On the dynamic properties of Okishio’s Monetarist model

Abstract

In this paper, we reconsider the Monetarist model that was formulated by Okishio (Econ Rev 30:289–299, 1979; Econ Stud Q 31:1–9, 1980) from analytical point of view. The basic model is formulated by a three-dimensional system of nonlinear differential equations, and the long run equilibrium of this system satisfies the typical Monetarist properties, namely, the ‘natural rate of employment’ is attained, and the rate of price inflation and nominal growth rate are determined by the growth rate of nominal money supply that is set by the central bank at the long run equilibrium point. However, as Okishio (Econ Rev 30:289–299, 1979; Econ Stud Q 31:1–9, 1980) correctly pointed out, the dynamic stability of the long run equilibrium point is ensured only if particular conditions for crucial parameter values are satisfied. We further prove by means of Hopf bifurcation theorem that the business cycles which entail cyclical fluctuations of the main variables occur at some range of parameter values. We also investigate the dynamic behavior of the extended model that consists of four-dimensional nonlinear differential equations. Finally, we briefly argue that the Keynesian coordinated active fiscal and monetary stabilization policies are necessary if the inactive Monetarist fiscal and monetary policies cannot ensure the dynamic stability of the macroeconomic system.

Appendices

Appendix A

Proof of Lemma 2

Step 1

First, let us consider a special case of $g_{r} * = 0$ and $i_{{\pi^{e} }} * = 1.$ In this case, Eq. (61) becomes as follows.

$$ \begin{aligned} F_{11} = & - [\mathop {f_{E} *}\limits_{( + )} \mathop {E_{m} *}\limits_{( + )} + \mathop {g_{{i - \pi^{e} }} *}\limits_{( - )} \mathop {i_{m} *}\limits_{( - )} ]m* < 0, \\ F_{12} = & - \mathop {f_{E} *}\limits_{( + )} \mathop {E_{n} *}\limits_{( - )} m* > 0, \\ F_{13} = & - [\mathop {f_{E} *}\limits_{( + )} \mathop {E_{{\pi^{e} }} *}\limits_{( + )} + 1 - \mathop {g_{{i - \pi^{e} }} *}\limits_{( - )} ]m* < 0, \\ F_{21} = & - \mathop {g_{{i - \pi^{e} }} *}\limits_{( - )} \mathop {i_{m} *}\limits_{( - )} n* < 0, \\ F_{22} = & F_{23} = 0, \\ G_{31} = & \mathop {f_{E} *}\limits_{( + )} \mathop {E_{m} *}\limits_{( + )} > 0, \\ G_{32} = & \mathop {f_{E} *}\limits_{( + )} \mathop {E_{n} *}\limits_{( - )} < 0, \\ G_{33} = & \mathop {f_{E} *}\limits_{( + )} \mathop {E_{{\pi^{e} }} *}\limits_{( + )} > 0. \\ \end{aligned} $$

(114)

Substituting Eq. (114) into Eqs. (63), (64) and (65) in the text, we obtain the following results.

$$ A_{2} = - \mathop {F_{11} }\limits_{( - )} > 0. $$

(115)

$$ \begin{aligned} A_{3} = & - \mathop {F_{13} }\limits_{( - )} \mathop {G_{31} }\limits_{( + )} + (\mathop {F_{11} }\limits_{( - )} + \mathop {F_{22} }\limits_{( - )} )\mathop {G_{33} }\limits_{( + )} \\ = & \mathop {f_{E} *}\limits_{( + )} [ - \mathop {g_{{i - \pi^{e} }} *}\limits_{( - )} \{ \mathop {i_{m} *}\limits_{( - )} \mathop {E_{{\pi^{e} }} *}\limits_{( + )} (m* + n*) + \mathop {E_{m} *}\limits_{( + )} m*\} + \mathop {E_{m} *}\limits_{( + )} m*] \\ = & \mathop {f_{E} *}\limits_{( + )} \left| {g_{{i - \pi^{e} }} *} \right|[\mathop {i_{m} *}\limits_{( - )} \mathop {E_{{\pi^{e} }} *}\limits_{( + )} (m* + n*) + \mathop {E_{m} *}\limits_{( + )} m* + (\mathop {E_{m} *}\limits_{( + )} m*/\left| {g_{{i - \pi^{e} }} *} \right|)]. \\ \end{aligned} $$

(116)

$$ A_{4} = - \mathop {F_{12} }\limits_{( + )} \mathop {F_{21} }\limits_{( - )} > 0. $$

