Kaldorian Assumptions and Endogenous Fluctuations in the Dynamic Fixed-Price IS-LM Model

Abstract

With the aim of better understanding the conditions which determine endogenous fluctuations at business cycle frequencies, recent literature has revived interest in the Schinasi’s variant of the dynamic, intermediate-run, IS-LM model (Schinasi 1981, 1982). Results, however, remain confined to Kaldorian-type economies, namely to those economies which present a greater-than-unity marginal propensity to spend out of income. This paper contributes to the debate by showing that, in the case of a negative interest rate sensitivity of savings, stable endogenous cycles can actually emerge as equilibrium solutions of the model also in the case of non Kaldorian-type economies. To this end, we combine the instruments of the global analysis, specifically the homoclinic bifurcation Theorem of Kopell and Howard (1975), with numerical methods.

Appendices

Appendix A

Linearization matrix associated with system $\mathcal {M}$.

As shown in the text, Schinasi’s model (1981) and (1982) gives rise to the following system of first-order differential equations

Let J $_{\mathcal {M}}^{*}$ be the Jacobian of the right hand side of system $\mathcal {M}$ evaluated at the steady state. The single elements of J $_{\mathcal {M}}^{*}$ are

$$\begin{aligned} j_{11}^{*} =&{\small \partial \dot{R}/\partial R|}_{ss}=-\alpha \frac{L_{Y}^{*}}{L_{R}^{*}}H_{R}^{*} \\ j_{12}^{*} =&{\small \partial \dot{R}/\partial Y|}_{ss}=-\alpha \frac{L_{Y}^{*}}{L_{R}^{*}}H_{Y}^{*}+\tfrac{\alpha L_{Y}^{*}-1}{L_{R}^{*}}\tau \\ j_{21}^{*} =&{\small \partial \dot{Y}/\partial R|}_{ss}=\alpha H_{R}^{*} \\ j_{22}^{*} =&{\small \partial \dot{Y}/\partial Y|}_{ss}=\alpha (H_{Y}^{*}-\tau ) \end{aligned}$$

where, for the sake of a simple representation, the arguments of the functions have been dropped. Therefore, we have

$$\begin{aligned} {\mathbf {J}}_{\mathcal {M}}^{*}{\small =}\left[ \begin{array}{cc} -\alpha \frac{L_{Y}^{*}}{L_{R}^{*}}H_{R}^{*} &{} -\alpha \frac{L_{Y}^{*}}{L_{R}^{*}}H_{Y}^{*}+\tfrac{\alpha L_{Y}^{*}-1}{L_{R}^{*}}\tau \\ \alpha H_{R}^{*} &{} \alpha (H_{Y}^{*}-\tau ) \end{array} \right] \end{aligned}$$

(A.1)

The eigenvalues of (A.1) are the solutions of the characteristic equation

$$\begin{aligned} \det \left( \lambda \mathbf {I-J}_{\mathcal {M}}^{*}\right) =\lambda ^{2}-\text {Tr}({\mathbf {J}}_{\mathcal {M}}^{*})\lambda +\text {Det}({\mathbf {J}}_{\mathcal {M}}^{*}) \end{aligned}$$

where I is the identity matrix. Tr$({\mathbf {J}}_{\mathcal {M}}^{*})$ and Det$({\mathbf {J}}_{\mathcal {M}}^{*})$ are Trace and Determinant of ${\mathbf {J}}_{\mathcal {M}}^{*}$, respectively. We obtain

$$\begin{aligned} \text {Tr(}{\mathbf {J}}_{\mathcal {M}}^{*}\text {)}=\alpha \left( H_{Y}^{*}-\tau -\frac{L_{Y}^{*}}{L_{R}^{*}}H_{R}^{*}\right) \end{aligned}$$

$$\begin{aligned} \text {Det(}{\mathbf {J}}_{\mathcal {M}}^{*}\text {)}=\alpha \tau \frac{H_{R}^{*}}{L_{R}^{*}} \end{aligned}$$

Appendix B

Linearization matrix associated with system $\mathcal {S}$.

