Household debt and housing bubbles: a Minskian approach to boom-bust cycles

Abstract

This paper examines macroeconomic dynamics of household debt and housing prices in a two-class economy. Drawing on Minsky’s insights into financial instability and cycles, our framework combines household debt dynamics with behavioral asset price dynamics in a Keynesian macro model. We show that endogenous boom-bust cycles can emerge through the interaction between household debt and housing price dynamics. In this model, a long period of housing bubbles is characterized by increases in the profit share and the workers’ indebtedness for most of the time. The long waves are combined with a Kaldorian model of short-run business cycles.

Appendices

Appendix A: Proof of Propositions 3 and 7

Proof

of Proposition 3 Let c ^w(m(t)) = c ^w. Equation 17 and Assumption (37) implies

$$ \kappa\underline \eta c^{w} < \dot h (s)+\kappa h(s)<\kappa\overline \eta c^{w} $$

Multiplying this by exp(κ s) and integrating it over [0,t] gives us

$$ [\exp(\kappa t)-1]\underline \eta c^{w}< \exp(\kappa t)h^{w}(t)-h(0)<[\exp(\kappa t)-1]\overline \eta c^{w} $$

Mutiplying by exp(−κ t) and rearranging the terms, we have:

$$ [1-\exp(-\kappa t)]\underline \eta c^{w}+\exp(-\kappa t)h(0) < h^{w}(t)<[1-\exp(-\kappa t)]\overline \eta c^{w}+\exp(-\kappa t)h(0) $$

Since 0< exp(−κ t)≤1 over t∈[0,∞), we have:

$$ \underline h \leq h^{w}(t) \leq \overline h $$

where $\underline h =\min \{ \underline \eta c^{w}, h(0)\}$ and $\overline h =\max \{ \overline \eta c^{w}, h(0)\}$. There h ^w(t) is bounded. Since $\dot h^{w}(t)$ is continuous in h ^w(t) and η is bounded by assumption, $\dot h^{w}(t)$ is bounded as well. To prove the boundedness of ρ ^e(t), consider

$$ \rho(t) =\dot h^{w}(t)/h^{w}(t)+n $$

Since $\dot h^{w}(t)$ and h ^w(t) are bounded, ρ(t) is clearly bounded as long as h(0)>0. Note that, if h ^w(t) = 0, ρ(t) may explode but if h(0)>0, this case is ruled out: $\underline \eta >0$ by assumption and therefore $h^{w}(t)>\underline h \equiv \min \{\underline \eta c^{w}, h(0)\}>$0 if h(0)>0. Because ρ(t) is bounded, the same method as in the proof of the boundedness of h ^w(t) can be applied to prove that of ρ ^e(t) by using (15). Since the trajectories are bounded and (η(n)c ^w, n) is a unique unstable fixed point, if $-\kappa +\nu \left (\frac {\kappa \eta ^{\prime }}{\eta } -1\right ) >0$, the trajectories of h ^w(t) and ρ ^e(t) must converge to a closed orbit according to the Poincare-Bendixson theorem. □

Proof

of Proposition 7 To prove the boundedness of the trajectories of the 3D system, we first construct a set on the (m(t), h ^w(t)) space from which any trajectory cannot escape once it enters independently of the value of ρ ^e(t) (i.e., the projection onto the (m(t), h ^w(t)) space of a positively invariant set). We can confine our initial analysis to the (m(t), h ^w(t)) space thanks to the boundedness of η(ρ ^e), i.e., Assumption (37). Consider the following set:

$$\begin{array}{@{}rcl@{}} \mathbf{A}&\equiv&\{(m(t), h(t)) \in [m_{1}-a , m_{2}+a] \times [h_{1}-a, h_{2}+a]~\vert~ \\ && a>0, ~\underline \eta \tilde c^{w} ( m_{1} , h_{1})= h_{1}, \overline \eta c^{w}( m_{2} , h_{2})= h_{2}, m_{i} = \tilde m (h_{i}), i=1,2 \} \end{array} $$

where a sufficiently small positive a can be chosen for A to include {[m ₁,m ₂]×[h ₁,h ₂]} as its proper subset and to ensure m ₁−a > 0 and h ₁−a > 0. It can be easily shown that the gradient at any point on the boundaries of A points inward the set. It can be also shown that any trajectory from an arbitrary initial condition eventually enter A. The intuitive explanation is as follows: for any given ρ ^e(t), the 2D sub-system (m(t), h(t)) has a unique fixed point and the fixed point is locally stable since the trace and the determinant of the subsystem are negative and positive, respectively. The fixed point depends continuously and monotonically on the value of η(ρ ^e(t)). The set of all fixed points of the 2D subsystem is a finite and closed segment on the $\dot m(t)$-nullcline (Note that the $\dot h^{w}(t)$-nullcline depends continuously and monotonically on the value of ρ ^e(t), but the area it spans is limited by the boundedness of the η-function). We can choose a set that includes the set of fixed points as a proper subset. Such a set has the desired property: any trajectory cannot escape from it. A is an example of those sets with such a property. In our proof, we used the fact that the determinant of the 2D sub-system is positive, which can be checked:

