Stability analysis of the Kaldor model with time delays: monetary policy and government budget constraint

Nonlinear Analysis: Real World Applications 5 (2004) 277 – 308 www.elsevier.com/locate/na

Stability analysis of the Kaldor model with time delays: monetary policy and government budget constraint Yasuhiro Takeuchi∗;1 , Tatsuya Yamamura Department of Systems Engineering, Faculty of Engineering, Shizuoka University, Hamamatsu 432-8561, Japan Received 23 August 2002; accepted 10 June 2003

Abstract We analyze the model with monetary policy based on the Kaldor’s business cycle theory. We introduce the government sector, which conducts the 2scal policy and monetary policy to stabilize the economy. The execution of such a policy needs legislation, and generally, the legislative process is time consuming. We investigate in this paper how the 2scal policy with a time delay a4ects stability of the economy. We assume that the monetary policy is conducted as a countermeasure of the 2scal de2cit by the government, and we consider two extreme cases, namely money 2nance and bond 2nance case. In each case, when no time delay exists for the 2scal policy, Keynesian 2scal policy is the preferred method for preventing the economic 6uctuations. However, it is not so simple when the time delay exists in the 2scal policy. There exists the policy, which stabilizes the economy under any time delay in the money 2nance case. On the other hand, in the bond 2nance case, such a policy does not exist and as the time delay increases the economy becomes unstable. However in both cases, contrary to the expectations of the government, the stronger the 2scal policy, the more unstable the economy becomes for the short time delay. ? 2003 Elsevier Ltd. All rights reserved.

1. Introduction Goodwin was probably the 2rst economist to realize the importance of the nonlinear mechanism of the economy and to introduce nonlinear di4erential equations into ∗

Corresponding author. Tel.: +81-53-478-1200; fax: +81-53-478-1200. E-mail address: [email protected] (Y. Takeuchi).

1 The research was partly supported by the Ministry, Science and Culture in Japan, under Grand-in-Aid for Scienti2c Research(A) 13304006.

1468-1218/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S1468-1218(03)00039-7

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economics. Goodwin [4], based on the Marxian view, built a growth cycle model which generates closed cycles caused by the class warfare between capitalists and workers. His model is based on the Lotka–Volterra equations well known in mathematical biology [5,7,8]. Goodwin assumed the following situation: all wages were consumed, further all pro5ts were saved and invested. This assumption is called Say’s law in economics, which means that the goods market equilibrium always holds and the e4ective demand problem does not appear [3]. Such an assumption is suitable for the analysis of classical capitalist economies, where there is no shortage of e4ective demand. However, we must dismiss this assumption if we consider about the business cycle theory in modern capitalist economies. That is to say, we need to consider the situation where the discrepancy between demand and supply exists in the goods market. We introduce the Kaldor’s business cycle theory [6]. His model does not assume the balance in the goods market. For the simpli2cation, we disregard the government expenditure and trade. Then the real output is equal to the national income (Y ), and the demand is composed of only consumption (C) and investment (I ). In this case, an excess demand YD is given by YD = (C + I ) − Y = I − (Y − C) = I − S, where S is the saving, and Kaldor assumed that KY ¿ 0 ⇔ YD ¿ 0. The assumption shows the mechanism of the Keynesian quantity adjustment. Note that in the Kaldor model, 6exible price adjustment mechanism is not considered. Furthermore, we can obtain from the above assumption that KY ¿ 0 ⇔ I ¿ S. Hence, the investment (not the saving) is important in order to increase the national income by the government. Next, we describe a Kaldor’s investment function I . Let us express the real pro2t as P and the capital stock as K. He assumed that the increase of the real pro2t stimulates the investment volition of the investor, on the other hand, investment is controlled by the accumulation of capital. Therefore, @I @I I = f(P; K); ¿ 0; ¡ 0: @P @K Since we can consider P is increasing with respect to national income, we have @I @I ¿ 0; I = F(Y; K); ¡ 0: @K @Y Following Wolfstetter [9], we introduce the government sector which responds to the condition of the economy. If the economy is in prosperity (the national income increases) then the government decreases the expenditure, on the other hand, if there exist indications that the economy is in recession then the expenditure is increased by the government. Further, it is also necessary for the government to stimulate consumption and investment by the monetary ease. Hence, in the extended Kaldor model, the government will be able to conduct monetary policy which a4ects stability of the economy. In this paper, the economy is called stable if it is less 6uctuating. However, the e4ect of the stabilization policy by the government depends on the length of the policy lag, which is divided into recognition, decision, action lags. Suppose the situation where there exist indications that the economy is in recession. Some time will elapse before the policy makers recognize a recession (recognition lag). Then they intend an expansionary policy. The execution of such a policy needs legislation, and generally, the legislative process is time consuming (decision lag). Furthermore,

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there exists a time lag between the policy decision and its implementation (action lag). Hence from the above time lags, contrary to the expectations of the government, the stabilization policy may destabilize the economy. We investigate in this paper how the 2scal policy with a time delay a4ects stability of the economy. This paper is organized as follows. In the next section, we describe two Kaldor models, the basic Kaldor model and the extended one with monetary policy. In Section 3, we consider the uniqueness of the equilibrium point for the extended model, and investigate its local stability for the case where no time delay exists in 2scal policy. We consider in Section 4 the extended models with a time delay in 2scal policy by using stability switch theorem. In Section 5, we execute numerical simulations. Finally, the conclusion is given in Section 6. 2. The Kaldor model The symbols used through this paper are de2ned as follows: Y : real output(national income), C: consumption, I : investment function, T : tax revenue, M : money supply, G: government spending, B: (government) interest in bonds market, (i.e., market price of bonds B=r), K: capital stock, r: interest rate, p: price level, : tax rate. Note that Y; C; I; T; G, and K are measured in real terms. 2.1. Basic Kaldor model 2.1.1. Goods market We 2rst describe the discrepancy between demand and supply in the goods market. If there exists an excess demand in the goods market, it is necessary to adjust its di4erence from supply. Generally, two adjustments are considered. One is quantity adjustment, another is 6exible price adjustment. However, in the modern capitalist economy, the goods market is not always in an equilibrium attained by 6exible price adjustment, because capitalists usually determine the price level. Thus, it is supposed that the discrepancy between demand and supply in the goods market is absorbed by inventory 6uctuations: when there exists an excess demand in the goods market, inventories are falling, and in the other case, inventories are piling up. In this framework, the demand side is always realized. Thus, the production decision of the 2rms depends on the excess demand in the goods market: Y˙ (t) = {C(t) + I (t) + G(t) − Y (t)};  ¿ 0; (1) where  is an adjustment speed in the goods market. Through this paper, x˙ ≡ d x=dt denotes the derivative of x with respect to time t. In the Kaldor model, consumption C is expressed as follows: C(t) = c{Y (t) − T (t)} + C0 ;

0 ¡ c ¡ 1; C0 ¿ 0;

(2)

where c is the marginal propensity to consume, and C0 is the constant consumption which is independent of national income. The 2rst term represents disposable income which is expressed by the di4erence between national income Y and tax T .

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If we consider that supply is determined by demand, we need to determine an investment function which is independent of savings. Hence, we should adopt a Kaldor’s investment function as follows: I (t) = I (Y (t); M (t));

IY ≡

@I ¿ 0; @Y

IM ≡

@I ¿ 0: @M

(3)

Note that investment is a monotone increasing function for output and money supply. As money and capital market, we 2rst assume that the price level is a monotone increasing function for output: p(t) = p(Y (t));

pY ≡

@p ¿ 0; p(0) ¿ 0: @Y

(4)

We assume the following on the interest rate r(t) = r(Y (t); M (t));

@r ≡ rY ¿ 0; @Y

@r ≡ rM ¡ 0; @M

r(0; M ) ¿ 0:

(5)

This assumption suggests the economic situation like the following. If the money supply increases in the money market, it becomes the in6ationary situation, and it is necessary to raise demand for money. Therefore, the interest rate decreases. On the other hand, if the national income increases, that is, the economy is in prosperity, then the interest rate rises. 2.1.2. Government and 5scal policy A fraction, , of national income is collected in the form of taxes: T (t) = Y (t) − T0 ;

 ¿ 0;

(6)

where T0 is the tax revenue which is independent of national income. The government spending G is speci2ed as follows: G(t) = G0 +  (Y ∗ − Y (t)) ;

 ¿ 0:

(7)

The 2rst term indicates a regular expenditure which is constant. The second represents a discretionary expenditure. Here,  is the strength of the policy. The government conducts 2scal policy on the basis of the equilibrium national income (Note that in this framework, the government can always know the equilibrium national income Y ∗ ): the counter-cyclical policy is adopted when  ¿ 0, on the other hand the pro-cyclical policy is selected when  ¡ 0. According to Wolfstetter [9], the latter is called the classical policy rule and the former is termed as the Keynesian rule. Here, we assume  ¿ 0, that is, we only deal with Keynesian policy. The dynamical system of the basic Kaldor model is rewritten as follows: Y˙ (t) = [I (Y (t)) − {1 − c(1 − ) + }Y (t) + C0 + G0 + Y ∗ + cT0 ]:

(8)

Eq. (8) is obtained from quantity adjustment mechanism (1), where we assumed that I is independent of money supply. We give the local stability analysis of this model in the appendix.