(117)

$$ \begin{aligned} A_{5} = & \left| {\begin{array}{*{20}c} {F_{11} } & {F_{12} } & {F_{13} } \\ {F_{21} } & 0 & 0 \\ {G_{31} } & {G_{32} } & {G_{33} } \\ \end{array} } \right| = - \mathop {F_{21} }\limits_{( - )} (\mathop {F_{12} }\limits_{( + )} \mathop {G_{33} }\limits_{( + )} - \mathop {F_{13} }\limits_{( - )} \mathop {G_{32} }\limits_{( - )} ) \\ = & - \mathop {F_{21} }\limits_{( - )} \{ (1 - \mathop {g_{{i - \pi^{e} }} *}\limits_{( - )} )\mathop {f_{E} *}\limits_{( + )} \mathop {E_{n} *}\limits_{( - )} m*\} < 0. \\ \end{aligned} $$

(118)

Step 2

Equations (115), (117) and (118) imply that we have $A_{2} > 0,$ $A_{4} > 0,$ and $A_{5} < 0$ in a special case of $g_{r} * = 0$ and $i_{{\pi^{e} }} * = 1.$ However, by continuity, these inequalities are also satisfied as long as $g_{r} *$ is sufficiently small and $i_{{\pi^{e} }} *$ is sufficiently close to 1, even if $g_{r} > 0$ and $i_{{\pi^{e} }} * \ne 1.$ Incidentally, we have the following expression in case of $g_{r} * = 0$ from Eqs. (28) and (34).

$$ \begin{aligned} i_{{\pi^{e} }} * = & \frac{{ - \{ 1 - (1 - \tau )y_{E} *\} y*r_{E} * - (1 - \tau )c_{{\pi^{e} }} *y_{E} *\phi * - \left| {g_{{i - \pi^{e} }} *} \right|y_{E} *\phi *}}{{\{ 1 - (1 - \tau )c*\} y*\phi_{i} * - \left| {g_{{i - \pi^{e} }} *} \right|y_{E} *\phi *}} \\ = & \frac{{[ - \{ 1 - (1 - \tau )y_{E} *y*r_{E} * - (1 - \tau )c_{{\pi^{e} }} *y*y_{E} *\phi *]/\left| {g_{{i - \pi^{e} }} *} \right| - y_{E} *\phi *}}{{\{ 1 - (1 - \tau )c*\} y*\phi_{i} */\left| {g_{{i - \pi^{e} }} *} \right| - y_{E} *\phi *}} \\ \end{aligned} $$

(119)

It follows from Eq. (119) that

$$ \mathop {\lim }\limits_{{\left| {g_{{i - \pi^{e} }} *} \right| \to + \infty }} i_{{\pi^{e} }} * = ( - y_{E} *\phi *)/( - y_{E} *\phi *) = 1. $$

(120)

From continuity, Eq. (120) implies that $i_{{\pi^{e} }} *$ is sufficiently close to 1 if $\left| {g_{{i - \pi^{e} }} *} \right|$ is sufficiently large.

Step 3

Finally, we shall turn to the question whether $A_{3}$ is positive or not. From Eqs. (28), (29), (30) and (33), we obtain the following expressions in case of $g_{r} * = 0.$

$$ \begin{aligned} E_{m} * = & \frac{{ - \left| {g_{{i - \pi^{e} }} *} \right|}}{{\{ 1 - (1 - \tau )c*\} y*\phi_{i} * - \left| {g_{{i - \pi^{e} }} *} \right|y_{E} *\phi *}} \\ = & \frac{ - 1}{{\{ 1 - (1 - \tau )c*\} y*\phi_{i} */\left| {g_{{i - \pi^{e} }} *} \right| - y_{E} *\phi *}} \\ \end{aligned} $$

(121)

$$ i_{m} * = \frac{{\{ 1 - (1 - \tau )c*\} y_{E} *}}{{\{ 1 - (1 - \tau )c*\} y*\phi_{i} * - \left| {g_{{i - \pi^{e} }} *} \right|y_{E} *\phi *}} $$

(122)

$$ \begin{aligned} E_{{\pi^{e} }} * = & \frac{{(1 - \tau )y*\phi_{i} * + (\phi_{i} * + \phi_{{\pi^{e} }} *)\left| {g_{{i - \pi^{e} }} *} \right|}}{{\{ 1 - (1 - \tau )c*\} y*\phi_{i} * - \left| {g_{{i - \pi^{e} }} *} \right|y_{E} *\phi *}} \\ = & \frac{{(1 - \tau )y*\phi_{i} */\left| {g_{{i - \pi^{e} }} *} \right| + (\phi_{i} * + \phi_{{\pi^{e} }} *)}}{{\{ 1 - (1 - \tau )c*\} y*\phi_{i} */\left| {g_{{i - \pi^{e} }} *} \right| - y_{E} *\phi *}} \\ \end{aligned} $$

(123)

From these equations, we have the following results.

