Fiscal policy lag and equilibrium determinacy in a continuous-time New Keynesian model

Abstract

It has been suggested that local equilibrium determinacy can be achieved by applying active monetary policies combined with passive fiscal policies in discrete-time New Keynesian (NK) models that include a fiscal policy rule with a time lag in the policy response. However, indeterminacy can occur even under such policies in models with money in the production function. In this paper, we first confirm that these results hold in a continuous-time NK model without a policy lag. Furthermore, we present the case of a fiscal policy lag that is capable of avoiding indeterminacy.

Appendix: Direction of crossing

Differentiating equation $F(\lambda )=0$ with respect to $\theta$, we obtain the following:

$$\begin{aligned} \frac{\mathrm{d}\lambda }{\mathrm{d}\theta }(1-\theta \tau 'e^{-\theta \lambda })=\lambda \tau 'e^{-\theta \lambda }, \end{aligned}$$

or equivalently,

$$\begin{aligned} \left( \frac{\mathrm{d}\lambda }{\mathrm{d}\theta }\right) ^{-1}=\frac{1}{\lambda \tau 'e^{-\theta \lambda }}-\frac{\theta }{\lambda }. \end{aligned}$$

Next, we use equation $F(\lambda )=0$ to express $e^{-\theta \lambda }$ as follows:

$$\begin{aligned} e^{-\theta \lambda }=\frac{r^*-\lambda }{\tau '}. \end{aligned}$$

Thus,¹⁴

$$\begin{aligned} {\mathrm {Re}}\left. \left( \frac{\mathrm{d}\lambda }{\mathrm{d}\theta }\right) ^{-1}\right| _{\lambda =i\omega _+}&={\mathrm {Re}}\left[ \frac{1}{i\omega _+(r^*-i\omega _+)}\right] \\&=\frac{1}{{r^*}^2+\omega _+^2}>0. \end{aligned}$$