تجزیه و تحلیل پایداری مدل کالدور با تاخیر زمان: سیاست های پولی و محدودیت های بودجه دولت

We analyze the model with monetary policy based on the Kaldor's business cycle theory. We introduce the government sector, which conducts the fiscal 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 fiscal policy with a time delay affects stability of the economy. We assume that the monetary policy is conducted as a countermeasure of the fiscal deficit by the government, and we consider two extreme cases, namely money finance and bond finance case. In each case, when no time delay exists for the fiscal policy, Keynesian fiscal policy is the preferred method for preventing the economic fluctuations. However, it is not so simple when the time delay exists in the fiscal policy. There exists the policy, which stabilizes the economy under any time delay in the money finance case. On the other hand, in the bond finance 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 fiscal policy, the more unstable the economy becomes for the short time delay.

Goodwin was probably the first economist to realize the importance of the nonlinear mechanism of the economy and to introduce nonlinear differential equations into 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] and [8]. Goodwin assumed the following situation: all wages were consumed, further all profits were saved and invested. This assumption is called Say's law in economics, which means that the goods market equilibrium always holds and the effective demand problem does not appear [3]. Such an assumption is suitable for the analysis of classical capitalist economies, where there is no shortage of effective 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 simplification, 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 ΔY⩾0⇔YD⩾0. The assumption shows the mechanism of the Keynesian quantity adjustment. Note that in the Kaldor model, flexible price adjustment mechanism is not considered. Furthermore, we can obtain from the above assumption that ΔY⩾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 profit as P and the capital stock as K. He assumed that the increase of the real profit stimulates the investment volition of the investor, on the other hand, investment is controlled by the accumulation of capital. Therefore, Since we can consider P is increasing with respect to national income, we have 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 affects stability of the economy. In this paper, the economy is called stable if it is less fluctuating. However, the effect 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, 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 fiscal policy with a time delay affects 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 fiscal policy. We consider in Section 4 the extended models with a time delay in fiscal policy by using stability switch theorem. In Section 5, we execute numerical simulations. Finally, the conclusion is given in Section 6.

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 effective demand and we considered the extended models with monetary policy. Two extreme cases were considered what the government finances deficit, the first was a money finance case where the government deficit is financed only by the issue of money, and the second was a bond finance case where the government deficit is financed 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 fiscal policy, and the effect of the stabilization policy depends on its length. In money finance case, we obtain the following. When no time delay exists in the fiscal policy, it was shown that countercyclical fiscal policy (Keynesian rule) is the preferred method for preventing the economic fluctuations, that is, the equilibrium point is stable for all β>0 (β: the strength of the fiscal policy). However, it is not so simple when the time delay exists in the fiscal policy. If β is sufficiently 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 fiscal 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 finance case, we obtain the following. When no time delay exists for the fiscal policy, if the Keynesian policy is sufficiently active (β>βc), then the government can stabilize the economy. In this point, the Keynesian policy is effective. However, if the time delay exists for the fiscal 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 finance case, the stronger the fiscal 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 fiscal 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 finance case, the government must conduct strong policy to attain stability of the economy by comparison with the money finance case. Further the time delay strongly affects the stability of the economy in the bond finance case. We also showed that the increasing the marginal propensity to consume of wealth (c2) is effective in order to stabilize the economy. This implies that these models are based on Keynesian view. Finally, we describe our future problem. For the simplification, we assumed two extreme cases in this paper. However, this assumption is not realistic. Therefore, we must consider a new model where the government deficit is financed 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, 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 first consider the case where no time delay exists for the fiscal policy. Denote the right-hand side of (8) as F1(Y) and the equilibrium point as Y∗, of which existence is assumed. Since Y∗ is stable if β>a and unstable if β<a, where . 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 fiscal policy satisfying β>a(>0) is necessary in order to stabilize the economy. Next we consider system (31) where time delay exists for the fiscal policy. We assume that the economy without time delay is stable, that is, β>a. The characteristic equation evaluated at Y∗ is given by Following Kuang [7], we have that Y∗ is uniformly asymptotically stable when τ<τ0 and unstable when τ>τ0, where τ0=θ/ω, and . Therefore, the stable economy without time delay becomes unstable by increasing τ.