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2.2. Kaldor model with monetary policy In the previous subsection, we introduced the basic Kaldor model. We described in the introduction that the government also conducts monetary policy to stabilize the economy. Naturally, the basic Kaldor model contains monetary policy, but as long as we use previous consumption C (2) and tax revenue T (6), monetary policy cannot a4ect the dynamical system. Hence, we introduce the extended model with corrected consumption and tax revenue. In this model, the government will be able to conduct monetary policy which a4ects stability of the economy. In general the government cannot sel2shly conduct 2scal policy because of the budget constraint, and the government expenditure is not always equal to tax revenue. Therefore, we need to consider the government de2cit and budget constraint. We 2rst assume that issue of money and selling bonds 2nance the government de2cit. Thus, the government budget constraint is expressed as follows: M˙ B˙ B + =G+ − T: p(Y ) r(Y; M )p(Y ) p(Y )

(9)

Here, we assume all bonds are consol. The 2rst and second terms of the left-hand side of (9) express the revenue for the government by the issue of money and bonds, respectively. The right-hand side gives the government spending and the public bond interest by the government. Following Blinder and Solow [2], we consider two extreme cases such as (a) money N (b) bond 2nance case; M˙ = 0(M = MN ). For the case (a), 2nance case; B˙ = 0(B = B): the government de2cit is 2nanced only by the issue of money, and in the other case, 2nanced only by selling bonds. In the basic Kaldor model, investment is only part of demand. However, we should deal with investment as quantity, therefore we adopt capital stock K. We assume that all investment is used to augment capital stock: K˙ = I (Y; K; M ):

(10)

Here, we assume that investment is controlled by the accumulation of capital stock: IK ≡

@I ¡ 0: @K

We modify consumption C (2) as follows: 

 B B M C(t) = c1 Y (t) + + C0 : − T + c2 + p(Y ) r(Y; M )p(Y ) p(Y )

(11)

(12)

The 2rst term indicates the disposable income. Note that the public bond interest is added in the income. In the second term, B=r expresses the market price of bonds, therefore, the term represents that the consumption depends on wealth e4ect de2ned in public bond and money supply. The c1 and c2 are the marginal propensity to consume, and C0 is the constant consumption which is independent of national income and wealth.

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Further the public bond interest is also taxed, thus the tax revenue becomes as follows: 
B − T0 : (13) T (t) =  Y (t) + p(Y ) When we assume B˙ = 0 in (9), the government budget constraint for the money 2nance case is expressed as follows: M˙ BN =G+ − T: (14) p(Y ) p(Y ) The dynamical system of the extended model for the money 2nance case is written in the reduced form as follows:   Y˙ (t) =  I (Y; K; M ) − {1 − c1 (1 − ) + }Y + c1 (1 − )   BN c2 M ∗ + c2 + G0 + C0 + c1 T0 + Y ; + p(Y ) r(Y; M ) p(Y ) ˙ = I (Y; K; M ); K(t) (15)   BN M˙ (t) = p(Y ) −( + )Y + (1 − ) + G0 + T0 + Y ∗ : p(Y ) Substituting (3), (7), (12) and (13) into quantity adjustment mechanism (1) leads to the 2rst equation of (15). The second is equal to (10). Finally, from (6), (7) and (14), we obtain the last. The government budget constraint for the bond 2nance case (M˙ = 0 in (9)) is expressed as follows: B B˙ =G+ − T: N p(Y ) r(Y; M )p(Y ) The extended model for the bond 2nance case is written in the reduced form as follows:   Y˙ (t) =  I (Y; K) − {1 − c1 (1 − ) + }Y + c1 (1 − ) c2 + r(Y ) ˙ = I (Y; K); K(t)



 B MN ∗ + c2 + G0 + C0 + c1 T0 + Y ; p(Y ) p(Y ) 

˙ = r(Y )p(Y ) −( + )Y + (1 − ) B(t)



(16)

B + G0 + T0 + Y ∗ : p(Y )

3. Local stability of the extended model 3.1. Uniqueness of the equilibrium point We 2rst consider the uniqueness of the positive equilibrium point. For the money or bond 2nance case, the positive equilibrium point is denoted as (Y ∗ ; K ∗ ; M ∗ ) or

Y. Takeuchi, T. Yamamura / Nonlinear Analysis: Real World Applications 5 (2004) 277 – 308 283

(Y ∗ ; K ∗ ; B∗ ), respectively. For the both cases, we assume for the points that the real output Y is 2xed at the equilibrium national income Y ∗ . 3.1.1. Money 5nance case We 2rst show that Y ∗ is unique. From the right-hand side of the last equation in (15), we de2ne as a3 (Y ) ≡ −( + )Y + (1 − )

BN + G0 + T0 + Y ∗ : p(Y )

Since p(0) ¿ 0 from (4), we have that a3 (0) ¿ 0; @a3 (Y )=@Y ¡ 0 and limY →∞ a3 (Y )= −∞. Therefore, we can determine a unique Yˆ such that a3 (Yˆ ) = 0. We assume in this paper that Yˆ = Y ∗ . We assume the following on the investment function in order to determine a unique K ∗ . (H1): I (Y; 0; M ) ¿ 0 and limK→∞ I (Y; K; M ) ¡ 0 for all Y ¿ 0 and M ¿ 0. If we adopt an investment function satisfying (H1), we can determine a unique K ∗ such that I (Y ∗ ; K ∗ ; M ∗ )=0 for each Y ∗ ¿ 0 and M ∗ ¿ 0 (note that IK ¡ 0 from (11)). Finally, we consider the uniqueness of M ∗ . Substituting I = 0 and Y = Y ∗ into the right hand side of the 2rst equation in (15), we have   BN M c2 a1 (M ) ≡ −uY ∗ + c1 (1 − ) + + c2 + v; ∗ r(Y ; M ) p(Y ∗ ) p(Y ∗ ) where u ≡ {1 − c1 (1 − )} and v ≡ G0 + C0 + c1 T0 . By (5), we can obtain @a1 (M )= @M ¿ 0; limM →∞ a1 (M ) = ∞. If a1 (0) ¡ 0, we can determine the unique M ∗ such that a1 (M ∗ ) = 0. Hence we assume the following, (H2): a1 (0) ¡ 0. Under (H1) and (H2), the money 2nance case has a unique positive equilibrium point. 3.1.2. Bond 5nance case We now consider the uniqueness of the positive equilibrium point (Y ∗ ; K ∗ ; B∗ ). We can prove the uniqueness of K ∗ as well as the money 2nance case by choosing an investment function which satis2es (H1). Now, we consider the following simultaneous equations:   B∗ MN c2 − uY ∗ + c1 (1 − ) + + c2 + v = 0; (17) ∗ ∗ r(Y ) p(Y ) p(Y ∗ ) − Y ∗ + (1 − )

B∗ + G0 + T0 = 0: p(Y ∗ )

(18)

From (18), we have B∗ =

{Y ∗ − (G0 + T0 )}p(Y ∗ ) : 1−

(19)

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Since 0 ¡  ¡ 1 and p(Y ∗ ) ¿ 0, B∗ is positive for any Y ∗ ¿ YN , where G0 + T0 YN ≡ : 

(20)