$$ \mathop {\lim }\limits_{{\left| {g_{{i - \pi^{e} }} *} \right| \to + \infty }} E_{m} * = 1/(\mathop {y_{E} *}\limits_{( + )} \mathop {\phi *}\limits_{( + )} ) > 0 $$

(124)

$$ \mathop {\lim }\limits_{{\left| {g_{{i - \pi^{e} }} *} \right| \to + \infty }} i_{m} * = 0 $$

(125)

$$ \mathop {\lim }\limits_{{\left| {g_{{i - \pi^{e} }} *} \right| \to + \infty }} E_{{\pi^{e} }} * = - (\mathop {\phi_{i} *}\limits_{( - )} + \mathop {\phi_{{\pi^{e} }} *}\limits_{( - )} )/(\mathop {y_{E} *}\limits_{( + )} \mathop {\phi *}\limits_{( + )} ) > 0 $$

(126)

Eqs. (116), (124), (125) and (126) imply that $A_{3}$ is positive for all sufficiently large values of $\left| {g_{{i - \pi^{e} }} *} \right|.$ We obtained this conclusion under the assumption of $g_{r} * = 0.$ However, this conclusion applies even if $g_{r} * > 0$ as long as $g_{r} *$ is sufficiently small, by continuity. This completes the proof of Lemma 2. $\quad \square$

Appendix B

Proof of Proposition 3

Suppose that $g_{r} * > 0$ is sufficiently small and $\left| {g_{{i - \pi^{e} }} *} \right|$ is sufficiently large. Then, we have the following results from Eqs. (63)–(66) and Lemma 2.

$$ a_{1} = - \mathop {A_{1} }\limits_{( + )} \gamma + \mathop {A_{2} }\limits_{( + )} \equiv \Phi_{1} (\gamma );\;\Phi_{1} (0) = \mathop {A_{2} }\limits_{( + )} > 0. $$

(127)

$$ a_{3} = - \gamma \mathop {A_{5} }\limits_{( - )} > 0\;{\text{for all}}\;\gamma > 0. $$

(128)

$$ \begin{aligned} a_{1} a_{2} - a_{3} = & - \mathop {A_{1} }\limits_{( + )} \mathop {A_{3} }\limits_{( + )} \gamma^{2} + ( - \mathop {A_{1} }\limits_{( + )} \mathop {A_{4} }\limits_{( + )} + \mathop {A_{2} }\limits_{( + )} \mathop {A_{3} }\limits_{( + )} + \mathop {A_{5} }\limits_{( - )} )\gamma + \mathop {A_{2} }\limits_{( + )} \mathop {A_{4} }\limits_{( + )} \equiv \Phi_{2} (\lambda ) \\ \Phi_{2} (0) = & \mathop {A_{2} }\limits_{( + )} \mathop {A_{4} }\limits_{( + )} > 0. \\ \end{aligned} $$

(129)

Eq. (128) implies that one of a series of Routh-Hurwitz conditions (67) is always satisfied for all $\gamma > 0.$ In this case, the local stability conditions of the equilibrium point of the dynamic system (54a)–(54c) can be reduced to the following set of inequalities.

$$ a_{1} \equiv \Phi_{1} (\gamma ) > 0,\;a_{1} a_{2} - a_{3} \equiv \Phi_{2} (\gamma ) > 0 $$

(130)

Eq. (127) means that we have $\Phi_{1} (\gamma ) > 0$ for all $\gamma \in (0,\gamma_{1} ),$ $\Phi_{1} (\gamma_{1} ) = 0,$ and $\Phi_{1} (\gamma ) < 0$ for all $\gamma \in (\gamma_{1} , + \infty ),$ where $\gamma_{1} = A_{2} /A_{1} > 0.$

On the other hand, Eq. (129) means that $\Phi_{2} (\gamma )$ is a quadratic function of $\gamma$ such that $\Phi_{2} (0) > 0$ and the coefficient of $\gamma^{2}$ is negative. This means that there exists a positive parameter value $\gamma_{0}$ such that we have $\Phi_{2} (\gamma ) > 0$ for all $\gamma \in (0,\gamma_{0} ),$ $\Phi_{2} (\gamma_{0} ) = 0,$ and $\Phi_{2} (\gamma ) < 0$ for all $\gamma \in (\gamma_{0} , + \infty ).$ Incidentally, we have $\Phi_{2} (\gamma_{1} ) = - a_{3} < 0,$ so that we obtain the inequality

$$ 0 < \gamma_{0} < \gamma_{1} . $$

(131)

These results imply the following conclusion.