$$\begin{array}{@{}rcl@{}} F_{m} G_{h}-F_{h} G_{m}&=& \frac{\kappa}{1+\mu_{y} -f_{y} (1-s_{f})} \times [\{rf_{y}s_{f}+f_{\omega} (1+\alpha)\}( \mu_{y} +\eta \mu_{\omega}) \\ &&~~~~~~~~~~~+\mu_{\omega} \{1-(1-s_{f}) f_{y} \}+n \{\mu_{y} +\eta \mu_{\omega} f_{y} (1-s_{f})\}]>0 \end{array} $$

Since m(t) and h ^w(t) are bounded, the boundedness of ρ ^e(t) follows from the argument similar to that in Proposition 3. □

Appendix B: Proof of Proposition 5 and 6

Let us consider the following Jacobian Matrix evaluated at the stationary point.

$$ J=\left[\begin{array}{lll} F_{m}& F_{h^{w}} & 0 \\ G_{m}&G_{h} &G_{\rho^{e}} \\ \nu \frac{G_{m}}{h^{w*}}& \nu \frac{G_{h}}{h^{w*}} & \nu \left( \frac{G_{\rho^{e}}}{h^{w*}}-1\right) \\ \end{array}\right] $$

(73)

We have seen F _m < 0 and $\phantom {\dot {i}\!}F_{h^{w}}>0$ in Eqs. 31 and 32. We also have: $G_{h}= \kappa (\eta \tilde {c^{w}_{h}} -1)<0$ and $G_{\rho ^{e}}=\kappa \eta ^{\prime }\tilde c^{w}(\cdot )>0$.

Let us define $b_{1}\equiv \frac {G_{\rho ^{e}}}{h^{w*}}-1$, $\phantom {\dot {i}\!}b_{2}\equiv F_{m} G_{h}-F_{h^{w}} G_{m}>0$, b ₃≡F _m + G _h < 0, and $b_{4}\equiv \left (\frac {G_{\rho ^{e}}}{h^{w*}}-1\right )F_{m}- G_{h}$. Using the definition of b _i ’s, we have

$$\begin{array}{@{}rcl@{}} &&\text{tr}(J)=b_{3}+b_{1}\nu, ~~~~~{\Sigma}_{i=1}^{3} J_{i}=b_{2}+ b_{4}\nu, ~~~~~\det(J)=-b_{2}\nu <0\\ &-&\text{tr}(J)({\Sigma}_{i=1}^{3} J_{i})+\det(J)=A_{0} +A_{1} \nu+A_{2}\nu^{2} \end{array} $$

where A ₀=−b ₂ b ₃ > 0, A ₁=−b ₁ b ₂−b ₃ b ₄−b ₂, A ₂=−b ₁ b ₄, and J _i ’s are the first principal minors of J.

Proof

of Proposition 5 The Routh-Hurwitz necessary and sufficient condition for the asymptotic local stability is:

$$\text{tr}(J)<0,~~{\Sigma}_{i=1}^{3} J_{i} >0, ~~ \det(J)<0,~\text{and}~ -\text{tr}(J)({\Sigma}_{i=1}^{3} J_{i})+\det(J)>0 $$

As ν→0, we have:

$$\text{tr}(J) \rightarrow b_{3}<0, ~ {\Sigma}_{i=1}^{3} J_{i} \rightarrow b_{2}>0, ~\text{and}~ -\text{tr}(J)({\Sigma}_{i=1}^{3} J_{i})+\det(J) \rightarrow A_{0}>0 $$

It is readily seen that for a sufficiently small positive value of ν, the signs of tr(J), ${\Sigma }_{i=1}^{3} J_{i}$ and $-\text {tr}(J)({\Sigma }_{i=1}^{3} J_{i})+\det (J)$ should be retained while det(J) being negative. □

Proof

of Proposition 6 To prove the existence of a limit cycle for the system of Eqs. 38–40 , we will show that the Jacobian matrix (73) evaluated at ( m ^∗(ν), h ^∗(ν), ρ ^e∗(ν), ν), where ( m ^∗(ν), h ^∗(ν), ρ ^e∗(ν)) is a fixed point of the system, has a negative real root and a pair of imaginary roots. If we denote the eigenvalues of the Jacobian matrix as λ(ν) and β(ν)±𝜃(ν)i, we need to show that λ(ν ^b) < 0, β(ν ^b) = 0, and 𝜃(ν ^b) ≠ 0. ν ^b is called a Hopf bifurcation point. The Routh-Hurwitz criterion states that the Jacobian matrix will have a negative real root and a pair of pure imaginary roots if and only if:

$$\begin{array}{@{}rcl@{}} \text{tr}(J)<0,~&~&{\Sigma}_{i=1}^{3} J_{i} >0\\ \det(J)<0,&~& -\text{tr}(J)({\Sigma}_{i=1}^{3} J_{i})+\det(J)=0 \end{array} $$