Finally, the left-hand side of (17) is denoted as b1 (Y ) and we have @b1 (Y )=@Y ¡ 0 and limY →∞ b1 (Y ) = −∞. If b1 (YN ) ¿ 0, then b1 (Y ) = 0 has a unique positive root Yˆ , and we assume Yˆ = Y ∗ in this paper. Furthermore from (19) and (20), we can determine the positive unique B∗ . Hence we assume the following, (H3): b1 (YN ) = −uYN + c2 MN =p(YN ) + v ¿ 0. The bond 2nance case has a unique positive equilibrium point under (H1) and (H3). Note that the positive equilibrium point is independent of  in each case. 3.2. Local stability analysis 3.2.1. Money 5nance case The Jacobian matrix evaluated at the positive equilibrium point (Y ∗ ; K ∗ ; M ∗ ) is given by     N 2 p)} f11 f12 f13 () IK {IM + c2 =p − c2 rM B=(r      IY  =  f21 f22 f23  ; IK IM     0 0 f () 0 0 31 where     rY BN c2  p Y pY c1 (1 − ) + () =  IY − u −  − + c2 2 − c2 M 2 ; r p r p p () = −( + )p − (1 − )BN

pY p

and the sign of the component of the matrix is f12 = IK ¡ 0; f21 = IY ¿ 0;

N 2 p)} ¿ 0; f13 = {IM + c2 =p − c2 rM B=(r f22 = IK ¡ 0;

f23 = IM ¿ 0;

f31 = () ¡ 0:

Note that the asterisk (∗) is omitted. Its corresponding characteristic equation is "3 + a1 "2 + a2 " + a3 = 0;

(21)

where a1 ≡ a1 () = −( () + f22 ), a2 ≡ a2 () = ()f22 − f12 f21 − f13 () and a3 ≡ a3 () = − ()(f12 f23 − f13 f22 ). We are interested in whether the stability of the equilibrium point changes by the strength of 2scal policy . Therefore, we express each coeQcient as a function of . Note that the equilibrium value (Y ∗ ; K ∗ ; M ∗ ) is not dependent on .

Y. Takeuchi, T. Yamamura / Nonlinear Analysis: Real World Applications 5 (2004) 277 – 308 285

From the Routh–Hurwitz conditions for (21), the equilibrium point is stable if and only if a1 ¿ 0; a3 ¿ 0 and a1 a2 − a3 ¿ 0. Now we replace that () = − + #1 ; where

() = −p + #2 ;

(22)

    rY BN c 2  pY pY #1 =  IY − u − c1 (1 − ) + + c2 2 − c2 M 2 ; r p r p p p Y #2 = −p − (1 − )BN ¡ 0: p

(23) (24)

Since () ¡ 0 and from f12 = f22 ¡ 0, we have a3 () = −f22 ()(f23 − f13 ) ¿ 0. If #1 ¡ 0 then () ¡ 0 and we have a1 () ¿ 0 for all  ¿ 0. Substituting (22) into a1 a2 − a3 gives 2 ) −f13 (p#1 + #2 ) −pf12 f23 } −(f22 −pf13 )2 + {2#1 f22 −(f12 f21 −f22 2 × + {−#12 f22 + #1 (f12 f21 − f22 ) + f13 #1 #2 + f12 (f21 f22 + #2 f23 )}:

If #1 ¡ 0, then the coeQcient of  and the constant term are positive and a1 a2 − a3 ¿ 0 for all  ¿ 0. Hence we can obtain the following theorem. Theorem 3.1. If #1 ¡ 0, then the equilibrium point (Y ∗ ; K ∗ ; M ∗ ) of (15) is stable for all  ¿ 0. We consider the assumption #1 ¡ 0 from the economic point of view. If the marginal propensity to consume of wealth (c2 ) is large, and there exists no much interest in the increasing real pro2t by the investor (that is, IY is small), then this condition is easily satis2ed. Furthermore, if the monetary policy is frequently conducted as a countermeasure of the 2nancial de2cit by the government (that is, M or BN is large), then it also holds true easily. Under these conditions, the theorem implies that the equilibrium of the economy stays stable under any 2scal policy by the government for money 2nance case. Note that the economy without monetary policy is unstable for  ¡ a (a = IY∗ − u, a ¿ 0: see the appendix). Therefore, when the economy is unstable ( ¡ a) in the basic Kaldor model (without monetary policy), if M or BN is large (#1 ¡ 0), then the economy is always stable for any  in the extended model. That is, even if the 2scal policy is weak and the economy is unstable, the government can supplement the shortage of 2scal policy by conducting the monetary policy, and stabilize the economy. 3.2.2. Bond 5nance case Next we consider local stability of the equilibrium point in (16). The Jacobian matrix at the equilibrium point is given by    IK =p{c1 (1 − ) + c2 =r} g11 g12 2 ()     IY  =  g21 g22 0 IK    g31 () 0 r(1 − ) 0 2

the bond 2nance case g13



 0  ; g33

286 Y. Takeuchi, T. Yamamura / Nonlinear Analysis: Real World Applications 5 (2004) 277 – 308

where

    rY B pY c2  p Y N c1 (1 − ) + + c2 2 − c2 M 2 ; (25) 2 () =  IY − u −  − r p r p p   pY (26) 2 () = −rp ( + ) + (1 − )B 2 p

and the sign of the component of the matrix is g12 = IK ¡ 0; g22 = IK ¡ 0;

g13 = =p{c1 (1 − ) + c2 =r} ¿ 0; g31 =

2 () ¡ 0;

g21 = IY ¿ 0;

g33 = r(1 − ) ¿ 0:

Again the asterisk (∗) is omitted. Its corresponding characteristic equation is "3 + b1 "2 + b2 " + b3 = 0; where b1 ≡ b1 () = −( 2 () + g22 + g33 ), b2 ≡ b2 () = 2 ()g22 − g12 g21 − g13 2 () + g33 ( 2 () + g22 ) and b3 ≡ b3 () = g13 g22 2 () − g33 ( 2 ()g22 − g12 g21 ). Again we replace as 2 () = − + %1 and 2 () = −rp + %2 , where     rY B c 2  pY pY N + c2 2 − c2 M 2 ; %1 =  IY − u − c1 (1 − ) + r p r p p   pY %2 = −rp  + (1 − )B 2 ¡ 0: (27) p Since there exists g33 = 0, we cannot immediately obtain the sign of all coeQcients b1 , b2 and b3 . Now we assume %1 ¡ 0 (which is the corresponding condition to #1 ¡ 0 for the money 2nance case). If %1 + g22 + g33 ¡ 0 then b1 ¿ 0 for all  ¿ 0. Hence we assume as follows: %1 + g22 + g33 ¡ 0:

(28)

Next, we express b3 as a function of  such as b3 () = (g22 g33  − g13 g22 pr) + g12 g21 g33 − g22 g33 %1 + g13 g22 %2 . We obtain the coeQcient of  as g22 g33  − g13 g22 pr = g22 {(1 − c1 )(1 − )r − c2 }, which is positive if c2 ¿ (1 − c1 )(1 − )r. On the other hand, the constant term of b3 () is given by
  B ∗ rY c2 (1 − ) r MN pY + : (29) IK (1 − c1 )(1 − )r − c2 + p p r If (29) is positive and c2 ¿ (1 − c1 )(1 − )r, then b3 () ¿ 0 for all  ¿ 0. However from the constraint of c2 and , it is diQcult to take the parameter values ensuring for the above term positive. Therefore, we assume the constant term (29) is negative. Under this condition, we have b3 () ¿ 0 when  ¿ c , where c =

−g21 g33 + g33 %1 − g13 %2 : {(1 − c1 )(1 − )r − c2 }

(30)

Finally, we require the condition of b1 b2 − b3 ¿ 0. Denote as b1 b2 − b3 = d1 2 + d2  + d3 , where d1 = prg13 − 2 g22 − 2 g33 = 2 {c2 − (1 − c1 )(1 − )r − g22 } ¿ 0,