Consider now system $\mathcal {S}$ in the text

$$\begin{aligned} \dot{Y}&= \alpha \left[ \varepsilon _{1}\sqrt{Y^{3}}-\varepsilon _{2}R-\varepsilon _{3}(1-\tau )^{2}Y^{2}-\varepsilon _{4}R-\varepsilon _{5}- \tfrac{\delta Y}{R}+G-\tau Y\right] \end{aligned}$$

Let J $_{\mathcal {S}}^{*}$ be the Jacobian of the right hand side of system $\mathcal {S}$ evaluated at the steady state. The single elements of $\mathbf J _{\mathcal {S}}^{*}$ are the following

$$\begin{aligned} j_{11}^{*} =&{\small \partial \dot{R}/\partial R|}_{ss}=-\tfrac{\gamma \alpha }{\beta }\left( -\varepsilon _{2}-\varepsilon _{4}+\tfrac{\delta }{R^{*}{}^{2}}\right) \\ j_{12}^{*} =&{\small \partial \dot{R}/\partial Y|}_{ss}=-\tfrac{\gamma \alpha }{\beta }\left( \frac{3}{2}\varepsilon _{1}\sqrt{\tfrac{G}{\tau }}-2\varepsilon _{3}(1-\tau )^{2}\tfrac{G}{\tau }\right) -\tfrac{\gamma \alpha -1}{\beta }\tau \\ j_{21}^{*} =&{\small \partial \dot{Y}/\partial R|}_{ss}=\alpha \left[ -\varepsilon _{2}-\varepsilon _{4}+\tfrac{\delta }{R^{*}{}^{2}}\right] \\ j_{22}^{*} =&{\small \partial \dot{Y}/\partial Y|}_{ss}=\alpha \left( \frac{3}{2}\varepsilon _{1}\sqrt{\tfrac{G}{\tau }}-2\varepsilon _{3}(1-\tau )^{2}\frac{G}{\tau }-\tau \right) \end{aligned}$$

where, for the sake of a simple representation, the arguments of the functions have been dropped. Therefore, we have

$$\begin{aligned} {\mathbf {J}}_{\mathcal {S}}^{\mathbf {*}}{\small =}\left[ \begin{array}{cc} \frac{\gamma \alpha }{\beta }H_{R}^{*} &{} \frac{\gamma \alpha }{\beta } H_{Y}^{*}-\frac{\gamma \alpha -1}{\beta }\tau \\ \alpha H_{R}^{*} &{} \alpha (H_{Y}^{*}-\tau ) \end{array} \right] \end{aligned}$$

(B.1)

where $H_{R}^{*}=-\varepsilon _{2}-\varepsilon _{4}+\tfrac{\delta Y^{*}}{R^{*}{}^{2}}$ and $H_{Y}^{*}=\frac{3}{2}\varepsilon _{1} \sqrt{\frac{G}{\tau }}-2\varepsilon _{3}(1-\tau )^{2}\frac{G}{\tau }-\tau$. Therefore,

$$\begin{aligned} \text {Tr(}{\mathbf {J}}_{\mathcal {S}}^{\mathbf {*}}\text {)}=\alpha (H_{Y}^{*}-\tau )+\tfrac{\gamma \alpha }{\beta }H_{R}^{*} \end{aligned}$$

$$\begin{aligned} \text {Det(}{\mathbf {J}}_{\mathcal {S}}^{\mathbf {*}}\text {)}=-\tfrac{\alpha \tau H_{R}^{*}}{\beta } \end{aligned}$$

Appendix C

For the sake of a simple discussion, we shall refer to the original version of the two-parameter homoclinic bifurcation Theorem in Kopell and Howard (1975) (Theorem 7.1, p. 334).

Let $(\delta ,\tau )$ be our control parameters. Posit $\mu =\delta -\bar{\delta }$ and $\nu =\tau -\bar{\tau }$ where $\bar{\delta }$ and $\bar{\tau }$ be the critical values of our bifurcation parameters. Let also $\bar{R}^{*}$ and $\bar{Y}^{*}$ be the particular steady state values of the interest rate and income implied by $\mu =\nu =0$.

