Financial structure, financial instability, and inflation targeting

Abstract

Minsky, the first to propose the financial instability hypothesis, stressed the importance of the lender-of-last-resort for preventing financial instability. Overall, however, most of the research on financial instability has focused little on measures to prevent instability. Japan was trapped in a prolonged recession after the collapse of the bubble economy. The government promoted market-oriented economic reforms to cope. The recent international monetary crisis, triggered by the subprime loan crisis of 2007 in the US, cast a dark shadow over the world economy. Some developed nations, most notably New Zealand have been successful in implementing inflation-targeting policies. The Bank of Japan and the US Federal Reserve have adopted the inflation-targeting measures after the crisis. The main purpose of this paper is to examine financial instability, financial cycles, and the effects of inflation targeting in a mixed competitive–oligopolistic system. The results of this paper demonstrate that inflation targeting stabilizes an economy in both competitive and oligopolistic systems.

Appendix

The proof of Proposition 3

Assume that $m_{y} < m_{y0}$ and $\beta$ is sufficiently small, such that $a_{2} > 0$. The proof of Proposition 1 demonstrates that $a_{1} a_{2} - a_{3} < 0$ if $\varepsilon$ is sufficiently small, while the proof of Proposition 2 demonstrates that $a_{1} a_{2} - a_{3} > 0$ if $\varepsilon$ is sufficiently large.

Given that $a_{1} a_{2} - a_{3}$ is a smooth and continuous function with $\varepsilon$, we find at least one value $\varepsilon_{0}$ at which $a_{1} a_{2} - a_{3} = 0$ and $\partial (a_{1} a_{2} - a_{3} )/\partial \varepsilon |_{{\varepsilon = \varepsilon_{0} }} \ne 0$. Furthermore, it also follows that $a_{2} > 0$.

One of the conditions of the Hopf bifurcation theorem is satisfied when $a_{2} > 0$ and $a_{1} a_{2} - a_{3} = 0$. The characteristic equation of dynamic system ($S$) has a pair of purely imaginary roots $\lambda_{1} = \sqrt {a_{2} } i$ and $\lambda_{2} = - \sqrt {a_{2} } i$ at $\varepsilon = \varepsilon_{0}$.

From the Orlando formation, we obtain

$$a_{1} a_{2} - a_{3} = - (\lambda_{1} + \lambda_{2} )(\lambda_{2} + \lambda_{3} )(\lambda_{3} + \lambda_{1} ) = - 2h_{1} (\lambda_{3}^{2} + 2h_{1} \lambda_{3} + h_{1}^{2} + h_{2}^{2} ) ,$$

where $h_{1}$ is the real part of two complex conjugate numbers and $h_{2}$ is the absolute value of the imaginary part. By differentiating this equation with $\varepsilon$, we obtain

$$\frac{{\partial (a_{1} a_{2} - a_{3} )}}{\partial \varepsilon } = - 2\left[ {\frac{{\partial h_{1} }}{\partial \varepsilon }(\lambda_{3}^{2} + 2h_{1} \lambda_{3} + h_{1}^{2} + h_{2}^{2} ) + h_{1} \frac{{\partial (\lambda_{3}^{2} + 2h_{1} \lambda_{3} + h_{1}^{2} + h_{2}^{2} )}}{\partial \varepsilon }} \right]$$

When $h_{1} = 0$ and $h_{2} = h$ are substituted into the above equation, we obtain

$$\frac{{\partial (a_{1} a_{2} - a_{3} )}}{\partial \varepsilon }|_{{\varepsilon = \varepsilon_{0} }} = - 2(\lambda_{3}^{2} + h^{2} )\left[ {\frac{{\partial h_{1} }}{\partial \varepsilon }|_{{\varepsilon = \varepsilon_{0} }} } \right]$$

Therefore, if

$$\frac{{\partial (a_{1} a_{2} - a_{3} )}}{\partial \varepsilon }|_{{\varepsilon = \varepsilon_{0} }} \ne 0$$

then

$$\frac{{\partial h_{1} }}{\partial \varepsilon }|_{{\varepsilon = \varepsilon_{0} }} \ne 0$$

From the above discussion, all of the conditions in which Hopf bifurcation occurs are satisfied at the point $\varepsilon = \varepsilon_{0}$. Q.E.D.