Y. Takeuchi, T. Yamamura / Nonlinear Analysis: Real World Applications 5 (2004) 277 – 308 287

if c2 ¿ (1 − c1 )(1 − )r. Since we have d2 = −prg13 (%1 + g33 ) − g12 g21 + (g22 + g33 )2 + 2%1 (g22 + g33 ) − g13 %2 , d2 ¿ 0 if g22 + g33 ¡ 0 and %1 + g33 ¡ 0 are satis2ed. Note that these two conditions also ensure that d3 = g12 g21 (%1 + g22 ) − (g22 + g33 )(%1 + g33 ) (%1 + g22 ) + g13 %2 (%1 + g33 ) ¿ 0. Hence we obtain: Theorem 3.2. The equilibrium point (Y ∗ ; K ∗ ; B∗ ) of (16) is stable for  ¿ c , if g22 + g33 ¡ 0, %1 + g33 ¡ 0 and c2 ¿ (1 − c1 )(1 − )r. Note that the 2rst two conditions ensure %1 ¡ 0 and (28). Remember that the economy is always stable for any  ¿ 0 in the money 2nance case, but in the bond 2nance case we need the strong control by the government (in the sense of  ¿ c ) to attain stability of the economy. Now we discuss the last condition in Theorem 3.2 from an economic point of view. Its left-hand side expresses “wealth e4ect of consumption,” and the right-hand side implies the marginal propensity to saving. Hence, it is necessary to raise the consumption, not the saving, in order to stabilize the economy by the government. This shows that the model is essentially based on the Keynesian view, of which character is the existence of an investment function which is independent of the saving. 4. Models with a time delay in scal policy We described in the introduction that time delay, namely the delay for the execution of the policy, exists in the 2scal policy. It is natural to expect that the e4ect of the stabilization policy depends on the length of the policy lag. Therefore, we consider the model including the time delay. We assume that the time delay is involved only in the discretionary 2scal policy of the government expenditure and we change (7) as follows: G(t) = G0 + (Y ∗ − Y (t − '));

 ¿ 0;

where ' is the policy lag, which means that the government executes the 2scal policy on the basis of the past national income. If we introduce this time delay in (8), then the basic model is revised as Y˙ (t) = [I (Y (t)) − {1 − c(1 − )}Y (t˙) − Y (t − ') +C0 + G0 + Y ∗ + cT0 ]:

(31)

The local stability analysis given in the appendix shows that large time delay can make the economy unstable. Now we analyze the following revised model, which involves the monetary policy. Money 5nance case:   Y˙ (t) =  I (Y; K; M ) − {1 − c1 (1 − )}Y − Y (t − ') + c1 (1 − ) c2 + r(Y; M )



 BN M ∗ + c2 + G0 + C0 + c1 T0 + Y ; p(Y ) p(Y )

288 Y. Takeuchi, T. Yamamura / Nonlinear Analysis: Real World Applications 5 (2004) 277 – 308

˙ = I (Y; K; M ); K(t)

(32)

 ˙ M (t) = p(Y ) −Y − Y (t − ') + (1 − )

 BN ∗ + G0 + T0 + Y : p(Y )

Bond 5nance case:   ˙ Y (t) =  I (Y; K) − {1 − c1 (1 − )}Y − Y (t − ') + c1 (1 − ) +

c2 r(Y )



 B MN + c2 + G0 + C0 + c1 T0 + Y ∗ ; p(Y ) p(Y )

˙ = I (Y; K); K(t)  ˙ = r(Y )p(Y ) −Y − Y (t − ') + (1 − ) B(t)

(33)  B + G0 + T0 + Y ∗ : p(Y )

Note that (32) and (33) have the same equilibrium point as in (15) and in (16), respectively. 4.1. Local stability of the money 5nance case Theorem 3.1 shows that the condition (#1 ¡ 0) ensures for the economy to be stable under any Keynesian policy of the government when no time delay exists in 2scal policy. In this subsection, we consider the local stability for model (32) with #1 ¡ 0 when time delay exists in 2scal policy. Now we investigate the stability of the steady state by writing (y; k; m) = (Y − Y ∗ ; K − K ∗ ; M − M ∗ ). The linearized system for (32) can be written as       y(t) ˙ y(t) y(t − ')        k(t) ˙  = G  k(t)  + H  k(t − ')  ;       m(t) m(t − ') m(t) ˙ where #i are de2ned by (23) and (24) and    N 2 p)} F11 #1 IK {IM + c2 =p − c2 rM B=(r     =  F21 G= IK IM  IY   F31 #2 0 0   − 0 0   0 0 H=  0 : −p 0 0

F12 F22 0

F13



 F23  ; 0

Note that F11 =#1 =f11 +; F31 =#2 =f31 +p ¡ 0; F12 =f12 ¡ 0, F13 =f13 ¿ 0; F21 = f21 ¿ 0; F22 = f22 ¡ 0 and F23 = f23 ¿ 0.

Y. Takeuchi, T. Yamamura / Nonlinear Analysis: Real World Applications 5 (2004) 277 – 308 289

Its corresponding characteristic equation is P(") + Q(") e−"' = 0;

(34)

where P(") = "3 + A"2 + B" + C and Q(") = D"2 + E" + F, of which coeQcients are N 2 p)}¿0, A=−(#1 +F22 )¿0, B=#1 F22 −F12 F21 −#2 F13 ¿0, C=−c2 #2 F22 {−1=p+rM B=(r D() =  ¿ 0, E() = (pF13 − F22 ) ¿ 0, F() = p(F12 F23 − F13 F22 ) ¿ 0. Note that all coeQcients are positive since #1 ¡ 0. Now we can use the following stability switch theory: Theorem 4.1 (See Kuang [7]). Consider (34), where P(") and Q(") are analytic functions in Re " ¿ 0 and satisfy the following conditions: (i) (ii) (iii) (iv) (v)

P(") = 0 and Q(") = 0 have no common imaginary root; P(−iy) = P(iy), Q(−iy) = Q(iy) for real y; P(0) + Q(0) = 0; lim sup{|Q(")=P(")| : |"| → ∞; Re " ¿ 0} ¡ 1; FS (y) ≡ |P(iy)|2 − |Q(iy)|2 = 0 for real y has at most a 5nite number of real zeros.

Then the following statements are true: (a) If FS (y) = 0 has no positive roots, then no stability switch may occur. (b) If FS (y) = 0 has at least one positive root and each of them is simple, then as ' increases, a 5nite number of stability switch may occur, and eventually the considered equation becomes unstable. In our model, (i) is clearly satis2ed because the real parts of the solution to Q(")=0 are negative. If P(") and Q(") are functions with real coeQcients, then (ii) is always true. Clearly P(0) + Q(0) = C + F() = 0, which implies " = 0 can never be a root of (34); otherwise, the considered equation is not uniformly asymptotically stable. (iv) is true, and FS (y) = 0 has clearly at most a 2nite number of real zeros. Hence, our model satis2es all the above conditions. Now we calculate FS (y): FS (y) = |P(iy)|2 − |Q(iy)|2 = y6 + (A2 − D2 − 2B)y4 + (B2 − E 2 − 2AC + 2DF)y2 + C 2 − F 2 : Let us de2ne z = y2 and FS (z) = z 3 + (A2 − D2 − 2B)z 2 + (B2 − E 2 − 2AC + 2DF)z + C 2 − F 2 :

(35)

If FS (z)=0 has no positive roots, then FS (y)=0 also has no positive roots. We rewrite (35) as FS (z) = z 3 + a1 z 2 + a2 z + a3 .

