From Sunspots to Black Holes: Singular Dynamics in Macroeconomic Models

Abstract

We present conditions for the emergence of singularities in DGE models. We distinguish between slow-fast and impasse singularity types, review geometrical methods to deal with both types of singularity and apply them to DGE dynamics. We find that impasse singularities can generate new types of DGE dynamics, in particular temporary determinacy/indeterminacy. We illustrate the different nature of the two types of singularities and apply our results to two simple models: the Benhabib and Farmer (1994) model and one with a cyclical fiscal policy rule.

We acknowledge, without implicating, the useful comments by Alain Venditti, Nuno Barradas, and an anonymous referee. This work is part of UECE’s strategic project PEst-OE/EGE/UI0436/2014. UECE has financial support from national funds by FCT (Fundação para a Ciência e a Tecnologia).

Appendix

To simplify notation let $x=(x_{1},x_{2})\equiv (K,L)$ and consider functions $f_{1}(x)$, $f_{2}(x)$, and $\delta (x)$. In addition, consider the two vector fields F(x), as in Eq. (3.13), and $F^{r}(x)$, as in Eq. (3.27). Then, at a regular steady-state ($\bar{ x}$), we have $f_{1}(\bar{x})=f_{2}(\bar{x})=0$ and $\delta (\bar{x})\ne 0$. Furthermore, at a generic impasse-transversal point ($x^{i}$), we have $\delta (x^{i})=f_{2}(x^{i})=0$ and $f_{1}(x^{i})\ne 0$.

Lemma 4

Assume there is one generic impasse-transversal point $x^{i}$ and one regular steady-state $\bar{x}$, both belonging to a set X, such that $f_{2,x_{2}}(x)$ has the same sign for any point $x\in X$. Therefore, $\mathrm {sign}\left( \det DF^{r}(x^{i})\right) =-\mathrm {sign} \left( \det DF(\bar{x})\right)$.

Proof

For sake of simplicity, let us set $\epsilon (\varphi ) = 1$. The determinants of the Jacobian for F(x) evaluated at the steady-state and $F^{r}(x)$ at an impasse-transversal point are respectively given by

$$\begin{aligned} \det DF (\bar{x})= & {} \dfrac{f_{1,x_1} (\bar{x})\, f_{2,x_2} (\bar{x}) - f_{1,x_2} (\bar{x})\, f_{2,x_1} (\bar{x})}{\delta (\bar{x})}, \\ \det DF^r (x^i)= & {} f_1 (x^i) \left( \delta _{x_1} (x^i)\, f_{2,x_2} (x^i) - \delta _{x_2} (x^i)\, f_{2,x_1} (x^i) \right) . \end{aligned}$$

First, note that both points share a common condition $f_2 (x_1, x_2) = 0$. If this function is differentiable, we can write $\nabla f_2 (x) \cdot dx = 0$. By computing Taylor approximations to $f_1 (x^i)$ in a neighbourhood of $\bar{x}$ and to $\delta (\bar{x})$ in a neighbourhood of $x^i$, and considering the differentiability of $f_2\left( \cdot \right)$, we obtain:

$$\begin{aligned} f_1 (x^i)= & {} \dfrac{ \det DF (\bar{x}) \delta (\bar{x} )}{f_{2,x_2} (\bar{x})} (x^i_1 - \bar{x}_1), \\ \delta (\bar{x})= & {} \dfrac{\det DF^r (x^i) }{f_1 (x^i) f_{2,x_2} (x^i)} (\bar{x}_1-x^i_1). \end{aligned}$$

Thus,

$$ \dfrac{\det DF^r (x^i) }{\det DF (\bar{x}) } = - (\delta (\bar{x}))^2 \dfrac{f_{2,x_2} (x^i)}{f_{2,x_2} (\bar{x})}. $$

$\square$