• (1) The long run equilibrium point of the dynamic system (54a)–(54c) is locally stable if $\gamma \in (0,\gamma_{0} ),$ because a set of inequalities (130) is satisfied at the open interval $(0,\gamma_{0} ).$

• (2) It is locally unstable if $\gamma \in (\gamma_{0} , + \infty ),$ because we have $\Phi_{2} (\gamma ) < 0$ at the open interval $(\gamma_{0} , + \infty ).$

This completes the proof of Proposition 3. $\quad \square$

Appendix C: Two useful theorems for the analysis of cyclical fluctuations

In this appendix, we present two theorems that are useful for the analysis of cyclical fluctuations in the dynamic system (54a)–(54c) in the text. Theorem 1 presents a set of sufficient conditions for the existence of the closed orbits in an n-dimensional system of nonlinear differential equations. Theorem 2 presents some useful results in case of three-dimensional system.

Theorem 1

(Hopf bifurcation theorem for an n-dimensional dynamic system, cf. Gandolfo (2009) Chap. 24).

Let $\dot{x} = f(x;\varepsilon ),$ $x \in R^{n} ,$ $\varepsilon \in R$ be an n-dimensional system of differential equations depending upon a parameter $\varepsilon .$ Suppose that the following conditions (H1)–(H3) are satisfied.

• (H1) The system has a smooth curve of equilibria given by $f(x*(\varepsilon );\varepsilon ) = 0.$

• (H2) The characteristic equation $\left| {\lambda I - Df(x*(\varepsilon_{0} );\varepsilon_{0} )} \right| = 0$ has a pair of pure imaginary roots $\lambda (\varepsilon_{0} ),$ $\overline{\lambda }(\varepsilon_{0} )$ and no other roots with zero real parts, where $Df(x*(\varepsilon_{0} );\varepsilon_{0} )$ is the Jacobian matrix of the above system at $(x*(\varepsilon_{0} ),\varepsilon_{0} )$ with the parameter value $\varepsilon_{0} .$

• (H3) $\left. {\frac{{d\{ {\text{Re}} \lambda (\varepsilon )\} }}{d\varepsilon }} \right|_{{\varepsilon = \varepsilon_{0} }} \ne 0,$ where ${\text{Re}} \lambda (\varepsilon )$ is the real part of $\lambda (\varepsilon ).$

Then, there exists a continuous function $\varepsilon (\xi )$ with $\varepsilon (0) = \varepsilon_{0} ,$ and for all sufficiently small values of $\xi \ne 0$ there exists a continuous family of non-constant periodic solutions $x(t,\xi )$ for the above dynamic system, which collapses to the equilibrium point $x*(\varepsilon_{0} )$ as $\xi \to 0.$ The period of the cycle is close to $2\pi /{\text{Im}} \lambda (\varepsilon_{0} ),$ where ${\text{Im}} \lambda (\varepsilon_{0} )$ is the imaginary part of $\lambda (\varepsilon_{0} ).$

Theorem 2

(A theorem that is useful for the analysis of Hopf bifurcation in three-dimensional dynamic system, cf. Asada (1995))

• (1) Suppose that the coefficients of the characteristic equation

  $$ \lambda^{3} + a_{1} \lambda^{2} + a_{2} \lambda + a_{3} = 0 $$

  (132)

  satisfies the following set of conditions.

  $$ a_{1} > 0,\;a_{3} > 0,\;a_{1} a_{2} - a_{3} = 0 $$

  (133)

  Then, Eq. (132) has a pair of pure imaginary roots and a negative real root.²³

• (2) Suppose that the coefficients in Eq. (132) are differentiable functions of a parameter $\varepsilon ,$ namely,

  $$ a_{1} = a_{1} (\varepsilon ),\;a_{2} = a_{2} (\varepsilon ),\;a_{3} = a_{3} (\varepsilon ). $$

  (134)

Furthermore, suppose that a set of conditions

$$ a_{1} (\varepsilon_{0} ) > 0,\;a_{3} (\varepsilon_{0} ) > 0,\;a_{1} (\varepsilon_{0} )a_{2} (\varepsilon_{0} ) - a_{3} (\varepsilon_{0} ) = 0 $$

(135)

is satisfied at the parameter value $\varepsilon = \varepsilon_{0} .$²⁴ Theorem 2 (1) implies that the characteristic equation (132) has a pair of pure imaginary roots $\lambda (\varepsilon_{0} ),$ $\overline{\lambda }(\varepsilon_{0} )$ and a negative real root at the point $\varepsilon = \varepsilon_{0} .$ Then, the condition

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

(136)

is equivalent to the condition

$$ \left. {\frac{{\partial \{ {\text{Re}} (\lambda (\varepsilon )\} }}{\partial \varepsilon }} \right|_{{\varepsilon = \varepsilon_{0} }} \ne 0, $$

(137)

where ${\text{Re}} \lambda (\varepsilon )$ is the real part of $\lambda (\varepsilon ).$

Proof

See Asada (1995). $\quad \square$