actual income tax, is government expenditure, is nominal currency supply, is the market adjustment coefficient, and is the degree of capital movement.
Furthermore, we denote , , , and ; then the model can be transformed toConsidering that the right-hand sides of (4) should be analytical, we bring about the transform , and the system is class which is shown as
The time delay of money supply is a crucial factor which influences the effects of monetary policy. Monetary policy changes normally take a certain amount of time to have an effect on the economy. The time delay could span anywhere from nine months up to two years. Fiscal policy and its effects on output have a shorter time delay. When monetary policy attempts to stimulate the economy by lowering interest rates, it may take up to 18 months for evidence of any improvement in economic conditions to show up. However, there are few papers considering the macroeconomic model with currency supply delay. In this paper, we propose that there is a finite delay between the rate and the currency supply in a frame work. We study how the delay affects the macroeconomic stability. By using the theory of normal and center manifold theorem, an explicit algorithm for determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions is derived. To the best of our knowledge, this paper is the first one to deal with Hopf bifurcation about the dynamic Kaldor-Kalecki model with discrete delay in currency supply. Considering currency supply delay to macroeconomic model is very important to implement the monetary policy and the government’s stabilization policy.
This paper is organized as follows. In Section 2, we investigate the local stability of the equilibrium point of system (6). By choosing the delay as the bifurcation parameter, some sufficient conditions for the existence of Hopf bifurcation are found. In Section 3, the formulas determining the direction and the stability of the bifurcating periodic solutions are obtained by the normal form theory and center manifold theorem introduced by Hassard et al. . In Section 4, some numerical simulations consequences are given. Finally, some conclusions are made.
2. Equilibrium Stability and Existence of the Hopf Bifurcation
Bring the time delay in currency supply; system (5) becomes
At first, we will find the equilibrium of system (6) with , and the equilibrium point is equal to the solutions of the equation sets:
Then, the equilibrium of system (6) is , where
Through the linear transformation,
The equilibrium point of system (6) can be translated into the original point, and the system is transformed as
Considering Taylor expansion of the right members from (10) until the third order, we derive thatwhere is the nonlinear part of system (10) and its expression iswhere and are the bilinear and trilinear functions about the variation shown as
According to the expressions above, we denote that where
Furthermore, the characteristic equation of the linear part of system (10) iswhereWhen , according to the Routh-Hurwitz criteria, if the system meets
then all roots of (16) are negative, and at this time the equilibrium , is stable; when , according to the Hopf bifurcating condition, we assume that (16) has a pair of conjugate pure imaginary characteristic roots, and let , and substitute it into (16); we getSeparate the real and imaginary parts of (19), and we getthenSquare both sides of (21) and add them all:For simplicity, we denote thatwhere , , and .
Let ; we obtain thatthen, we discuss the roots of (24).
() When and , (24) has no positive roots.
() When , (24) has three positive roots; that is, and , where , , and .
With regard to any above, we denote
Thus, according to (21),and we will discuss transversality condition; we have the following lemmas.
Lemma 1. Assume that (16) has roots which is shown as , where and meet the conditions ; then .
Proof. Substitute into the characteristic equation (16), and differentiate it with respect to ; we haveand thenwith respect to (16). Considersubstitute it into the equation above; we havewhereThus, we haveAccording to Lemma 1, we know that if system (10) meets hypothesis () or (), then ; and if system (10) meets hypothesis (), then and ; if system (10) meets hypothesis (), then , , and , where , defined by (25).
With regard to Lemma 1 we can conclude the following.
Lemma 2. With respect to (6),(1)if system (6) meets () and (), for any , then the real parts of the roots of (16) are negative,(2)if system (6) meets () and () or () and (), then there exists a series of ; when , all roots of (16) have negative part; and when , (16) has a pair of conjugate pure imaginary roots; and when , (16) has pairs of characteristic roots with positive real part,(3)if system (6) meets conditions () and (), then there exists a series of ; when all of characteristic roots are negative and when , (16) has a pair of conjugate pure imaginary roots ,(4)if system (6) meets conditions () and (), there exists a series of , , and if , then when , all the characteristic roots of (16) have negative real part; when , (16) has a pair of characteristic roots with real part.
Theorem 3. If system (6) meets hypothesis (), one can conclude the following:(1)If system (6) meets hypothesis (), for all , the system is stable near the equilibrium .(2)If system (6) meets hypothesis () or (), for all , the system is stable near the equilibrium ; and when , system (6) undergoes Hopf bifurcation at .(3)If system (6) meets hypothesis () or (), one denotes ; system (6) is stable near the equilibrium for all , and when , system (6) undergoes Hopf bifurcation at ; that is, system (6) has a branch of periodic solutions bifurcation the zero solution near .
3. Stability of Bifurcating Periodic Solutions
In this section, formulation for determining the direction of Hopf bifurcation periodic solutions of system (6) at will be presented following the normal form method and center manifold theorem introduced by Hassard et al. in .
Denote as the criteria value of any and assume that the pure imaginary characteristic roots of system (6) are with respect to . For convenience, through linear transformation , we normalize the time delay based on system (10), and we get
Assume that ; when , the system undergoes Hopf bifurcation, and (33) can be transformed to the equation on ; that is,For , we obtainwhereand we easily know that the characteristic equation of isAccording to Riesz theorem, we take the function of bounded variation; that is, . ConsiderThen, we can define infinitely small generating element with respect to (34); we havewhere andthus, (34) is equal to abstract ordinary differential equation:
Assume that are the solutions to the characteristic equation of system (37) when , and are the characteristic value of , and the corresponding characteristic vector is , where meets the characteristic equation . That is to say that , and we havewhere; denote as the formally adjoint operator of :and assume that is the characteristic vector with respect to the characteristic value of , where meets the characteristic equation . ConsiderBy calculating, ; thus, according to bilinear form ; that is,where .
According to the solution of (33) at , we define and denotethus, on center manifold , where can be expressed into power series form with variations and ,
We can obtain the form of (34) on center manifold which is shown as follows:and we denotefor simplicity, we denote (49) as the form shown asand denotewhere (51) is the norm form of (34) on center manifold.
Furthermore, we obtain that
Substitute into formula and compare the coefficient with formula (50); andwhere
Substitute into formula (53); then
However, there are some unknown terms, , in formula , and thus in order to obtain we must calculate firstly, and with respect to formula (52) we know that ; that is,
On the other hand, the system can be transferred into , the center manifold near the origin point:
Substitute into formula above; then
Furthermore, we have