290 Y. Takeuchi, T. Yamamura / Nonlinear Analysis: Real World Applications 5 (2004) 277 – 308

We 2rst notice the constant term, a3 = C 2 − F 2 () = f31 (#2 + p)(F12 F23 − F13 F22 )2 . Since f31 ¡ 0, the constant term is negative if ¿−

pY #2 N =  + (1 − )BN 2 ≡ : p p

(36)

It is trivial that FS (z) = 0 has at least one positive root, therefore stability switch N certainly occurs for  ¿ . We consider the sign of a1 = A2 − D2 () − 2B = −2 2 + 01 , where 01 ≡ (#12 + 2 F22 + 2F12 F21 + 2#2 F13 ). Hence if 01 ¡ 0 then a1 is always negative; otherwise, we can obtain 1 which satis2es a1 (1 ) = 0 and a1 () ¿ 0 for 0 ¡  ¡ 1 . Finally we consider the sign of a2 . Here we can rewrite as a2 = d1 2 + d2 . Note that 2 2 + 2pF12 F23 ¡ 0. the coeQcient of the quadratic term is given by d1 = −p2 F13 − 2 F22 On the other hand, the constant term is d2 = (#1 F22 − F12 F21 − #2 F13 )2 − 2#2 (#1 + F22 )(F12 F23 − F13 F22 ) 2 2 2 2 − 2#1 F12 F21 F22 = #12 F22 + F12 F21 + #22 F13 

 1 rM BN 2 − F12 F13 F21 ¿ 0: −2#2 #1 F12 F23 + c2 F22 − 2 p r p

This implies that a positive 2 satisfying a2 (2 )=0 exists and a2 () ¿ 0 for 0 ¡  ¡ 2 . However, we cannot show the relationship between the size of 1 and 2 , because we speci2ed only the qualitative form of I (Y; K; M ); r(Y; M ) and p(Y ). Therefore, we execute numerical simulations. The investment function, interest rate, price level and other parameter values are identical to those in Section 5.1. In this case, the unique positive equilibrium point is given by Y ∗ = 100:285; K ∗ = 454:145; M ∗ = 671:203, and each coeQcient of FS (z) is a1 = −50:8293 − 0:252 ;

a2 = 729:554 − 28832 ;

a3 = 0:0528 − 0:212 :

In this situation, 01 is negative, that is a1 is always negative. Furthermore, the values of N and 2 are given by N = 0:5013; 2 = 0:503, therefore we may consider only the N Since a1 ¡ 0 and a2 ¿ 0 for  ¡ , N FS (z) = 0 can have 0 or 2 positive case  ¡ . N the constant term of FS (z) = 0 is positive and the minimal value roots. Since  ¡ , of FS (z) for z ¿ 0 is a decreasing function with respect to  (see Fig. 1). Hence, FS (z) = 0 has one positive root at ∗ = 0:1702. From Theorem 4.1, no stability switch may occur for  ¡ ∗ , and the equilibrium point is still stable. If ∗ ¡  then as time delay ' increases, a 2nite number of stability switch may occur, and eventually the equilibrium point becomes unstable. Examples of the solutions are given in Section 5. Now we discuss the above situation from an economic point of view. As we described in Section 3, our interest is whether the stability of the equilibrium point changes by increasing strength of the 2scal policy. When no time delay exists for the 2scal policy, it was shown that if the Keynesian policy is active ( ¿ 0), the government can stabilize the economy under any strength  ¿ 0. From the above result, when  is suQciently small, that is, the Keynesian policy is very weak, it is still available

Y. Takeuchi, T. Yamamura / Nonlinear Analysis: Real World Applications 5 (2004) 277 – 308 291

minimal value of

FS(z) 0.1

0.2

0.3

0.4

0.5

beta -5000 -10000 -15000

Fig. 1. The minimal value of FS (z) is a decreasing function of .

for stabilization under any time delay. However it is not so simple when the Keynesian policy is suQciently active ( ¿ ∗ ). Thus, we discuss the relationship between the strength of the policy and time delay in Section 4.3. 4.2. Local stability of the bond 5nance case Similarly to the money 2nance case, we assume for the model (33) the existence of the positive equilibrium point. Further the equilibrium point is assumed to be stable ( ¿ c ) for the case where no time delay exists for the 2scal policy (see Theorem 3.2). Now we examine the stability of the steady state by writing (y; k; b) = (Y − Y ∗ ; K − K ∗ ; B − B∗ ). The linearized system for (33) is given as       y(t) ˙ y(t) y(t − ')        k(t) ˙  = G2  k(t)  + H2  k(t − ')  ;       ˙ b(t) b(t − ') b(t) where %i are de2ned by (27) and    %1 IK {c1 (1 − ) + c2 =r}=p G11     =  G21 IK 0 G2 =   IY   %2 G31 0 r(1 − )   − 0 0   0 0 H2 =   0 : −rp 0 0

G12 G22 0

G13



 0  ; G33

Note that G11 =%1 =g11 +; G31 =%2 =g31 +rp ¡ 0; G12 =g12 ¡ 0, G13 =g13 ¿ 0; G21 = g21 ¿ 0; G22 = g22 ¡ 0, and G33 = g33 ¿ 0.

292 Y. Takeuchi, T. Yamamura / Nonlinear Analysis: Real World Applications 5 (2004) 277 – 308

Its characteristic equation is P2 (")+Q2 (") e−"' =0, where P2 (")="3 +A2 "2 +B2 "+C2 and Q2 (")=D2 "2 +E2 "+F2 , of which coeQcients are A2 =−(%1 +G22 +G33 ) ¿ 0, B2 = G11 G22 −G12 G21 −G13 G31 +G33 (G11 +G22 ), C2 =G13 G22 G31 +G33 (G12 G21 −G11 G22 ) ¡ 0, D2 () =  ¿ 0, E2 () = −(G22 + G33 ) + rpG13 ¿ 0, F2 () = G22 {(1 − c1 )(1 − ) r − c2 } ¿ 0. Clearly this model satis2es conditions (i) – (v) of Theorem 4.1. Now we calculate FS2 (y) = |P2 (iy)|2 − |Q2 (iy)|2 . Let us de2ne z = y2 and FS2 (z) = z 3 + (A22 − D22 − 2B2 )z 2 + (B22 − E22 − 2A2 C2 + 2D2 F2 )z + C22 − F22 :

(37)

We 2rst notice the constant term of (37), C22 −F22 ()=(C2 +F2 ())(C2 −F2 ())=b3 (C2 − F2 ()) ¡ 0, where b3 is given by b3 = C2 + F2 ¿ 0 when  ¿ c . Hence, F2S (z) = 0 has at least one positive root and stability switch certainly occurs for  ¿ c . In the bond 2nance case ( ¿ c ), there exists no policy which stabilizes the economy (this means the equilibrium point (Y ∗ ; K ∗ ; B∗ ) is stable) under any time delay. Now we rewrite (37) as FS2 (z)=z 3 +b21 ()z 2 +b22 ()z+b23 . We would like to know the relationship between the size of 1 (which is a positive solution of b21 () = 0), and 2 (which is a positive solution of b22 () = 0). Again, we execute numerical simulations. The investment function, interest rate, price level and other parameter values are identical to those in Section 5.2. In this case, the unique positive equilibrium point is given by Y ∗ = 194:276; K ∗ = 21:5753; M ∗ = 43126, and c = 1:405. Then each coeQcient of FS2 (z) is b21 = 7:14338 − 42 , b22 = 10:7566 − 25:17882 , and we can obtain 1 = 1:336, 2 = 0:648 and b21 ¡ 0 and b22 ¡ 0 for  ¿ c (¿ 1 ¿ 2 ). Hence, FS2 (z) = 0 can have one positive root. Hence, the Keynesian policy satisfying  ¿ c can destabilize the economy. We consider in Section 4.3 the process of this instabilization as we increase . 4.3. Detailed analysis of stability switch Consider the characteristic equation given by P(")+Q(") e−"' =0, where P(")="3 + A"2 + B" + C, Q(") = D"2 + E" + F. When " = 0 cannot be a root of the characteristic equation, a stability switch necessarily occurs with " = ±iy with y ¿ 0. Without loss of generality assume " = iy, y ¿ 0. Now we replace that P(iy) = PR (iy) + iPI (iy) and Q(iy) = QR (iy) + iQI (iy) where PR (iy) = −Ay2 + C; PI (iy) = −y3 + By; QR (iy) = −Dy2 + F; QI (iy) = Ey. Substituting them in P(iy) + Q(iy)(cos y ' − i sin y') = 0, we have PR (iy) + iPI (iy) + (QR (iy) + iQI (iy))(cos y' − i sin y') = 0. Hence, y must satisfy PR + QR cos y' + QI sin y' = 0; which gives cos y' = −

’ ; |Q(iy)|2

sin y' =

PI + QI cos y' − QR sin y' = 0; 3 ; |Q(iy)|2

(38)

where ’ = PR QR + PI QI = (−Ay2 + C)(−Dy2 + F) + Ey(−y3 + By);

(39)

3 = −PR QI + PI QR = −Ey(−Ay2 + C) + (−y3 + By)(−Dy2 + F):

(40)