Preliminarily, we translate our system of differential equation to the origin and provide a second-order Taylor expansion.

Let $\tilde{R}=R-\bar{R}^{*}$ and $\tilde{Y}=Y-\bar{Y}^{*}$. We have, from system $\mathcal {S}$

$$\begin{aligned} \overset{\cdot }{\tilde{R}}&= \tfrac{\gamma \alpha }{\beta }\tilde{H}\left( (\bar{Y}^{*}+\tilde{Y}),(\bar{R}^{*}+\tilde{R}),(\bar{\delta }+v),(\bar{\tau }+\mu )\right) \nonumber \\&\quad +\tfrac{\gamma \alpha -1}{\beta }\left[ G-(\bar{\tau } +\mu )(\bar{Y}^{*}+\tilde{Y})\right] \\ \overset{\cdot }{\tilde{Y}}&= \alpha \tilde{H}\left( (\bar{Y}^{*}+ \tilde{Y}),(\bar{R}^{*}+\tilde{R}),(\bar{\delta }+v),(\bar{\tau }+\mu )\right) +\alpha [ G-(\bar{\tau }+\mu )(\bar{Y}^{*}+\tilde{Y}))] \end{aligned}$$

(C.1)

where

$$\begin{aligned} \tilde{H}&=\varepsilon _{1}\sqrt{(\bar{Y}^{*}+\tilde{Y})^{3}}-\varepsilon _{2}(\bar{R}^{*}+\tilde{R})-\varepsilon _{3}(1-\bar{\tau }-\mu )^{2}\left( \bar{Y}^{*}+\tilde{Y}\right) ^{2} \\&\quad -\varepsilon _{4}(\bar{R}^{*}+\tilde{R})-\varepsilon _{5}-\tfrac{\left( \bar{\delta }+v\right) \left( \bar{Y}^{*}+\tilde{Y}\right) }{(\bar{R}^{*}+\tilde{R})} \end{aligned}$$

System C.1 corresponds to the generic two-parameter family of ordinary differential equations $\dot{X}=F_{\mu ,v}\left( X\right)$ in Kopell-Howard’s original Theorem. We present now, in sequence, the computation necessary to apply the homoclinic bifurcation Theorem 7.1 in Kopell and Howard to system C.1.

• 1. Computation of $dF_{\mu ,\nu }\left( \mathbf {0}\right)$. We obtain

  $$\begin{aligned} dF_{\mu ,\nu }\left( \mathbf {0}\right) =\left[ \begin{array}{cc} \frac{2\alpha \gamma }{\beta }\varepsilon _{3}(1-\bar{\tau }-\mu )\bar{Y}^{*}{}^{2}-\frac{\alpha \gamma -1}{\beta }\bar{Y}^{*} &{} -\frac{\alpha \gamma }{\beta }\frac{\bar{Y}^{*}}{\bar{R}^{*}} \\ 2\alpha \varepsilon _{3}(1-\bar{\tau }-\mu )\bar{Y}^{*}{}^{2}-\alpha \bar{Y}^{*} &{} -\frac{\alpha \bar{Y}^{*}}{\bar{R}^{*}} \end{array} \right] \end{aligned}$$

  (C.2)

  Simple algebra gives

  $$\begin{aligned} Tr\ dF_{\mu ,v}(0)&= \tfrac{2\alpha \gamma }{\beta }\varepsilon _{3}(1-\bar{\tau }-\mu )\bar{Y}^{*2}-\tfrac{\alpha \gamma -1}{\beta }\bar{Y}^{*}-\tfrac{\alpha \bar{Y}^{*}}{\bar{R}^{*}} \\ \det dF_{\mu ,v}(0)&= -\tfrac{\alpha \bar{Y}^{*2}}{\beta \bar{R}^{*}} \end{aligned}$$

