6.Analysis of the policy mix of monetary and fiscal stabilization policies

Finally, we shall try to construct a theoretical framework that can analyze the dynamic effect of the policy mix of monetary and fiscal policies.For this purpose, the following two equations play important roles.

where M=mH=nominal money stock, H=nominal high-powered money, m=money multiplier＞1, p=price level, T=real tax, G=real government expenditure, and B=stock of nominal public debt.

Eq.（16）is nothing but a standard type of LM equation that describes the equilibrium condition of the money market.We can see, however, that the high-powered money per capital stock H/（pK）in Eq.（16）becomes an endogenous variable if the central bank controls nominal rate of interestρrather than money stock according to the Taylor rule that is expressed by Eq.（9）.Eq.（17）is the budget constraint of the consolidated government that includes the central bank.This equation shows that the government expenditure including the interest payment of the public debt must be financed by（1）tax,（2）new issue of the public debt, and（3）money financing of the central bank.

It follows from Eq.（17）that we can obtain the following dynamic of the public debtcapital ratio b=B/（pK）.

Rewriting this equation, we have the following relationship.

If we neglect the indirect effects of the change of b onthrough the changes of T/K and other variables, we have

We can see from this result that the inequality

or equivalently, the inequality

acts as a destabilizing factor of the system, and the inequality

or equivalently, the inequality

acts as a stabilizing factor of the system.

Needless to say, these inequalities are only partial conditions for instability/stability. Nevertheless, these conditions provide us some insight concerning instability/stability of the system.The（partial）stability condition（23）or（24）is called“Domar condition”after Domar（1957）.It must be noted that the nominal rate of interestρcannot become negative.Therefore, it is likely that the destabilizing“anti Domar condition”（21）or（22）is satisfied if the rate of inflationπis negative and the real rate of capital accumulation g is negative or almost zero.In fact, this is the case of the Japanese economy in the 1990s and the 2000s under long term deflationary depression, during which the public debtcapital ratio b increased rapidly.

Now, let us formulate the fiscal policy rule as follows.

where v=G/K and-b is the target public debt-capital ratio that is set by the government. This is the feedback rule of the government expenditure that considers both of the stabilization of the real national income（employment）and the stock of public debt.The parameterθis the weight of the relative importance of national income（employment）consideration compared with the public debt consideration that is attached by the government.

The equations（16）,（19）and（25）together with Eq.（13）consists of sixdimensional system of nonlinear differential equations with six endogenous variables（d, y, πe, ρ, b, v）. Equations（13）（iv）and（25）formalize the policy mix of monetary and fiscal policies in this model.

Proposition 1 in the previous section also applies to this extended system.Furthermore, we can show analytically that the large value of the parameterθ（value ofθthat is close to 1）is a stabilizing factor and the small value ofθ（value ofθthat is close to 0）is a destabilizing factor.This means that the appropriate policy mix is the combination of the active monetary policy with sufficient credibility and the fiscal policy that attaches importance to the employment consideration rather than public debt consideration.The opposite policy mix is inappropriate.If the policy mix is inappropriate, the equilibrium point becomes dynamically unstable, and in this case the deflationary depression, in which the nominal rate of interest is stuck in its lower bound, is likely to occur like the Japanese economy in the 1990s and the 2000s.

The intuitive explanation of this conclusion is as follows. Suppose that the government expenditure responds actively to the changes of real national income（employ-ment）.Then, the following stabilizing negative feedback mechanism will work.

On the other hand, the following two destabilizing positive feedback mechanisms will work if the government expenditure responds to the changes of the stock of public debt excessively.

These destabilizing feedback mechanisms produce the paradoxical situation in which the decrease of the government expenditure per capital stock induces the decrease of real national income per capital stock（employment）and the increase of debt-capital ratio, contrary to the government's subjective intention.

This theoretical reasoning is consistent with the experience of the Japanese economy in the 1990s and the 2000s, which is called the“lost twenty years”.We already noted that the destabilizing“anti Domar condition”was satisfied in Japan in this period.In fact, in Japan in this period of serious deflationary depression with high unemployment, the sharp increase of the public debt-capital ratio b coexisted with the decrease rather than increase of both of government expenditure-capital ratio v and income-capital ratio y together with the decline of the growth rate of money stock（cf.Asada,2011）.These apparently paradoxical behaviors of some key variables in Japan in recent twenty years are quite consistent with the theoretical reasoning of the model in this section.