Y. Takeuchi, T. Yamamura / Nonlinear Analysis: Real World Applications 5 (2004) 277 – 308 293

Here |Q(iy)|2 = 0, because |Q(iy)|2 =0 implies Q(iy)=0 which contradicts to Q(iy)= −Dy2 + iEy + F = 0 (note that Ey = 0). On the other hand, cos y' and sin y' can be written as     P(iy) P(iy) ; sin y' = Im : cos y' = −Re Q(iy) Q(iy) If y satis2es (38), then y must satisfy that |P(iy)|2 = |Q(iy)|2 . Now we de2ne FS (y) as follows: FS (y) ≡ |P(iy)|2 − |Q(iy)|2 = 0:

(41)

If FS (y) = 0 has at least one positive root (simple), a 2nite number of stability switch may occur, and eventually the considered equation becomes unstable. Now we de2ne the angle  ∈ [0; 24), as the solution of (38) 3 ’ ; sin  = (42) cos  = − |Q(iy)|2 |Q(iy)|2 and the relation between the arguments  in (42) and y' in (38) for ' ¿ 0 must be y' =  + 2n4;

n ∈ N0 := {0; 1; 2; : : :}:

Hence we can obtain as follows:  + 2n4 ; 'n ¿ 0; n ∈ N0 : 'n = y

(43)

Then we can use the following stability switch theorem. Theorem 4.2 (Beretta and Kuang [1]): Assume that y is a positive real root of (41) and at some '∗ ¿ 0, Sn ('∗ ) = '∗ − 'n = 0 for some n ∈ N0 . Then a pair of simple conjugate pure imaginary roots " = ±iy∗ of the characteristic equation exists at ' = '∗ which crosses the imaginary axis according to  
 d"(')  = sign FS (y): S = sign Re d' "=iy∗ By Theorem 4.2, we can know the direction of the stability switch. That is, the switch to instability occurs when FS (y∗ ) ¿ 0, and the switch to stability occurs when FS (y∗ ) ¡ 0. Remark 4.3. Assume that  ∈ (0; 24), where  is de2ned by (42). Then we have  = arctan(−3=’)  = 4=2

if sin  ¿ 0; cos  ¿ 0;

if sin  = 1; cos  = 0;

 = 4 + arctan(−3=’)  = 34=2

if cos  ¡ 0;

if sin  = −1; cos  = 0;

 = 24 + arctan(−3=’)

if sin  ¡ 0; cos  ¡ 0:

294 Y. Takeuchi, T. Yamamura / Nonlinear Analysis: Real World Applications 5 (2004) 277 – 308

4.3.1. Money 5nance case Now we will use Theorem 4.2 in our model. Since y and  are dependent on , we denote y() and (). yi∗ () is a positive solution of (41) and from (43), we have 'in =

 ∗ () + 2n4 ; yi∗ ()

(44)

therefore Sn is Sin = ' −

 ∗ () + 2n4 ; yi∗ ()

(45)

where  ∗ () is de2ned as in Remark 4.3. We 2rst consider the sign of ’. Let us de2ne y2 ≡ X in (39), and consider ’(X ) = (AD − E)X 2 + (BE − AF − CD)X + CF = 0:

(46)

Note that X ∗ which is a solution of (46) is independent of , since D; E and F are linear functions of  and A, B and C are independent of . Now we take all functions and parameters which are identical √ to those given in Section 5.1. In this case, (46) has a positive solution, X ∗ ∼ N and we can determine the sign = 27:4735, X ∗ = 5:2415 ≡ y, of cos  ∗ as follows from the size of yi∗ ,  ’∗  ¿ 0 (yN ¡ yi∗ ); −    |Q(iyi∗ )|2     ’∗ ∗ (yN = yi∗ ); (47) cos  = − ∗ )|2 = 0 |Q(iy  i      ’∗  − ¡ 0 (yN ¿ yi∗ ): |Q(iyi∗ )|2 From (40), we have 3 = y{Dy4 + (AE − BD − F)y2 + BF − CE}. Since y ¿ 0, we denote y2 = X2 , and consider 3(X2 ) = DX22 + (AE − BD − F)X2 + BF − CE = 0:

(48)

Note that X2∗ which is a solution of (48) is independent of . Substituting all functions and parameters given in Section 5.1 to (48) implies that all coeQcients of 3(X2 ) are positive, therefore we have 3 ¿ 0 for all yi∗ ¿ 0. Hence we can obtain as follows: sin  ∗ =

3∗ ¿0 |Q(iyi∗ )|2

(for all yi∗ ¿ 0):

(49)

Now we give the examples for the case  = 0:18 and 0.3. Since these values are larger than ∗ = 0:1702, FS (z) = 0 has two positive roots (see Section 4.1). We 2rst consider for the case  = 0:18. In this case, two positive roots of FS (z) = 0 are z1 = 22:262 and z2 = 28:575. FS (z) = 0 has two positive roots and FS (z) ¡ 0 for 8:28 ¡ z ¡ 25:613. Hence, if zi∗ which is a positive solution of FS (z) = 0 is smaller

Y. Takeuchi, T. Yamamura / Nonlinear Analysis: Real World Applications 5 (2004) 277 – 308 295 unstable

3

S20=0 S10=0

2

S21=0

1

S11=0 S22=0 0.5

1

1.5

2

2.5

3 S12=0

-1 -2 -3

Fig. 2. Money 2nance case: Sin and the stability switch of equilibrium point ( = 0:18). When ' increases, the equilibrium point becomes unstable (or stable) when ' enters from the white (or shaded) interval to the shaded (or white) interval.

than 8.28 or bigger than 25.613, then the crossing the imaginary axis of the root is always to the right direction, that is, stability switch occurs only toward instability. Inversely if 8:28 ¡ zi∗ ¡ 25:613, then the crossing is always to the left direction, that is, stability switch occurs only toward stability. We can determine  from (47), (49) and Remark 4.3, furthermore we can obtain the function Sin in (45). Now we consider the function S1n with respect to z1 (the stability switch occurs toward stability), and S2n with respect to z2 (the stability switch occurs toward instability). The graph of Sin (') versus ' in Fig. 2 shows how stability switch occurs and the stability of the equilibrium point changes. From Fig. 2, we know that the stability switch occurs toward instability at the 2rst value '20 = 0:273, and the switch occurs toward stability at the second value '10 = 0:451 and so on. Fig. 2 shows that the stability switch alternatively occurs, and the interval of the time delay where the equilibrium point is stable reduces. Next we consider for the case  = 0:3 in the same way. Then FS (z) = 0 has two positive roots z1 = 12:15 and z2 = 38:72. FS (z) = 0 has two positive roots and FS (z) ¡ 0 for 5:521 ¡ z ¡ 28:38. We determine  from Remark 4.3 and calculate the function Sin similar to the case =0:18. In Fig. 3, we show the relationship between the stability of the equilibrium point and the time delay '. At the 2rst value '20 =0:131 the stability switch occurs toward instability, at the second value '10 = 0:796 the switch occurs toward stability. Note that the frequency of the stability switch toward instability surpasses the frequency of the switch toward stability, and the equilibrium point becomes unstable as the time delay increases. In fact, we have '20 ¡ '10 ¡ '21 ¡ '22 ¡ '11 ¡ '23 ¡ '12 ¡ · · · in Fig. 3. Remember that the stability switch occurs toward instability at '2j and toward stability at '1j (j = 0; 1; 2; 3; : : :). Hence, we know from the sequence of 'kj that the model seems to remain unstable for ' ¿ '21 . This is a qualitatively different point of the stability switch for the case  = 0:18 (Fig. 2) where we have '20 ¡ '10 ¡ '21 ¡ '11 ¡ '22 ¡ '12 ¡ · · · and the alternative switch between the intervals ensuring stability and instability of the model occurs.

296 Y. Takeuchi, T. Yamamura / Nonlinear Analysis: Real World Applications 5 (2004) 277 – 308 unstable

S20=0 S10=0 S21=0 S22=0 S11=0

Sin 4

2

S23=0 1

2

3

4

S12=0

-2

-4

Fig. 3. Money 2nance case: Sin and the stability switch of equilibrium point ( = 0:3). When ' increases, the equilibrium point becomes unstable (or stable) when ' enters from the white (or shaded) interval to the shaded (or white) interval. Note that as time delay increases, the frequency of the stability switch toward instability surpasses the frequency of the switch toward stability, and the equilibrium point becomes unstable.