  At $(\mu ,v)=(0,0)$ (C.2) becomes

  $$\begin{aligned} dF_{0,0}(0)=\left[ \begin{array}{cc} \frac{2\gamma \alpha }{\beta }\varepsilon _{3}(1-\bar{\tau })\bar{Y}^{*}{}^{2}-\frac{\gamma \alpha -1}{\beta }\bar{Y}^{*} &{} -\frac{\gamma \alpha }{\beta }\frac{\bar{Y}^{*}}{\bar{R}^{*}} \\ 2\alpha \varepsilon _{3}(1-\bar{\tau })\bar{Y}^{*}{}^{2}-\alpha \bar{Y}^{*} &{} -\frac{\alpha \bar{Y}^{*}}{\bar{R}^{*}} \end{array} \right] \end{aligned}$$

Since $dF_{0,0}(0)$ has a double zero eigenvalue, the first requirement of the Theorem is satisfied.

• 2. Computation of the mapping $\left( \mu ,v\right) \rightarrow \left( \det dF_{\mu ,v}(0),\ \text {Tr\ }dF_{\mu ,v}(0)\right)$.

  We have

  $$\begin{aligned} \left[ \begin{array}{cc} \frac{\partial }{\partial \mu }\det dF_{\mu v}(0) &{} \frac{\partial }{\partial v}\det dF_{\mu v}(0) \\ \frac{\partial }{\partial \mu }\text {Tr}dF_{\mu v}(0) &{} \frac{\partial }{\partial v}\text {Tr}dF_{\mu v}(0) \end{array} \right] \end{aligned}$$

  which reduces to

  $$\begin{aligned} \left[ \begin{array}{cc} 0 &{} 0 \\ -2\frac{\alpha \gamma }{\beta }\varepsilon _{3}\bar{Y}^{*2} &{} 0 \end{array} \right] \ne 0 \end{aligned}$$

  Therefore the second requirement of the Theorem is satisfied.

  $$\begin{aligned} \tilde{H}&=\varepsilon _{1}\sqrt{(\bar{Y}^{*}+\tilde{Y})^{3}}-\varepsilon _{2}(\bar{R}^{*}+\tilde{R})-\varepsilon _{3}(1-\bar{\tau }-\mu )^{2}\left( \bar{Y}^{*}+\tilde{Y}\right) ^{2} \\&\quad -\varepsilon _{4}(\bar{R}^{*}+\tilde{R})-\varepsilon _{5}-\tfrac{\left( \bar{\delta }+v\right) \left( \bar{Y}^{*}+\tilde{Y}\right) }{\bar{R}^{*}+\tilde{R}} \end{aligned}$$

• 3. Computation of the $Q(e,e)$ matrix.

Let $\mathbf {P}_{i}$, $i=1,2$ be the matrices of the second order derivatives of system C.1 evaluated at $\left( \mu ,v\right) =\left( 0,0\right)$. We have

$$\begin{aligned} \mathbf {P}_{1}=\tfrac{\alpha \gamma }{\beta }\left[ \begin{array}{cc} -\frac{2\bar{\delta }\bar{Y}^{*}}{\bar{R}^{*3}} &{} \frac{\bar{\delta }}{\bar{R}^{*2}} \\ \frac{\bar{\delta }}{\bar{R}^{*2}} &{} \varepsilon _{1}\frac{3}{4}\frac{1}{\sqrt{\bar{Y}^{*}}}-2\varepsilon _{3}(1-\bar{\tau })^{2} \end{array} \right] \end{aligned}$$

$$\begin{aligned} \mathbf {P}_{2}=\alpha \left[ \begin{array}{cc} -\frac{2\bar{\delta }\bar{Y}^{*}}{\bar{R}^{*3}} &{} \frac{\bar{\delta }}{\bar{R}^{*2}} \\ \frac{\bar{\delta }}{\bar{R}^{*2}} &{} \frac{3\varepsilon _{1}}{4}\frac{1}{\sqrt{\bar{Y}^{*}}}-2\varepsilon _{3}(1-\bar{\tau })^{2} \end{array} \right] \end{aligned}$$