(74)

Let us suppose that b ₁ > 0 and consider two cases: b ₄ > 0 and b ₄ < 0

• Case 1. b ₄ > 0. We then have A ₂ < 0. Since A ₀ > 0, the quadratic equation, A ₀ + A ₁ ν + A ₂ ν ²=0, has one positive and one negative roots. Choose the positive root and denote it as ν ^∗. ν ^∗ is given by

  $$\nu^{*} \equiv (A_{1}+\sqrt{{A_{1}^{2}}+4A_{0}\vert A_{2}\vert})/(2\vert A_{2}\vert)>0 $$

  where −tr(J)(J ₁ + J ₂ + J ₃)+ det(J) = 0. Because b ₂ > 0 and b ₄ > 0, J ₁ + J ₂ + J ₃ = b ₂ + b ₄ ν ^∗>0. tr(J) = 0 if $\nu =\frac {b_{3}}{b_{1}}>0$. It implies that if $\nu =\frac {b_{3}}{b_{1}}$, then −tr(J)(J ₁ + J ₂ + J ₃)+ det(J) = det(J) < 0. For any ν > 0, −tr(J)(J ₁ + J ₂ + J ₃)+ det(J) = det(J) < 0 only if ν > ν ^∗. Therefore, $\nu ^{*}<\frac {b_{3}}{b_{1}}$. Since tr(J) is increasing in ν ( b ₁ > 0), $\nu ^{*}<\frac {b_{3}}{b_{1}}$ implies that tr(J) < 0 at ν = ν ^∗. Therefore, we conclude that if ν = ν ^∗, the Routh-Hurwitz criterion (74) is satisfied and, therefore, λ(ν ^∗) < 0, β(ν ^∗∗) = 0, and 𝜃(ν ^∗) ≠ 0. For a later purpose, note that the first derivative of −tr(J)(J ₁ + J ₂ + J ₃)+ det(J) with respect to ν is negative at ν = ν ^∗, i.e. A ₁ + 2A ₂ ν ^∗ < 0.

• Case 2. Next suppose that b ₄ < 0. We then have A ₂ > 0, A ₁ < 0 and A ₀ > 0. Furthermore, a straightforward calculation shows that:

  $${A_{1}^{2}}-4A_{0}A_{2} =(b_{1}b_{2}-b_{3} b_{4})^{2}+2(b_{1}b_{2}+b_{3} b_{4})b_{2}+{b_{2}^{2}}>0 $$

  The last inequality follows from the fact that b ₁ > 0, b ₂ > 0, b ₃ < 0 and b ₄ < 0. Therefore, the quadratic equation, A ₀ + A ₁ ν + A ₂ ν ²=0, has two distinct positive roots. Denote the smaller as ν ^∗∗.

  $$\nu^{**} \equiv (\vert A_{1}\vert-\sqrt{{A_{1}^{2}}-4A_{0} A_{2}})/(2 A_{2})>0 $$

  It is simple to show that tr(J) < 0 and J ₁ + J ₂ + J ₃ > 0 at ν = ν ^∗∗. Therefore, we conclude that if ν = ν ^∗∗, Eq. 74 is satisfied.

It remains to show that β ^′(ν ^∗) ≠ 0 and β ^′(ν ^∗∗) ≠ 0. Tedious algebra shows:

$$\begin{array}{@{}rcl@{}}\beta^{\prime}(\nu^{*})&=&\frac{2\theta(\nu^{*})[b_{1}b_{2}+b_{3}b_{4}+b_{2}+2b_{1} b_{4} \nu^{*}]}{4\lambda(\nu^{*})^{2}\theta(\nu^{*})+4\theta(\nu^{*})^{3}}=\frac{-2\theta(\nu^{*})[A_{1}+2A_{2}\nu^{*}]}{4\lambda(\nu^{*})^{2}\theta(\nu^{*})+4\theta(\nu^{*})^{3}}>0\\ \beta^{\prime}(\nu^{**})&=&\frac{2\theta(\nu^{**})\left[b_{4}\lambda(\nu^{**})+ b_{1}\theta(\nu^{**})^{2}+b_{2}\right]}{4\lambda(\nu^{**})^{2}\theta(\nu^{**})+4\theta(\nu^{**})^{3}}>0 \end{array} $$

□