Stability , Instability and Complex Behavior in Macrodynamic Models with Policy Lag

We construct simple macrodynamic models with policy lag by means of mixed difference and differential equations, and study how lags in policy response affect the macroeconomic (in)stability. Local dynamics of the prototype model are studied analytically, and the global dynamics of the prototype and the extended models are studied by means of numerical simulations. We show that the government can stabilize the intrinsically unstable economy if the policy lag is sufficiently short, but the system becomes locally unstable when the policy lag is too long. We also show the existence of cycles and complex behavior in some range of the policy lag.


INTRODUCTION
In a classical paper, Friedman (1948) expressed the view that the government's stabilization policy may be in fact destabilizing because of the existence of the lags in policy response.However, his argument is rather intuitive and his conclusion is not derived ana.lytically from the formal model of macroeconomic interdependency.Without doubt, the analysis of policy lag is important from the practical as well as theoretical point of view.Nevertheless, even now there exist only a few formal analyses of policy lag.In this paper, we construct simple macrodynamic models with policy lag and study how lags in policy response affect the macroeconomic (in)stability.In the next section, we formulate formal models with policy lag by means of nonlinear mixed difference and differential equations (delay differential equa- tions).Prototype model is reduced to the system with only one variable, real national income (Y).
study the local dynamics of the prototype model analytically, and the conditions for local stability, local instability, and cyclical movement around the equilibrium point are detected by means of the linearization method.In section four, we study the global dynamics of prototype and extended models by using the numerical simulations.We show that the government can stabilize the intrinsically unstable economy if the policy lag is sufficiently short, but the system becomes locally unstable when the policy lag is too long.We also show the existence of cycles and complex behavior in some range of the policy lag and other parameters.
2. THE MODEL Basic system of equations in our model is expressed as follows.
where Y= real national income, C= real private consumption expenditure, I= real private invest- ment expenditure, G= real government expenditure, T= real income tax, K= real capital stock, r nominal rate of interest, M= nominal money supply, p price level, time period, 0 policy lag.
Equation (1) is the quantity adjustment process in the goods market.This equation implies that the real output fluctuates according as the excess demand in the goods market is positive or negative.Equations (2) through (5) are consump- tion function, investment function, income tax function, and equilibrium condition for money market respectively.Eq. ( 6) implies that the real money supply (M/p) is fixed, which is merely a simplifying assumption.Eq. ( 7) is the govern- ment's policy function with the delay in policy response to national income.Solving Eq. ( 5) with respect to r, we have the following 'LM equation'.r(t) r(Y(t)); rr r'(r) -Ly/Lr > 0 (8)   Substituting Eq. (4) into Eq.( 2), and substituting Eq. ( 8) into Eq. (3),we obtain the following expressions.
which is a simple type of mixed difference and differential equation (delay differential equation).We shall call the model which is summarized in the system ($1) 'model 1'.
An extended version of our model is the 'intermediate run' model in which the capital stock becomes an endogenous variable.In this case, we allow for the fact that the investment contributes to change the level of capital stock, so that we replace Eq. ( 12) with the following equation.
'Model 2' is more akin to Kaldor (1940)'s business cycle theory than 'model 1' in spirit.We shall study 'model 1' analytically and numerically, but we shall study 'model 2' only numerically.

LOCAL DYNAMICS OF 'MODEL 1'" A MATHEMATICAL ANALYSIS
In this section, we shall investigate the local dynamics of 'model 1' analytically by means of the linear approximation method.Let us assume that there exists an equilibrium solution Y* > 0 of the system ($1) which satisfies 14)   Expanding the system (S) in a Taylor series around the equilibrium point Y* and neglecting the terms of higher order than the first order, we have the following linear approximation of the system (S).
4We need not assume that Y* is unique.In fact, we shall present a numerical example with multiple equilibria in Section 4. 5The asterisk (*) shows that the values are evaluated at the equilibrium point.
or equivalently, (1/0)A ca 4-c/3e -0 (17)   where A=_Op.If all the roots of Eq. ( 17) have negative real parts, the equilibrium point of the system ($1) is locally stable.On the other hand, it becomes locally unstable if at least one root of Eq. ( 17) has positive real part. 6First, let us consider the characteristics of the real roots of Eq. ( 17).