Let us now compute the right eigenvector $\mathbf {e} =(e_{1},e_{2})^{T}$ of ${\mathbf {J}}_{\mathcal {S}}^{*}$. A possible candidate is

$$\begin{aligned} \mathbf {e}= \left[ \begin{array}{c} e_{1} \\ e_{2} \end{array} \right] =\left[ \begin{array}{c} -\frac{\frac{\alpha \gamma }{\beta }\left( \frac{3}{2}\varepsilon _{1}\sqrt{\bar{Y}^{*}}-2\varepsilon _{3}(1-\tau )^{2}\bar{Y}^{*}-\frac{\delta }{\bar{R}^{*}}\right) -\frac{\alpha \gamma -1}{\beta }\bar{\tau }}{\frac{\gamma \alpha }{\beta }\left( -(\varepsilon _{2}+\varepsilon _{4})+\delta \frac{\bar{Y}^{*}}{\bar{R}^{*}{}^{2}}\right) } \\ 1 \end{array} \right] \end{aligned}$$

Therefore

$$\begin{aligned} Q(e,e)=\frac{1}{2}\left( \begin{array}{c} e^{T}\mathbf {P}_{1}e \\ e^{T}\mathbf {P}_{2}e \end{array} \right) =\frac{\alpha }{2}\left( \begin{array}{c} -\left( \frac{j_{12}^{*}}{j_{11}^{*}}\right) ^{2}\frac{\gamma }{\beta }\frac{2\bar{\delta }\bar{Y}^{*}}{\bar{R}^{*3}}+\frac{\gamma }{\beta }\left( \frac{3\varepsilon _{1}}{4}\frac{1}{\sqrt{\bar{Y}^{*}}}-2\varepsilon _{3}(1-\bar{\tau })^{2}\right) \\ -\left( \frac{j_{12}^{*}}{j_{11}^{*}}\right) ^{2}\frac{2\bar{\delta }\bar{Y}^{*}}{\bar{R}^{*3}}+\frac{3\varepsilon _{1}}{4}\frac{1}{\sqrt{\bar{Y}^{*}}}-2\varepsilon _{3}(1-\bar{\tau })^{2} \end{array} \right) \end{aligned}$$

Finally

$$\begin{aligned} \left[ dF_{0,0}(0),Q(e,e)\right] =\left[ \begin{array}{cc} &{}\frac{2\alpha \gamma }{\beta }\varepsilon _{3}(1-\bar{\tau })\bar{Y}^{*2}-\frac{\alpha \gamma -1}{\beta }\bar{Y}^{*} -\left( \frac{j_{12}^{*}}{j_{11}^{*}}\right) ^{2}\frac{\gamma }{\beta }\frac{2\bar{\delta }\bar{Y}^{*}}{\bar{R}^{*3}}\\ &{}+\frac{\gamma }{\beta }\left( \frac{3\varepsilon _{1}}{4}\frac{1}{\sqrt{\bar{Y}^{*}}}-2\varepsilon _{3}(1-\bar{\tau })^{2}\right) \\ &{}2\alpha \varepsilon _{3}(1-\bar{\tau })\bar{Y}^{*2}-\alpha \bar{Y}^{*} -\left( \frac{j_{12}^{*}}{j_{11}^{*}}\right) ^{2}\frac{2\bar{\delta }\bar{Y}^{*}}{\bar{R}^{*3}}\\ &{}+\varepsilon _{1}\frac{3}{4}\frac{1}{\sqrt{\bar{Y}^{*}}}-2\varepsilon _{3}(1-\bar{\tau })^{2} \end{array} \right] \end{aligned}$$

(C.3)

where $j_{11}^{*}=\frac{\gamma \alpha }{\beta }\left( -(\varepsilon _{2}+\varepsilon _{4})+\delta \frac{\bar{Y}^{*}}{\bar{R}^{*}{}^{2}}\right)$ and $j_{12}^{*}=\frac{\gamma \alpha }{\beta }\left( \frac{3}{2}\varepsilon _{1}\sqrt{\bar{Y}^{*}}-2\varepsilon _{3}(1-\tau )^{2}\bar{Y}^{*}\right.$ $\left. -\frac{\delta }{\bar{R}^{*}}\right) -\frac{\alpha \gamma -1}{\beta }\tau$. Since (C.3) has rank 2, the third requirement of the Theorem is satisfied.