Mathematical Modelling of Financial Instability and Macroeconomic Stabilisation Policies

Abstract

In this paper, we formulate a series of mathematical macrodynamic models that contribute to the theoretical analysis of financial instability and macroeconomic stabilisation policies. Two-dimensional model of fixed prices without active macroeconomic stabilisation policy, four-dimensional model of flexible prices with central bank’s monetary stabilisation policy, and six-dimensional model of flexible prices with monetary and fiscal policy mix are considered in order. In the final section, we provide an intuitive economic interpretation of the analytical results.

Appendices

Appendix 1: Proof of Proposition 4

Suppose that $\xi =\beta _1=\beta _2=\theta =0$. In this case, we have

$$\begin{aligned} \begin{array}{rl} d_2&{}={\text {sum of all principal second-order minors of}}\;J_4\\ &{}= -\gamma \alpha \varepsilon \underbrace{F_{23}}_{(+)}+A, \end{array} \end{aligned}$$

(100)

where $A$ is independent of the value of $\gamma$. This means that we have $d_2<0$ for all sufficiently large values of $\gamma$, which violates one of the necessary conditions for local stability (99). By continuity, this conclusion applies even if the parameters $\xi$, $\beta _1$, $\beta _2$, and $\theta$ are positive, as long as they are sufficiently small. $\square$

Appendix 2: Proof of Proposition 5

Step 1. Suppose that $\xi =s_3=1$. In this case, the characteristic Eq. (95) becomes

$$\begin{aligned} \Delta _4(\lambda )\equiv |\lambda I-J_4|=|\lambda I-J_5|(\lambda +\gamma )=0, \end{aligned}$$

(101)

$$\begin{aligned} J_5=\left[ \begin{array}{ccccc} F_{11} &{}F_{12} &{}F_{14} &{}0 &{}0\\ \alpha F_{21} &{}\alpha F_{22} &{}\alpha F_{24} &{}\alpha &{}0\\ 0 &{} \beta _1\varepsilon +\beta _2 &{} 0 &{} 0 &{} 0\\ 0 &{} -\beta _3\theta &{} 0 &{} 0 &{} -\beta _3(1-\theta )\\ F_{61} &{}F_{62} &{}F_{64} &{}F_{65} &{}F_{66}\\ \end{array}\right] \end{aligned}$$

(102)

Equation (101) has a negative real root $\lambda _6=-\gamma$ and other five roots are determined by the equation

$$\begin{aligned} \Delta _5(\lambda )\equiv |\lambda I-J_5|=0. \end{aligned}$$

(103)

Step 2. Next, suppose that $\theta =1$. In this case, Eq. (103) is reduced to

$$\begin{aligned} \Delta _5(\lambda )=|\lambda I-J_6|(\lambda -F_{66})=0, \end{aligned}$$

(104)

$$\begin{aligned} J_6=\left[ \begin{array}{cccc} F_{11} &{}F_{12} &{}F_{14} &{}0 \\ \alpha F_{21} &{}\alpha F_{22} &{}\alpha F_{24} &{}\alpha \\ 0 &{} \beta _1\varepsilon +\beta _2 &{} 0 &{} 0 \\ 0 &{} -\beta _3 &{} 0 &{} 0 \\ \end{array}\right] . \end{aligned}$$

(105)

Equation (104) has a negative real root $\lambda _5=F_{66}$ and other four roots are determined by the following equation.

$$\begin{aligned} \Delta _6(\lambda )=|\lambda I-J_6|=(\lambda ^3+z_1\lambda ^2+z_2\lambda +z_3)\lambda =0, \end{aligned}$$

(106)

$$\begin{aligned} z_1=-\underbrace{F_{11}}_{(-)}-\alpha \underbrace{F_{22}}_{(+)}, \end{aligned}$$

(107)

$$\begin{aligned} z_2=\alpha \{\underbrace{(F_{11}F_{22}-F_{12}F_{21})}_{(+)}-\underbrace{F_{24}}_{(-)}(\beta _1\varepsilon +\beta _2)+\beta _3\}>0, \end{aligned}$$

(108)

$$\begin{aligned} z_3=\alpha \{-\underbrace{F_{11}}_{(-)}\beta _3+(\beta _1\varepsilon +\beta _2)\underbrace{(F_{11}F_{24}-F_{14}F_{21})}_{(+)}\}>0, \end{aligned}$$

(109)

$$\begin{aligned} \begin{array}{rl} z_1z_2-z_3=&{}\alpha \{(-\underbrace{F_{11}}_{(-)}-\alpha \underbrace{F_{22}}_{(+)})\underbrace{(F_{11}F_{22}-F_{12}F_{21})}_{(+)}-\alpha \beta _3 \underbrace{F_{22}}_{(+)}\\ &{}+(\beta _1\varepsilon +\beta _2)(\underbrace{F_{14}}_{(-)}\underbrace{F_{21}}_{(-)}+\alpha \underbrace{F_{22}}_{(+)}\underbrace{F_{24}}_{(-)})\}. \end{array} \end{aligned}$$

(110)

Equation (106) has a real root $\lambda _4=0$, and other three roots are determined by the equation

$$\begin{aligned} \Delta _7(\lambda )\equiv \lambda ^3+z_1\lambda ^2+z_2\lambda +z_3=0. \end{aligned}$$

(111)

Step 3. It is easy to see that all of the following Routh-Hurwitz conditions for stable roots of Eq. (111) are satisfied if $\alpha$ is sufficiently small (cf. Gandolfo 2009 Chap. 16).

$$\begin{aligned} z_j>0~(j=1,2,3),~z_1z_2-z_3>0 \end{aligned}$$

(112)

Hence, we have just proved the following result. “Suppose that $\theta =1$. Then, the characteristic Eq. (103) has a real root $\lambda _4=0$ and other four roots of this equation have negative real parts under the conditions (1) and (3) of Proposition 5.” This means that Eq. (103) has at least four roots with negative real parts under the conditions (1) and (3) of Proposition 5 even if $0<\theta <1$, as long as $\theta$ is sufficiently close to 1 by continuity. On the other hand, in case of $0<\theta <1$, we have

$$\begin{aligned} \Delta _5(0)&=-\prod ^{5}_{j=1}\lambda _j=|-J_5|=-\text {det}J_5 \\&=\alpha (\beta _1\varepsilon +\beta _2)\beta _3(1-\theta )\{F_{65}\underbrace{(F_{11}F_{24}-F_{14}F_{21})}_{(+)}+\underbrace{(F_{14}F_{61}-F_{11}F_{64})}_{(+)}\},\nonumber \end{aligned}$$

(113)

and $F_{65}$ becomes positive if $\alpha$ is sufficiently small. Therefore, Eq. (113) becomes positive so that we have $\prod ^{5}_{j=1}\lambda _j<0$ if $0<\theta <1$ and $\alpha$ is sufficiently small. This means that all roots of Eq. (103) have negative real parts under the conditions (1), (3), and (4) of Proposition 5

Step 4. We have just proved the following result. “Suppose that $\xi =s_3=1$. Then, all of six characteristic roots of Eq. (95) in the text have negative real parts under the conditions (1), (3), and (4) of Proposition 5.” By continuity, this conclusion applies even if $0<\xi <1$ and $0<s_3<1$, as long as they are sufficiently close to 1. This proves Proposition 5. $\square$