Characteristics of the Real Roots 7
We can rewrite Eq. ( 17) as =_ -( We can see from Figure that Eq. ( 17) has one positive real root and one negative real root when 0</3<a.
Figure 2 illustrates the case of/3 a.In this case, A=0 is always one of the roots of Eq. ( 18).In addition, (i) we have one negative real root when 0 < 1/oa and/3 a, and (ii) we have one positive real root when 0 > 1lena and/3 a.
The case of/3 > a is illustrated in Figure 3.This figure shows that (i) Eq. ( 17) has two negative real roots when 0 is sufficiently small, (ii) it has two positive real roots when 0 is sufficiently large, and (iii) it has no real root at the intermediate values of 0.
Next, we shall consider the mathematical con- dition for the existence of the multiple real roots of Eq. ( 17).This condition is given by f  6As for the proof, see, for example, Bellman and Cooke (1963) Chap.11.As for the significance of the characteristic root approach to the mixed difference and differential equations, see Frisch and Holme (1935) and James and Belz (1938).
(24) above 3.2.Local Stability/Instability Analysis Table I shows that the equilibrium point of the system (S1) is locally unstable in the region A U B. But, it is necessary to obtain the information on the complex roots to study the local stability/ instability in the region CUD.For this pur- pose, we can utilize the following math- ematical result which is due to Hayes (1950) to get full information on the local stability of the system.
LEMMA (Hayes' theorem) All the roots of H(A) =pc + q-Ae O, where p and q are real, have negative real parts if and only if (i) p < 1, and (ii) p < q < v/(x .2+p2), where x* is the root of x=p tan x such that 0 < x < 7r.Ifp O, we take x* 7r/2.
Proof See Hayes (1950)  In fact, we can show that Eq. ( 17) has infinite number of complex roots.where x* is the solution of g(x)=(1/Oca)x= tan x _= g2(x) such that 0 < x < rv.
We can illustrate the solution x* as in Figure 5 when the inequality (26) (i) is satisfied. 9Further- more, we can see from Figure 5 that tan x* becomes a decreasing function of 0. Therefore, we have We can rewrite the characteristic Eq. ( 17) as d d(tanx*) < 0 (27) dO (x*/Oc) a dO H(A) Ooae + (-Ooefl) Ae -pe A + q Ae 0 (25) We can derive the following relationships from Eq. ( 26) (iii) and Eq. ( 27). 1 where p-Oaa and q--Ooefl.It follows from Lemma that we can express a set of the necessary and sufficient conditions for local stability as follows. (i) av/(tan 2X* + 1) a. 9Note that d(tan x)/dx= +tan20= when x=0, and the inequality (26)(i) implies that 1/Ooza > 1.