4.3.2. Bond 5nance case Now we consider only the case where  ¿ c = 1:405 (see Section 4.2). As well as the money 2nance case, we require the sign of cos  ∗ and sin  ∗ . We 2rst consider the sign of ’. Let us de2ne y2 ≡ X in (39), and consider ’(X ) = (A2 D2 − E2 )X 2 + (B2 E2 − A2 F2 − C2 D2 )X + C2 F2 = 0:

(50)

Note that X ∗ which is a solution of (50) is independent of , since D2 ; E2 and F2 are linear functions of  and A2 , B2 and C2 are independent of . Now we take all functions and parameters which are identical √ to those given in Section 5.2. In this case, ’(X ) has one positive root X ∗ ∼ N We can determine the sign = 0:248, X ∗ = 0:498 ≡ y. of cos  ∗ similar to the money 2nance case. From (40), we have 3=y{D2 y4 +(A2 E2 − B2 D2 − F2 )y2 + B2 F2 − C2 E2 }. Since y ¿ 0, we denote y2 = X2 , and consider 3(X2 ) = D2 X22 + (A2 E2 − B2 D2 − F2 )X2 + B2 F2 − C2 E2 = 0:

(51)

Note that X2∗ which is a solution of (51) is independent of . Substituting all functions and parameters given in Section 5.2 to (51) implies that all coeQcients of 3(X2 ) are positive, therefore we have 3 ¿ 0 for all yi∗ ¿ 0. Hence we have sin  ∗ ¿ 0 as (49). Now we give the example for the case  = 2:0 and 3.5. Note that in both cases, FS2 (z) = 0 has one positive root. We 2rst consider for the case  = 2:0, where z1 =  14:9117 is one positive root of FS2 (z) = 0. FS2 (z) = 0 has one positive root and  FS2 (z) ¿ 0 for z ¿ 9:178. Since z1 ¿ 9:178, the crossing the imaginary axis of the root is always to the right direction, that is, stability switch occurs only toward instability. From Fig. 4, we know that the stability switch occurs toward instability at the value '10 = 0:433.

Y. Takeuchi, T. Yamamura / Nonlinear Analysis: Real World Applications 5 (2004) 277 – 308 297

Sin

unstable

S10=0

0.4 0.2

0.2

0.4

0.6

0.8

1

-0.2 -0.4

Fig. 4. Bond 2nance case: Sin and the stability switch of equilibrium point ( = 2:0). At the value '10 = 0:433 the stability switch occurs toward instability, the equilibrium point becomes unstable.

S10=0

Sin

unstable

0.6 0.4 0.2

0.2

0.4

0.6

0.8

1

-0.2

Fig. 5. Bond 2nance case: Sin and the stability switch of equilibrium point ( = 3:5). At the value '10 = 0:235 the stability switch occurs toward instability, the equilibrium point becomes unstable.

Similarly, we consider for the case  = 3:5. Then FS2 (z) = 0 has one positive root   z1 =48:0594 and FS2 (z)=0 has one positive root and FS2 (z) ¿ 0 for z ¿ 31:097. Similar to the case  = 2:0, since z1 ¿ 31:097, the crossing is always to the right direction. The graph of Sin (') versus ' in Fig. 5 shows that the stability switch occurs toward instability at the value '10 = 0:235. 5. Numerical simulations In the previous section, we investigate the local stability of the equilibrium point for the extended model with a time delay by using stability switch theorem. We found

298 Y. Takeuchi, T. Yamamura / Nonlinear Analysis: Real World Applications 5 (2004) 277 – 308

that as the time delay increases, the equilibrium point becomes unstable in both cases with money or bond 2nance by the government. In this section, we execute numerical simulations for the purpose of investigating how the stability of the equilibrium point changes as the time delay increases for various values for the strength of 2scal policy . In both cases, we adopt the Runge–Kutta algorithm and numerical simulations are carried out for time step Kt = 0:005. 5.1. Money 5nance case We take the parameter values as follows:  = 0:5; c1 = 0:8; c2 = 0:6;  = 0:5; G0 = 30; T0 = 20; C0 = 10 and BN = 30. The investment function is speci2ed as  I (Y (t); K(t); M (t)) = Y (t) − 1:5K(t) + M (t) and for interest rate, we assume Y (t) + 1500 r(Y (t); M (t)) = : M (t) + 1500 Further we also assume for the price level, p(Y (t)) = Y (t) + 5: In this case, (H1) and (H2) (see Section 3) are satis2ed (a1 (0) = −3:898 ¡ 0). Hence, we have a unique positive equilibrium point Y ∗ = 100:285; K ∗ = 454:145; M ∗ = 671:203. Further the equilibrium point is stable for any  ¿ 0 when no time delay exists in the 2scal policy for money 2nance case, since #1 = −0:295 ¡ 0. Now we give the example for the case  = 1:0. Fig. 6 shows that the equilibrium point is stable, that is, the Keynesian policy is e4ective when no time delay exists in 2scal policy. Next we consider the system where the time delay exists in the 2scal policy. In this case, we know that if  ¿ ∗ = 0:1702, then as time delay increases, the equilibrium point becomes unstable (see Section 4.1). Now we give the examples for the case =0:3 and '=0:15; 0:9; 1:2. Figs. 7 and 9 show that there exists the economic oscillation (' = 0:15 and 1:2), and Fig. 8 shows that the equilibrium point is stable (' = 0:9). The graph of the time delay ' versus the strength of the 2scal policy  given in Fig. 10 shows the relationship between the stability of the equilibrium point and two parameters. Here we express the region S where the equilibrium point remains to be stable. From Fig. 10, we know that if  ¡ 0:17 (note that ∗ ¿ 0:17) then the equilibrium point is stable for 0 ¡ ' ¡ 4, and this result is consistent with the result in Section 4.1. Further if  ¿ 0:17 then the stability switch occurs alternatively and eventually the equilibrium point becomes unstable as the time delay increases. This is consistent with the result in Section 4.3.1. 5.2. Bond 5nance case We take the parameter values as follows:  = 2; c1 = 0:8; c2 = 0:6;  = 0:6; G0 = 10; T0 = 20; C0 = 10 and MN = 40. The investment function is speci2ed as  I (Y (t); K(t)) = Y (t) − 2:5K(t) + MN

Y. Takeuchi, T. Yamamura / Nonlinear Analysis: Real World Applications 5 (2004) 277 – 308 299

The graph of Y (t ) 130 120 110 100 90 80 70 60 50 0

500

1000

1500

2000

2500

The graph of K (t )

580 560 540 520 500 480 460 440 420 0

500

1000

1500

2000

2500

The graph of M (t ) 1400

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200 0

500

1000

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Fig. 6. The graph of the solution orbits of Y (t), K(t) and M (t) ( = 1:0) for money 2nance case without a time delay.

300 Y. Takeuchi, T. Yamamura / Nonlinear Analysis: Real World Applications 5 (2004) 277 – 308

The graph of Y (t) 140

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The graph of K (t) 400

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The graph of M (t) 800 700 600 500 400 300 200 100 0 16000

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Fig. 7. The graph of the solution orbits of Y (t), K(t) and M (t) for the money 2nance case with a time delay ( = 0:3, ' = 0:15).

Y. Takeuchi, T. Yamamura / Nonlinear Analysis: Real World Applications 5 (2004) 277 – 308 301

The graph of Y (t )

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The graph of K (t ) 700

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The graph of M (t ) 1500 1400 1300 1200 1100 1000 900 800 700 600 500 400

0

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10000 15000 20000 25000 30000 35000 40000 45000 50000

Fig. 8. The graph of the solution orbits of Y (t), K(t) and M (t) for the money 2nance case with a time delay ( = 0:3, ' = 0:9).

302 Y. Takeuchi, T. Yamamura / Nonlinear Analysis: Real World Applications 5 (2004) 277 – 308

The graph of Y (t )

120 115 110 105 100 95 90 85 80 4000

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The graph of K (t )

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The graph of M (t)

950 900 850 800 750 700 650 600 550 500 450 4000

4500

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Fig. 9. The graph of the solution orbits of Y (t), K(t) and M (t) for the money 2nance case with a time delay ( = 0:3, ' = 1:2).

