4.An extension to the model with flexible prices

In this section, we introduce the price flexibility into the model in the previous section following Asada's（2004）procedure.We can formulate this extended model by means of the following three dimensional system of differential equations.

where the meanings of the symbols d, y, andαare the same as those in the previous section, and the meanings of other symbols are as follows.πe=expected rate of price inflation.ε=parameter that reflects the price adjustment speed.γ=parameter that reflects the adjustment speed of price expectation. Furthermore, the following properties of the partial derivatives as well as a set of inequalities（2）in the previous section are assumed.

We can derive Eq.（4）（iii）as follows.In this extended model, the rate of price inflationπ becomes an endogenous variable that fluctuates according as the following standard type of the expectations-augmented price Phillips curve.

As for the price expectation formation, the following“adaptive”expectation formation hypothesis（or the“backward-looking”expectation formation hypothesis）is a dopted.

Under the absence of the central bank's active commitment to inflation targeting like the Japanese economy under deflationary depression during the 1990s and the 2000s, it is likely that the people who do not have any definite information form their inflation expectation adaptively.This is the rationale of the hypothesis such as Eq.（7）.Substituting Eq.（6）into Eq.（7）, we have

which is nothing but Eq.（4）（iii）.

In this model, the nominal rate of interest is still kept constant by the central banker's passive monetary policy. Nevertheless, the real rate of interest becomes a variable rather than constant through the variable expected rate of price inflation, so that the variable πe enters into equations（4）（i）and（4）（ii）through the investment function（3）.

Both of the rates of changeandbecomes increasing functions of πe（that is, f13＞0 and f23＞0）because of the following reasons.The increase of πe implies the decrease of ρ-πe, which induces the increase of the rate of investment and the increase of the debt financing of its expenditure as well as the increase of effective demand.

We can consider the increase of the parameter valueεorγas the increase of the“price flexibility”.Asada（2004）proved analytically, however, that the increase of price flexibility in this sense tends to destabilize rather than stabilize the economic system contrary to the teaching of the mainstream economic theory, because the following destabilizing positive feedback mechanism is intensified by the increase of the parameter values ε and γ, which contributes to intensify the amplitude of macroeconomic fluctuations.

Figures 3 and 4 summarize the result of Asada（2004）'s numerical simulations that support the above analytical conclusion.

Incidentally, we pointed out in the previous section that the variable d plays the role of“predator”and the variable y plays the role of“prey”in the dynamic interaction between these two variables like the Lotka-Volterra system of mathematical biology.The role of the variable πe is, however, not so simple.It follows from the system of equations（4）that both of the following results（A）and（B）apply.

（A）πe i s the food for d and y.

（B）y i s the food for πe.

This means that the relationship between two variables πe and y is not simple predator-prey relationship, and these variables share their destiny. This is the source of the destabilizing positive feedback mechanism that is described schematically by the causal relationships（FM1）.