FIGURE 6 Stable region.
3.3.Hopf-bifurcation and the Existence of the Closed Orbits Proposition says that (i) too long delay in policy response must fail to stabilize the economy, (ii) too strong policy as well as too weak policy is unsuccessful to stabilize the economy even if the policy lag (0) is relatively short, and (iii) the stabilization policy is successful at the intermediate range of the strength of the policy response (/3) if 0 is relatively short.These analytical results seem to suggest that the pure cyclical movements will occur at the intermediate values of/3 when 0 is not too large.11 Now, we shall prove mathematically that this conjecture is in fact correct.We can make use of the following version of the Hopf-bifurcation theorem. 12LEMMA 2 Let 5c(t) F(x(t),x(t-0); c), x R, c R be a mixed difference and differential equation with a parameter c.Suppose that the following properties are satisfied.
(i) This equation has smooth curve of equilibria F(x*, x*; c) O. (ii) The characteristic equation F(p) p-a- be-P=O has a pair of pure imaginary roots p(c0),/(c0) and no other root with zero real part, where a =_ (OF/Ox(t))* and b (OF/ Ox(t-O))* are partial derivatives of F which are evaluated at (x*(Co), Co) with the parameter (iii) d{Re p(c)}/de 0 at c Co, where Re p(c) is the real part of p Then, there exists a continuous function c('),) with c(0)=c0, and for all sufficiently small values of 70 there exists a continuous .family of non- constant periodic solutions x(t,',/) for the above dynamical equation, which collapses to the equili- brium point x* (Co) as "y -O.The period of the cycle is close to 27r/Im p(co), where Im p(co) is the imaginary part of p (Co).Now, it is clear from the analyses in Sections 3.1 and 3.2 that there exists the value 0 E [0, 1/ca) such that the following property (P) is satisfied (see Fig. 7).
(P) For all 0 E (0, 1/ca), COROLLARY OF PROPOSITION 2 There exist some non-constant periodic solutions of the system ($1) at some parameter values > 0 which are sufficiently close to o (0o).The period of the cycle is close to 27r/Zo > 200.Proof It directly follows from Lemma 2 and Proposition 2.
4. NUMERICAL SIMULATIONS In the previous section, we presented some analytical results on the local dynamics of 'model I' around the equilibrium point.However, we must resort to the study of the numerical simula- tions to get some information on the global dynamics of the system.Furthermore, it is difficult to get even the information on the local dynamics by means of analytical approach if we consider more complicated system such as 'model 2'.In this section, we shall present some results of the global dynamics of 'model 1' and 'model 2' by means of numerical simulations.'3The functions /(0) and (0) are given in Eq. ( 26)(iii) and ( 24) respectively.

Simulation of 'Model 1'
First, let us study 'model 1' by adopting the following specifications of the functional forms and the parameter values.
Figure 8 is the phase diagram of this system in (Y(t), Y(t+0.3))plane in the case of/3=6.6 and 0-0.2.14Because of the S-shaped investment function, there exist three equilibrium points and three limit cycles.One equilibrium point (Y* 400) is unstable and two equilibrium points (Y**< 400, Y*** >400) are locally stable.One (large) limit cycle is stable and two (small) limit cycles are unstable.Figure 9(a) and (b) are the bifurcation diagrams of this system with respect to the parameter corresponding to the initial conditions Y(0)= 420 and Y(0): 390 respectively.
These figures show that we have different bifurca- tion diagrams corresponding to different initial conditions because of the multiple equilibria, is In other words, this system has pathdependent characteristics.Figure 10 is the bifurcation dia- gram with respect to the policy lag (0) when/3 is fixed at/3 5.6.

Simulation of 'Model 2'
Next, we shall consider the numerical study of 'model 2' by using the following data.( -0.01vt 0.5X(t) C(1--T)--0.5, Co + cTo + Go 200, --0.9 Figure 11 shows that behavior of this system can be chaotic for some parameter values.This figure illustrates a strange attractor which is produced when/3-4.1 and 0-0.3.16 Figure 12(a) is the bifurcation diagram with respect to the parameter /3 when 0 is fixed at 0-0.3.Figure 12(b) is the same bifurcation diagram at the interval 4.0</3<4.2.These figures show that the limit cycles are produced for both of sufficiently small values and sufficiently large values of /3.At the intermediate values of /3, the equilibrium point becomes stable, and at the vicinity of the parameter value /3-4.1, the behavior of the system becomes complex.Finally, Figure 13 is the bifurcation diagram with respect to the policy lag (0) when , is fixed at/3-4.1.

CONCLUDING REMARKS
In this paper, we formulated simple macrodynamic models with policy lag by means of mixed difference and differential equations, and investi- gated the effects of the policy lag on the dynamic behavior of the system analytically and numeri- cally.We found that the too long lag must fail to stabilize the system, and in some situations cyclical movement occurs.Furthermore, we found that even the chaotic movement is possible for some parameter values in a model with variable capital stock.Nevertheless, it is not correct to say that the government's stabilization policy is entirely in- effective to stabilize the intrinsically unstable economy.In fact, the government can stabilize the economy when the policy lag is relatively short.In this sense, macroeconomic stabilization policy does not lose its significance even in the system with lags in policy response.