Y. Takeuchi, T. Yamamura / Nonlinear Analysis: Real World Applications 5 (2004) 277 – 308 303 4

3.5

3

time delay

2.5

S

2

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1

0.5

0 0

0.05

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0.15

0.2

0.25

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0.35

0.4

strength of fiscal policy

Fig. 10. Stability of the equilibrium point with respect to  and ' (the money 2nance case).

and for interest rate, we assume r(Y (t)) =

Y (t) : MN

Further we also assume for the price level p(Y (t)) = Y (t) + 5: In this case, (H3) is satis2ed as b1 (0) = 2:436 ¿ 0 and we have a unique positive equilibrium point Y ∗ = 194:276; K ∗ = 21:5753; M ∗ = 43126. Further if  ¿ c = 1:405 then the equilibrium point is stable for the case where no time delay exists in the 2scal policy (Theorem 3.2), since g22 + g33 = −0:557 ¡ 0, %1 + g33 = −0:585 ¡ 0, c2 − (1 − c1 )(1 − )r ∗ = 0:211 ¿ 0. Now we give the examples for the case  = 3:0. Fig. 11 shows that the equilibrium point is stable, that is, the Keynesian policy is e4ective when no time delay exists in 2scal policy. Next we consider the case where the time delay exists in the 2scal policy. In this case, we know that the stability switch occurs for all  ¿ c = 1:405 (see Section 4.2). Fig. 12 shows that there exists the economic oscillation. The graph of the time delay ' versus the strength of the 2scal policy  given in Fig. 13 shows the relationship between the stability of the equilibrium point and two parameters, and we express the region S where the equilibrium point becomes stable. From Fig. 13, we know that if  ¡ 1:48 then the equilibrium point is unstable for all '. Further if  ¿ 1:48 then as the time delay increases the stability switch occurs toward instability, therefore the equilibrium point becomes unstable. We also know that as  increases, the region where the equilibrium point is stable reduces. This is consistent with the result in Section 4.3.2.

304 Y. Takeuchi, T. Yamamura / Nonlinear Analysis: Real World Applications 5 (2004) 277 – 308

The graph of Y (t )

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Fig. 11. The graph of the solution orbits of Y (t), K(t) and M (t) ( = 3:0) for bond 2nance case without a time delay.

Y. Takeuchi, T. Yamamura / Nonlinear Analysis: Real World Applications 5 (2004) 277 – 308 305

The graph of Y (t)

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The graph of K (t) 23

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The graph of B (t ) 140000

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40000 20000 20500 21000 21500 22000 22500 23000 23500 24000 24500 25000

Fig. 12. The graph of the solution orbits of Y (t), K(t) and M (t) for the bond 2nance case with a time delay ( = 3:0 and ' = 0:3).

306 Y. Takeuchi, T. Yamamura / Nonlinear Analysis: Real World Applications 5 (2004) 277 – 308 0.7

0.6

time delay

0.5

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0

1

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strength of fiscal policy

Fig. 13. Stability of the equilibrium point with respect to  and ' (the bond 2nance case).

6. Conclusion In this paper, we analyzed two models, which are based on Kaldor’s business cycle theory and are developed by taking into account the government sector. The models emphasize the role of e4ective demand and we considered the extended models with monetary policy. Two extreme cases were considered what the government 2nances de2cit, the 2rst was a money 2nance case where the government de2cit is 2nanced only by the issue of money, and the second was a bond 2nance case where the government de2cit is 2nanced only by selling bonds. Further we considered the model including the time delay. The time delay, namely the delay for the execution of the policy exists in the 2scal policy, and the e4ect of the stabilization policy depends on its length. In money 2nance case, we obtain the following. When no time delay exists in the 2scal policy, it was shown that countercyclical 2scal policy (Keynesian rule) is the preferred method for preventing the economic 6uctuations, that is, the equilibrium point is stable for all  ¿ 0 (: the strength of the 2scal policy). However, it is not so simple when the time delay exists in the 2scal policy. If  is suQciently small, that is, the Keynesian policy is very weak, it is still available for stabilization under any time delay. In contrast, if the policy is implemented with a long lag, then a very active intervention tends to amplify disturbances from the equilibrium state and, contrary to the expectations of the government, the stabilization policy destabilizes the economy. Furthermore the stronger the 2scal policy, the more unstable the economy becomes even for the short time delay. In this point, Keynesian policy harms the stability of the economy. In bond 2nance case, we obtain the following. When no time delay exists for the 2scal policy, if the Keynesian policy is suQciently active ( ¿ c ), then the government can stabilize the economy. In this point, the Keynesian policy is e4ective. However, if

Y. Takeuchi, T. Yamamura / Nonlinear Analysis: Real World Applications 5 (2004) 277 – 308 307

the time delay exists for the 2scal policy, then the above may be false. Indeed, there exists no policy which stabilizes the economy under any time delay, furthermore as well as the money 2nance case, the stronger the 2scal policy, the more unstable the economy becomes for the short time delay. Therefore, the government must reduce the time necessary for the execution of the 2scal policy. From the above, we could show that the issue of money is better for the government than selling bonds in order to stabilize the economy. Because in the bond 2nance case, the government must conduct strong policy to attain stability of the economy by comparison with the money 2nance case. Further the time delay strongly a4ects the stability of the economy in the bond 2nance case. We also showed that the increasing the marginal propensity to consume of wealth (c2 ) is e4ective in order to stabilize the economy. This implies that these models are based on Keynesian view. Finally, we describe our future problem. For the simpli2cation, we assumed two extreme cases in this paper. However, this assumption is not realistic. Therefore, we must consider a new model where the government de2cit is 2nanced by both the methods. For example, we express the money demand as L(Y; M ), and we assume the money supply is determined by the excess money demand in money market. That is, M˙ = L(Y; M ) − M: But we could not analyze the model which includes the above, therefore this is our future problem. Appendix A. Analysis of basic Kaldor model We give the local stability analysis of the basic Kaldor model. We 2rst consider the case where no time delay exists for the 2scal policy. Denote the right-hand side of (8) as F1 (Y ) and the equilibrium point as Y ∗ , of which existence is assumed. Since  @F1  = [IY∗ − {1 − c(1 − ) + }]; @Y  ∗ Y =Y

∗

Y is stable if  ¿ a and unstable if  ¡ a, where a ≡ IY∗ − {1 − c(1 − )}. Now we assume a ¿ 0. This means that the marginal propensity to invest exceeds the propensity to save around the equilibrium state. Such a situation occurs in Kaldor’s business cycle model. Further this shows that 2scal policy satisfying  ¿ a(¿ 0) is necessary in order to stabilize the economy. Next we consider system (31) where time delay exists for the 2scal policy. We assume that the economy without time delay is stable, that is,  ¿ a. The characteristic equation evaluated at Y ∗ is given by " − a + e−"' = 0: '¡ '0 Following Kuang [7], we have that Y ∗ is uniformly asymptotically stable

when   2 2 2 2 and unstable when ' ¿ '0 , where '0 ==!, !=  − a and =arccot a=  − a . Therefore, the stable economy without time delay becomes unstable by increasing '.

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References [1] E. Beretta, Y. Kuang, Geometric stability switch criteria in delay di4erential systems with delay dependent parameters, SIAM J. Math. Anal. 33 (5) (2002) 1144–1165. [2] A. Blinder, R. Solow, Does 2scal policy matter? J. Public Econ. 2 (1973) 319–337. [3] J.K. Galbraith, Money, Whence It Came, Where It Went, Houghton MiWin Co, Boston, MA, 1975. [4] R.M. Goodwin, A growth cycle, in: C.H. Feinstein (Ed.), Socialism, Capitalism, and Economic Growth, Essays Presented to Maurice Dobb, Cambridge, 1967, pp. 54 –58. [5] J. Hofbauer, K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, MA, 1998. [6] N. Kaldor, A model of the trade cycle, Econ. J. 50 (1940) 78–92. [7] Y. Kuang, Delay Di4erential Equations with Applications in Population Dynamics, Academic Press, San Diego, 1993. [8] Y. Takeuchi, Global Dynamical Properties of Lotka–Volterra Systems, World Scienti2c, Singapore, 1996. [9] E. Wolfstetter, Fiscal policy and the classical growth cycle, Z. NationalYokon. 42 (1982) 375–393.